PLASMID ISOLATION AND ANALYSIS Part II Plasmid Purification and Isolation
Estimating the rate of plasmid transfer with an adapted ...
Transcript of Estimating the rate of plasmid transfer with an adapted ...
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Estimating the rate of plasmid transfer with an adapted LuriaโDelbrรผck fluctuation
analysis and a case study on the evolution of plasmid transfer rate
Olivia Kosterlitz1,2*, Clint Elg2,3, Ivana Bozic4, Eva M. Top2,3, Benjamin Kerr1,2*
1 Biology Department, University of Washington, Seattle, WA, USA.
2 BEACON Center for the Study of Evolution in Action, East Lansing, MI, USA.
3 Department of Biological Sciences and Institute for Bioinformatics and Evolutionary
Studies, University of Idaho, Moscow, ID, USA.
4 Department of Applied Mathematics, University of Washington, Seattle, WA, USA.
* e-mail: [email protected]; [email protected]
Author contributions
O.K. and B.K. conceived of the presented ideas. B.K. and I.B. developed the theory. O.K.
developed and performed the simulations. C.E. facilitated part of the simulations. O.K.
performed the experiments. O.K. and B.K. wrote the manuscript. E.M.T. and B.K.
supervised the project. All authors discussed the results and contributed to the final
manuscript.
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Abstract 1
To increase our basic understanding of the ecology and evolution of conjugative 2
plasmids, we need a reliable estimate of their rate of transfer between bacterial cells. 3
Accurate estimates of plasmid transfer have remained elusive given biological and 4
experimental complexity. Current methods to measure transfer rate can be confounded 5
by many factors, such as differences in growth rates between plasmid-containing and 6
plasmid-free cells. However, one of the most problematic factors involves situations 7
where the transfer occurs between different strains or species and the rate that one type 8
of cell donates the plasmid is not equal to the rate at which the other cell type donates. 9
Asymmetry in these rates has the potential to bias transfer estimates, thereby limiting our 10
capabilities for measuring transfer within diverse microbial communities. We develop a 11
novel low-density method (โLDMโ) for measuring transfer rate, inspired by the classic 12
fluctuation analysis of Luria and Delbrรผck. Our new approach embraces the stochasticity 13
of conjugation, which departs in important ways from the current deterministic population 14
dynamic methods. In addition, the LDM overcomes obstacles of traditional methods by 15
allowing different growth and transfer rates for each population within the assay. Using 16
stochastic simulations, we show that the LDM has high accuracy and precision for 17
estimation of transfer rates compared to other commonly used methods. Lastly, we 18
implement the LDM to estimate transfer on an ancestral and evolved plasmid-host pair, 19
in which plasmid-host co-evolution increased the persistence of an IncP-1 conjugative 20
plasmid in its Escherichia coli host. Our method revealed the increased persistence can 21
be at least partially explained by an increase in transfer rate after plasmid-host co-22
evolution. 23
24
Introduction 25
A fundamental rule of life and the basis of heredity involves the passage of genes 26
from parents to their offspring. Plasmids in bacteria can violate this rule of strict vertical 27
inheritance by shuttling DNA among neighboring, non-related bacteria through horizontal 28
gene transfer (HGT). Plasmids are independently replicating extrachromosomal DNA 29
molecules, and some encode for their own horizontal transfer, a process termed 30
conjugation. Bacteria containing such conjugative plasmids are governed by two modes 31
of inheritance: vertical and horizontal. In an abstract sense, conjugation in prokaryotes is 32
similar to the genetic recombination that occurs in sexually reproducing eukaryotes; 33
specifically, conjugation is a fundamental mechanism for producing new combinations of 34
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genetic elements. Since conjugation facilitates gene flow across vast phylogenetic 35
distances, the expansive gene repertoire in the โaccessoryโ genome is shared among 36
many microbial species1. While this complicates phylogenetic inference, it reinforces the 37
notion that conjugation has a central role in shaping the ecology and evolution of microbial 38
communities2,3. Notably, conjugative plasmids are commonly the vehicle for the spread 39
of antimicrobial resistance (AMR) genes and the emergence of multi-drug resistance 40
(MDR) in clinical pathogens4. While such plasmids may impose growth costs on their 41
bacterial hosts in the absence of antibiotics5, theoretical and experimental studies have 42
demonstrated how conjugation itself may be a critical process for the invasion and 43
maintenance of costly plasmids6,7,8. To model the spread of antibiotic resistance, improve 44
phylogenetic inference, and increase our understanding of the fundamental process of 45
conjugation and its impacts on the basic biology of bacteria, accurate measures of 46
conjugation are of the utmost importance. 47
The basic approach to measure conjugation involves mixing plasmid-containing 48
bacteria, called โdonorsโ, with plasmid-free bacteria, called โrecipientsโ. After donors and 49
recipients are incubated together for some period of time, which is hereafter denoted as 50
๏ฟฝฬ๏ฟฝ, the population is assessed for recipients that have received the plasmid from the donor; 51
these transformed recipients are called โtransconjugantsโ. To describe conjugation the 52
earliest and some of the most widely used approaches measure the densities of donors, 53
recipients, and transconjugants after the incubation time ๏ฟฝฬ๏ฟฝ (๐ท๐ก, ๐ ๐ก, and ๐๐ก, respectively) to 54
calculate various ratios (i.e., conjugation frequency or efficiency) including ๐๐ก/๐ ๐ก, ๐๐ก/๐ท๏ฟฝฬ๏ฟฝ, 55
๐๐ก/(๐ท๐ก + ๐ ๐ก + ๐๏ฟฝฬ๏ฟฝ)9, see Zhong et. al.10 and Huisman et. al.11 for a comprehensive 56
overview. These conjugation ratios are descriptive statistics and have been shown to vary 57
with initial densities (๐ท0 and ๐ 0) and incubation time (๏ฟฝฬ๏ฟฝ)10,11. While these statistics may be 58
useful for within study comparisons of the combined processes of conjugation and 59
transconjugant growth, they are not estimating the rate of conjugation events. To see this, 60
consider two plasmids, A and B, and assume both have the same conjugation rate, but 61
plasmid A results in a higher transconjugant growth rate. Assuming all other parameters 62
(e.g., donor growth, recipient growth, etc.) are the same and growth occurs during the 63
assay, all of the above ratios would be higher for A, even though its conjugation rate 64
doesnโt differ from plasmid B. Clearly, an estimate of the rate of conjugation itself is 65
desirable. 66
The conjugation rate (i.e., plasmid transfer rate) is the โgold-standardโ 67
measurement to describe conjugation akin to measuring other fundamental rate 68
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parameters, such as growth rate or mutation rate. Such metrics are useful for cross-69
literature comparison to understand basic microbial ecology and evolution12. There are 70
only a few widely used methods to measure conjugation rate, which all are derived from 71
the Levin et. al.13 mathematical model describing the change in density of donors, 72
recipients and transconjugants over time (given by dynamic variables ๐ท๐ก, ๐ ๐ก, and ๐๐ก, 73
respectively). In this model, each population type grows exponentially at the same growth 74
rate ๐. In addition, the transconjugant density increases as a result of conjugation events 75
both from donors to recipients and from existing transconjugants to recipients at the same 76
conjugation rate ๐พ. The recipient density decreases due to these conjugation events. The 77
densities of these dynamic populations are described by the following differential 78
equations (where the ๐ก subscript is dropped from the dynamic variables for notational 79
convenience): 80
๐๐ท
๐๐ก= ๐๐ท, [1]
๐๐
๐๐ก= ๐๐ โ ๐พ๐ (๐ท + ๐), [2]
๐๐
๐๐ก= ๐๐ + ๐พ๐ (๐ท + ๐). [3]
Each current rate estimation method solved the set of ordinary differential 81
equations from the Levin et. al. model (or a slight variation) to find an estimate for the 82
conjugation rate ๐พ. These estimates differ by the assumptions used to find the analytical 83
solution. Two estimates for conjugation rate standout as widely used in the literature7,14,15. 84
Levin et. al.13 derived the first of these estimates for conjugation rate ๐พ in terms of the 85
density of donors, recipients, and transconjugants (๐ท๐ก , ๐ ๐ก , and ๐๐ก, respectively) after the 86
incubation time ๏ฟฝฬ๏ฟฝ: 87
๐พ =๐๐ก
๐ท๐ก๐ ๐ก ๏ฟฝฬ๏ฟฝ. [4]
We label the expression in equation [4] as the โTDRโ estimate for the conjugation rate, 88
where TDR stands for the dynamic variables used in the estimate. Besides the modelโs 89
homogeneous growth rates and conjugation rates, the most notable assumption used in 90
the TDR derivation is that there is little to no change in the population densities due to 91
growth. Thus, laboratory implementation that respects this assumption can be difficult; 92
albeit chemostats and other implementation methods have been developed to circumvent 93
this constraint7,13. 94
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Simonsen et. al.16 derived the second of these widely used estimates for 95
conjugation rate ๐พ, which importantly expands application beyond the TDR method by 96
allowing for population growth. In addition to ๐ท๐ก , ๐ ๐ก , and ๐๐ก, this estimate depends on the 97
initial and final density of all bacteria (๐0 and ๐๐ก, respectively) as well as the population 98
growth rate (๐). 99
๐พ = ๐ ln (1 +๐๐ก๐ ๐ก
๐๐ก๐ท๐ก)
1
(๐๐ก โ๐0) [5]
We term equation [5] as the โSIMโ estimate for the conjugation rate, where SIM stands 100
for โSimonsen et. al. Identicality Methodโ since the underlying model still assumes all 101
strains are identical with regards to growth rates and conjugation rates. The laboratory 102
implementation for the SIM is straightforward, requiring measurements of cell densities 103
over some period of time. Because several of these densities are acquired at the end of 104
an assay, the SIM is often referred to as โthe end-point methodโ. However, we forego this 105
term since other methods we discuss are also end-point methods. 106
Huisman et. al.11 recently updated the SIM estimate, further extending its 107
application by relaxing the assumption of identical growth and transfer rates for all strains. 108
Specifically, their model extension allowed each strain to have its own growth rate (๐๐ท, 109
๐๐ , and ๐๐ denote the growth rates for donors, recipients, and transconjugants, 110
respectively) and each plasmid-bearing strain to have a distinct transfer rate (๐พ๐ท and ๐พ๐ 111
denote the rate of conjugation to recipients from donors and transconjugants, 112
respectively). They derived an estimate of ๐พ๐ท by assuming that the rate of production of 113
transconjugants through cell division and conjugation from donors to recipients was much 114
greater than conjugation from transconjugants to recipients (e.g., ๐พ๐ โค ๐พ๐ท is sufficient to 115
ensure this assumption under many standard assay conditions): 116
๐พ๐ท = (๐๐ท + ๐๐ โ๐๐)๐๐ก
(๐ท๐ก๐ ๐ก โ ๐ท0๐ 0๐๐๐๐ก). [6]
We term equation [6] as the ASM estimate for donor-recipient conjugation rate, where 117
ASM stands for โApproximate Simonsen et. al. Methodโ. The descriptions provided for the 118
aforementioned conjugation rate methods are cursory but see the Supplemental 119
Information (SI section 1-6) for a more detailed historical overview, mathematical 120
derivations, laboratory implementations, and a fuller discussion of how methods compare. 121
Deterministic numerical simulations have been used to test the accuracy of the 122
aforementioned methods (TDR, SIM, and ASM), revealing important limitations of each 123
approach11,16,17. For instance, simulations have demonstrated that all the estimates can 124
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become inaccurate when the donor-recipient conjugation rate (๐พ๐ท, hereafter โdonor 125
conjugation rateโ for simplicity) differs from the transconjugant-recipient conjugation rate 126
(๐พ๐, hereafter โtransconjugant conjugation rateโ for simplicity). Such a situation is likely 127
when the donor and recipient populations are different strains or species. While 128
deterministic simulations have revealed estimate limitations, the deterministic framework 129
itself encompasses an additional limitation. Conjugation is an inherently stochastic 130
process (i.e., random process), and the effects of such stochasticity on the accuracy and 131
precision of rate estimates are currently unknown. 132
Here we derive a novel estimate for conjugation rate, inspired by the Luriaโ133
Delbrรผck fluctuation experiment, by explicitly tracking transconjugant dynamics as a 134
stochastic process (i.e., a continuous time branching process). In addition, our method 135
allows for unrestricted heterogeneity in growth rates and conjugation rates. Thus, our 136
method fills a gap in the methodological toolkit by allowing accurate estimation of 137
conjugation rates in a wide variety of strains and species. For example, over the past few 138
years several studies have called for and recognized the importance of evaluating the 139
biology of plasmids in microbial communities4,18,19. Our method is available to support this 140
direction of research by clearly disentangling cross-species and within-species 141
conjugation rates. Thus, our approach can provide accurate conjugation rate parameters 142
for mechanistic models describing plasmid dynamics in microbial communities, where key 143
assumptions of previous estimates can be violated. We used stochastic simulations to 144
validate our estimate and compare its accuracy and precision to the aforementioned 145
estimates. We also developed a high throughput protocol for the laboratory by using 146
microtiter plates to rapidly screen many mating cultures (i.e., donor-recipient co-cultures) 147
for the existence of transconjugants. Our protocol is unique compared to the other 148
estimates and circumvents problems involving spurious laboratory results that can bias 149
other approaches. Finally, we implement our approach to investigate the evolution of 150
conjugation rate in an Escherichia coli host containing an IncP-1 conjugative plasmid. 151
152
Theoretical Framework 153
Conjugation is a stochastic process that transforms the genetic state of a cell. This 154
is analogous to mutation, which is also a process that transforms the genetic state of a 155
cell. While mutation transforms a wild-type cell to a mutant, conjugation transforms a 156
recipient cell to a transconjugant. 157
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This analogy inspired us to revisit the way Luria and Delbrรผck handled the 158
mutational process in their classic paper on the nature of bacterial mutation20, outlined in 159
Figure 1a-d. For this process, assume that the number of wild-type cells, ๐๐ก, is expanding 160
exponentially. Let the rate of mutant formation be given by ๐. In Figure 1a, we see that 161
the number of mutants in a growing population increases due to mutation events 162
(highlighted purple cells) and due to faithful reproduction by mutants (non-highlighted 163
purple cells). The rate at which mutants are generated (highlighted purple cells) is ๐๐๐ก, 164
which grows as the number of wild-type cells increases (Figure 1b). However, the rate of 165
transformation per wild-type cell is the mutation rate ๐, which is constant (Figure 1c). 166
Since mutations are random, parallel cultures will vary in the number of mutants 167
depending if and when mutation events occur. As seen in Figure 1d, for sufficiently small 168
wild-type populations growing over sufficiently small periods, some replicate populations 169
will not contain any mutant cell (gray shading) while other populations exhibit mutants 170
(yellow shading). Indeed, the cross-replicate fluctuation in the number of mutants was a 171
critical component of the Luria-Delbrรผck experiment. 172
To apply this strategy to estimate the conjugation rate, we can similarly think about 173
an exponentially growing population of recipients (Figure 1e). But now there is another 174
important cell population present (the donors). The transformation of a recipient is simply 175
the generation of a transconjugant (highlighted purple cells) via conjugation with a donor. 176
If we ignore conjugation from transconjugants, the rate at which transconjugants are 177
generated is ๐พ๐ท๐ท๐ก๐ ๐ก (Figure 1f). However, when we consider the rate of transformation 178
per recipient cell, we do not have a constant rate. Rather, this transformation rate per 179
recipient is ๐พ๐ท๐ท๐ก, which grows with the donor population (Figure 1g). It is as if we are 180
tracking a mutation process where the mutation rate is exponentially increasing. Yet the 181
rate of transformation per recipient and donor is constant, which is the donor conjugation 182
rate ๐พ๐ท. As with mutation, conjugation is random which results in a distribution in the 183
number of transconjugants among parallel cultures depending on the time points at which 184
transconjugants arise. As seen in Figure 1h, under certain conditions, some replicate 185
populations will not contain any transconjugant cell (gray shading) while other populations 186
will exhibit transconjugants (yellow shading). 187
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Figure 1: Schematic comparing the process of mutation (a-d) to the process of conjugation (e-h). (a) In a growing population of wild-type cells, mutants arise (highlighted purple cells) and reproduce (non-highlighted purple cells). (b) The rate at which mutants are generated grows as the number of wild-type cells increases at rate ๐๐๐ก. (c) The rate of transformation per wild-type cell is the mutation rate ๐. (d) Wild-type cells growing in 9 separate populations where mutants arise in a proportion of the populations (yellow background) at different cell divisions. Here, for each tree, time progresses from top to bottom (rather than left to right as in part a). (e) In a growing population of donors and recipients, transconjugants arise (highlighted purple cells) and reproduce (non-highlighted purple cells). (f) The rate at which transconjugants are generated grows as the numbers of donors and recipients increase at rate ๐พ๐ท๐ท๐ก๐ ๐ก. (g) The rate of transformation per recipient cell grows as the number of donors increases at a rate ๐พ๐ท๐ท๐ก, where ๐พ๐ท is the constant conjugation rate. (h) Donor and recipient cells growing in 9 separate populations where transconjugants arise in a proportion of the populations (yellow background) at different points in time (which again, here, progresses top to bottom for each tree).
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Using this analogy, here we describe a new approach for estimating conjugation 188
rate which embraces conjugation as a stochastic process21. Let the density of donors, 189
recipients and transconjugants in a well-mixed culture at time ๐ก be given by the variables 190
๐ท๐ก, ๐ ๐ก, and ๐๐ก. In all that follows, we will assume that the culture is inoculated with donors 191
and recipients, while transconjugants are initially absent (i.e., ๐ท0 > 0, ๐ 0 > 0, and ๐0 =192
0). The donor and recipient populations grow according to the following standard 193
exponential growth equations 194
๐ท๐ก = ๐ท0๐๐๐ท๐ก , [7]
๐ ๐ก = ๐ 0๐๐๐ ๐ก , [8]
where ๐๐ท and ๐๐ are the growth rates for donor and recipient cells, respectively. With 195
equations [7] and [8], we are making a few assumptions, which also occur in some of the 196
previous methods (see SI Table 3 for a comparison). First, we assume that donors and 197
recipients grow exponentially. Given that we focus on low-density culture conditions 198
where nutrients are plentiful, this assumption will be reasonable. Second, we assume that 199
population growth for these two strains is deterministic (i.e., ๐ท๐ก and ๐ ๐ก are not random 200
variables). As long as the numbers of donors and recipients are not too small (i.e., ๐ท0 โซ201
0 and ๐ 0 โซ 0), this assumption is acceptable. Lastly, we assume the loss of recipient cells 202
to transformation into transconjugants can be ignored. For what follows, the rate of 203
generation of transconjugants per recipient cell (as in Figure 1g) is very small relative to 204
the per capita recipient growth rate, making this last assumption justifiable. 205
The population growth of transconjugants is modeled using a continuous-time 206
stochastic process. The number of transconjugants, ๐๐ก, is a random variable taking on 207
non-negative integer values. In this section, we will assume the culture volume is 1 ml 208
and thus the number of transconjugants (i.e., cfu) is equivalent to the density of 209
transconjugants (i.e., cfu per ml). For a very small interval of time, ฮ๐ก, the current number 210
of transconjugants will either increase by one or remain constant. The probabilities of 211
each possibility are given as follows: 212
Pr{๐๐ก+ฮ๐ก = ๐๐ก + 1} = ๐พ๐ท๐ท๐ก๐ ๐กฮ๐ก + ๐พ๐๐๐ก๐ ๐กฮ๐ก + ๐๐๐๐กฮ๐ก, [9]
Pr{๐๐ก+ฮ๐ก = ๐๐ก} = 1 โ (๐พ๐ท๐ท๐ก๐ ๐ก + ๐พ๐๐๐ก๐ ๐ก + ๐๐๐๐ก)ฮ๐ก. [10]
The three terms on the right-hand side of equation [9] illustrate the processes enabling 213
the transconjugant population to increase. The first term gives the probability that a donor 214
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transforms a recipient into a transconjugant via conjugation (๐พ๐ท). The second term gives 215
the probability that a transconjugant transforms a recipient via conjugation (๐พ๐). The third 216
term measures the probability that a transconjugant cell divides (๐๐ is the transconjugant 217
growth rate). Equation [10] is simply the probability that none of these three processes 218
occur. 219
We focus on a situation where there are no transconjugants. Therefore, the only 220
process that can change the number of transconjugants is conjugation of the plasmid 221
from a donor to a recipient. Using equation [10] with ๐๐ก = 0, we have 222
Pr{๐๐ก+ฮ๐ก = 0 | ๐๐ก = 0} = 1 โ ๐พ๐ท๐ท๐ก๐ ๐กฮ๐ก. [11]
We let the probability that we have zero transconjugants at time ๐ก be denoted by ๐0(๐ก) 223
(i.e., ๐0(๐ก) = Pr{ ๐๐ก = 0}). In SI section 7, we derive the following expression for ๐0(๐ก) at 224
time ๐ก = ๏ฟฝฬ๏ฟฝ: 225
๐0(๏ฟฝฬ๏ฟฝ) = exp {โ๐พ๐ท๐ท0๐ 0๐๐ท + ๐๐
(๐(๐๐ท+๐๐ )๐ก โ 1)}. [12]
Solving equation [12] for ๐พ๐ท yields a new measure for the donor conjugation rate: 226
๐พ๐ท = โ ln๐0(๏ฟฝฬ๏ฟฝ) (๐๐ท + ๐๐
๐ท0๐ 0(๐(๐๐ท+๐๐ )๐ก โ 1)
). [13]
This expression is similar in form to the mutation rate derived by Luria and Delbrรผck in 227
their classic paper on the nature of bacterial mutation, which is not a coincidence. 228
In SI section 8, we rederive the Luria-Delbrรผck result, which can be expressed as 229
๐ = โ ln๐0(๏ฟฝฬ๏ฟฝ) (๐๐
๐0(๐๐๐๐ก โ 1)). [14]
In the mutational process modeled by Luria and Delbrรผck, ๐0 is the initial wild-type 230
population size, which grows exponentially at rate ๐๐ . For Luria and Delbrรผck, ๐0(๏ฟฝฬ๏ฟฝ) refers 231
to the probability of zero mutants at time ๏ฟฝฬ๏ฟฝ (as in a gray-shaded tree in Figure 1d), whereas 232
๐0(๏ฟฝฬ๏ฟฝ) in the conjugation estimate refers to the probability of zero transconjugants (as in a 233
gray-shaded tree in Figure 1h). Comparing equation [14] to equation [13], conjugation 234
can be thought of as a mutation process with initial wild-type population size ๐ท0๐ 0 that 235
grows at rate ๐๐ท + ๐๐ . We explore further the connection between mutation and 236
conjugation in SI section 13. 237
We label the expression in equation [13] as the LDM estimate for donor 238
conjugation rate, where LDM stands for โLow Density Methodโ as the assumption of 239
exponential growth is well grounded when the cell populations are growing at low density. 240
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Indeed, this is how we conduct the assay to compute this estimate. The acronym can 241
alternatively stand for โLuria-Delbrรผck Methodโ given the connection to their approach. 242
243
Simulation Methods 244
To explore the accuracy and precision of the LDM estimate (as well as other 245
estimates), we developed a stochastic simulation framework using the Gillespie 246
algorithm. We extended the base model (equations [1] - [3]) to include plasmid loss due 247
to segregation at rate ๐. Specifically, transconjugants are transformed into plasmid-free 248
recipients due to plasmid loss via segregation at rate ๐๐. Similarly, the donors become 249
plasmid-free cells through segregative loss of the plasmid at rate ๐๐ท; therefore, the 250
extended model tracks the change in density of a new population type, plasmid-free 251
former donors (๐น). In total, the extended model describes the density changes in four 252
populations (๐ท, ๐ , ๐, and ๐น) influenced by the values of various biological parameters: 253
growth rates (๐๐ท, ๐๐ , ๐๐, and ๐๐น), conjugation rates (๐พ๐ท and ๐พ๐) and segregation rates 254
(๐๐ท and ๐๐). For each simulation, we specify starting cell densities (๐ท0 > 0 and ๐ 0 > 0, 255
but always ๐0 = ๐น0 = 0) and allow the four populations to grow and transform 256
stochastically for a specified incubation time ๏ฟฝฬ๏ฟฝ in a 1 ml culture volume. The simulations 257
occur in sets, where each set involves a sweep across values of some key parameter. 258
Each sweep varies a biological process (i.e., growth, conjugation, or segregation) in 259
isolation to determine the effect on accuracy and precision of the donor conjugation rate 260
estimates. For instance, the transconjugant conjugation rate (๐พ๐) could be varied while all 261
other parameters are held constant. For each parameter setting within a sweep, ten 262
thousand parallel populations were simulated, and then conjugation rate was calculated 263
to yield replicate estimates for each method of interest (TDR, SIM, ASM, and LDM). 264
Notably, the LDM requires an estimate of the probability that a population has no 265
transconjugants at the end of the assay, which we denote ๏ฟฝฬ๏ฟฝ0(๏ฟฝฬ๏ฟฝ). The maximum likelihood 266
estimate of this probability is the fraction of populations (i.e., parallel simulations) that 267
have no transconjugants at the specified incubation time ๏ฟฝฬ๏ฟฝ (see SI section 9 for details). 268
Thus, for LDM, we calculated ๏ฟฝฬ๏ฟฝ0(๏ฟฝฬ๏ฟฝ) at the specified incubation time ๏ฟฝฬ๏ฟฝ by reserving multiple 269
parallel populations for each conjugation rate estimate. For more details on the 270
simulations such as the extended model equations, selection of the specified incubation 271
time ๏ฟฝฬ๏ฟฝ, and calculation of each estimate see SI section 11. 272
273
274
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Simulation Results 275
Exploring accuracy and sensitivity of the LDM and other conjugation methods 276
The underlying assumptions used to derive the various estimates for conjugation 277
rate (i.e., TDR, SIM, ASM, and LDM) entail simplifications of biological systems. To 278
explore the effects of violating these assumptions on the LDM and other conjugation rate 279
estimates, we used a stochastic simulation framework to investigate their accuracy across 280
a range of theoretical conditions. All of our simulated populations increase in size over 281
the incubation time, so a fundamental assumption of the TDR approach (i.e., no change 282
in the density due to growth) is broken for all of the runs. Here we divide our simulations 283
into three scenarios that break additional assumptions of at least one conjugation model 284
considered to this point: unequal growth rates, unequal conjugation rates, and a non-zero 285
segregation rate. When growth rates are different between populations, a modeling 286
assumption is violated for the TDR and SIM estimates. When donor and transconjugant 287
conjugation rates are unequal, a modeling assumption can be violated for the TDR, SIM, 288
and ASM estimates. When the segregation rate is non-zero, a modeling assumption is 289
violated for all of the estimates considered. We note that some of our parameter values 290
fall outside of reported values; these choices were made either to improve computational 291
speed or to better illustrate the effects of differences (e.g., in growth or transfer rates). 292
However, many of the parameter values used to explore the effects of incubation time 293
were set to reported parameter values. 294
295
The effect of unequal growth rates 296
Plasmid-containing hosts often incur a cost or gain a benefit in terms of growth rate 297
for plasmid carriage relative to plasmid-free hosts. We simulated a range of growth-rate 298
effects on plasmid-containing hosts from large plasmid costs (๐๐ท = ๐๐ โช ๐๐ ) to large 299
plasmid benefits (๐๐ท = ๐๐ โซ ๐๐ ). The LDM estimate had high accuracy and precision 300
across all parameter settings (Figure 2a). The ASM estimate performed similarly to the 301
LDM estimate with regards to the mean value for each parameter setting (Figure 2a, black 302
horizontal lines). Given that the ASM was developed to allow for unequal growth rates, 303
its mean accuracy conforms to expectations. However, it is also worth noting that the 304
distribution of the ASM estimate was skewed for some of the parameter settings. In other 305
words, the mean (Figure 2a, black horizontal lines) is separated from the median (Figure 306
2a, green horizontal line); therefore, the average of a small number of replicates could 307
have led to a deviant mean estimate. The TDR and SIM estimates were inaccurate for 308
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some to all of the parameter settings and also had skewed distributions for some 309
parameter settings. With regards to precision, the LDM estimate outperformed the other 310
estimates due to a smaller variance at each parameter setting. 311
Other than unequal growth rates due to plasmid carriage, there are other biological 312
scenarios where growth rates can differ between donors, recipients and transconjugants. 313
In fact, often it is the case that donors and recipients are different strains or species. We 314
simulated donors growing faster (๐๐ท > ๐๐ = ๐๐ ) or slower (๐๐ท < ๐๐ = ๐๐ ) than 315
recipients and transconjugants. Similar to the plasmid carriage scenario, the LDM 316
exhibited high accuracy and precision relative to other metrics (SI Figure 1). 317
318
The effect of unequal conjugation rates 319
The conjugation rate can vary between different bacterial strains and species. We 320
explored different parameter settings again, ranging from a relatively low transconjugant 321
conjugation rate (๐พ๐ โช ๐พ๐ท) to a relatively high transconjugant conjugation rate (๐พ๐ โซ ๐พ๐ท) 322
compared to the donor conjugation rate. The LDM estimate had high accuracy and 323
precision across all parameter settings and outperformed the other estimates, especially 324
when transconjugant conjugation rate greatly exceeded donor conjugation rate (Figure 325
2b). This result aligns with our expectation given that a relatively high transfer rate from 326
transconjugants to recipients can violate a modeling assumption for all the other 327
estimates. 328
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Figure 2 : The effect of violating various modeling assumptions on estimating conjugation rate. The Gillespie algorithm was used to simulate population dynamics. The parameter values were ๐๐ท = ๐๐ = ๐๐ = 1, ๐พ๐ท = ๐พ๐ =1 ร 10โ6, and ๐๐ท = ๐๐ = 0 unless otherwise indicated. The dynamic variables were initialized with ๐ท0 = ๐ 0 = 1 ร 102 and ๐0 = ๐น0 = 0. All incubation times are short but are specific to each parameter setting (see SI section 11 for details).
Conjugation rate was estimated using 10,000 populations and summarized using boxplots. The gray dashed line indicates the โtrueโ value for the donor conjugation rate. The box contains the 25th to 75th percentile. The vertical lines connected to the box contain 1.5 times the interquartile range above the 75th percentile and below the 25th percentile with the caveat that the whiskers were always constrained to the range of the data. The colored line in the box indicates the median. The solid black line indicates the mean. The boxplots in gray indicate the baseline parameter setting, and all colored plots represent deviation of one or two parameters from baseline. (a) Unequal growth rates were explored over a range of growth rates for the plasmid-bearing strains, namely ๐๐ท = ๐๐ โ {0.0625, 0.125, 0.25, 0.5, 1, 2, 4, 8}. (b) Unequal conjugation rates were probed over a range of transconjugant conjugation rates, namely ๐พ๐ โ{10โ9, 10โ8, 10โ7, 10โ6, 10โ5, 10โ4, 10โ3, 10โ2}. (c) We also explored the process of segregation by considering a range of segregative loss rates ๐๐ท = ๐๐ โ {0, 0.0001, 0.001, 0.001, 0.01}.
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The effect of a non-zero segregation rate 329
Plasmid loss due to segregation is a common occurrence in plasmid populations 330
and violates a model assumption underlying all the conjugation rate estimates. We 331
simulated a range of segregative loss rates, ranging from low (๐๐ท = ๐๐ = 0.0001) to high 332
(๐๐ท = ๐๐ = 0.1). The LDM had high accuracy and precision across all parameter settings 333
(Figure 2c). The effect of segregation was undetectable even for an extremely high 334
segregation rate (๐๐ท = ๐๐ = 0.1). Similarly, the effect of segregative loss was undetectable 335
on the other conjugation estimates compared to their performance without any 336
segregative loss (Figure 2c, colored vs. grey box plots). Thus, we find that all estimates 337
appear robust with regards to an introduction of the process of segregation. 338
339
Case studies to explore the effects of incubation time 340
The incubation time ๏ฟฝฬ๏ฟฝ is an important consideration for executing all conjugation 341
rate assays in the laboratory. Often the incubation time is variable for estimates of 342
conjugation rate reported in the literature. We explored the effects of altering incubation 343
time by estimating conjugation rate at 30-minute intervals. Since these estimates rely on 344
measuring different quantities, the range for the 30-minute sampling intervals differ 345
between estimates. For the TDR, SIM, and ASM estimates, the range for these intervals 346
was determined by the condition that at least 90 percent of the simulated populations 347
contained transconjugants at the incubation time ๏ฟฝฬ๏ฟฝ. Specifically, we note that equations 348
[4], [5], and [6] all require ๐๐ก > 0 for a non-zero conjugation rate estimate. This contrasts 349
with the LDM since this estimate needs some parallel populations to be absent of 350
transconjugants. Specifically, equation [13] requires 0 < ๐0(๏ฟฝฬ๏ฟฝ) < 1 for a non-zero finite 351
conjugation rate estimate. Thus, for the LDM estimate, the range for the sampling 352
intervals was determined by the condition that at least one parallel population had zero 353
transconjugants. Therefore, estimates for the LDM are shown at earlier incubation times 354
than those for the other methods (Figure 3). 355
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Figure 3 : The effect of incubation time (๏ฟฝฬ๏ฟฝ) on estimating conjugation rate. The Gillespie algorithm was used to
simulate population dynamics with ๐๐ท = ๐๐ = ๐๐ = 1, ๐พ๐ท = ๐พ๐ = 1 ร 10โ14, and ๐๐ท = ๐๐ = 0 unless otherwise
indicated. The dynamic variables were initialized with ๐ท0 = ๐ 0 = 1 ร 105 and ๐0 = ๐น0 = 0. Conjugation rate was
estimated for 100 populations and summarized using boxplots. The gray dashed line indicates the โtrueโ value for the donor conjugation rate. Each box contains the 25th to 75th percentile. The vertical lines connected to the box contain 1.5 times the interquartile range above the 75th percentile and below the 25th percentile with the caveat that the whiskers were always constrained to the range of the data. The colored line in the box indicates the median. The solid black line indicates the mean. (a) An unequal growth rate was simulated with ๐๐ท = ๐๐ = 0.07. (b) An unequal conjugation rate
was simulated with ๐พ๐ = 1 ร 10โ9. (c) We simulated a positive segregation rate with ๐๐ท = ๐๐ = 0.01.
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We ran simulations with parameter values for growth and conjugation rates which 356
are reported in the literature (๐๐ท = ๐๐ = ๐๐ = ๐๐น = 1, ๐พ๐ท = ๐พ๐ = 1 ร 10โ14) and 357
reasonable initial densities (๐ท0 = ๐ 0 = 1 ร 105). As before, each biological scenario 358
examined a perturbation to a single biological parameter (i.e., growth, conjugation, or 359
segregation) to determine the effect on accuracy and precision. For the scenario exploring 360
growth differences, we added a plasmid cost (Figure 3a, ๐๐ท = ๐๐ = 0.7). For the scenario 361
highlighting unequal conjugation rates, we increased the conjugation rate of 362
transconjugants (Figure 3b, ๐พ๐ = 1 ร 10โ9). For the scenario incorporating segregation, 363
we simply chose a rather high segregation rate (Figure 3c, ๐๐ท = ๐๐ = 0.01). 364
The LDM estimate had high accuracy over all time points for all scenarios with 365
precision increasing through time. The other estimates also became more precise over 366
time. However, their greater precision over time was sometimes accompanied by 367
increasing inaccuracy (Figure 3b). Again, the LDM estimate performed as well or better 368
than other estimates across incubation times. 369
370
The LDM Implementation 371
In this section we describe the general experimental procedure for estimating 372
donor conjugation rate (๐พ๐ท) using the LDM estimate in the laboratory. The assay can 373
accommodate a wide variety of microbial species and conjugative plasmids since our 374
estimate allows for distinct growth and conjugation rates. In SI section 6, we rearrange 375
equation [13] to this alternative form for implementing the LDM in the laboratory: 376
๐พ๐ท = ๐ {1
๏ฟฝฬ๏ฟฝ[โ ln ๏ฟฝฬ๏ฟฝ0(๏ฟฝฬ๏ฟฝ)]
ln๐ท๐ก๐ ๐ก โ ln๐ท0๐ 0๐ท๐ก๐ ๐ก โ ๐ท0๐ 0
}. [15]
Similar to previous conjugation estimates, the LDM requires measurement of initial and 377
final densities of donors and recipients (๐ท0, ๐ 0, ๐ท๐ก, and ๐ ๐ก). In addition, the LDM requires 378
an estimate of the probability that a population has no transconjugants at the end of the 379
assay, which we denote ๏ฟฝฬ๏ฟฝ0(๏ฟฝฬ๏ฟฝ). The maximum likelihood estimate of this probability is the 380
fraction of populations (i.e., parallel mating cultures) that have no transconjugants at the 381
specified incubation time ๏ฟฝฬ๏ฟฝ (see SI section 9 for details). Cell densities are generally 382
measured in cfu
ml units and the conjugation rate in
ml
h โ cfu units, thus, the standard unit of 383
volume in these assays is a milliliter. If exactly 1 ml is used as the volume for each mating 384
culture (as assumed in previous sections), then the quantity in braces in equation [15] is 385
the estimate for the conjugation rate. However, smaller mating volumes can be 386
advantageous (e.g., 100 l in a well inside a microtiter plate). When the mating volume 387
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deviates from 1 ml, we must add a correction factor ๐ which is the number of experimental 388
volume units (evu) per ml (e.g., for a mating volume of 100 l, we would have ๐ = 10evu
ml, 389
see SI section 10 for details). 390
For a given incubation time (๏ฟฝฬ๏ฟฝ) and initial densities of donors (๐ท0) and recipients 391
(๐ 0), there will be some probability that transconjugants form (1 โ ๐0(๏ฟฝฬ๏ฟฝ)). The 392
combinations of times and densities where this probability is not close to zero or one is 393
hereafter referred to as โthe conjugation windowโ. That is, for these time-density 394
combinations, there is a good chance that some mating cultures will result in 395
transconjugants, while others do not. For fixed initial donor and recipient densities, the 396
conjugation window refers to the time period where transconjugants are first expected to 397
form. Determining the conjugation window provides the researcher with target initial 398
densities of donors (๐ท0โฒ ) and recipients (๐ 0
โฒ ) and target incubation times (๏ฟฝฬ๏ฟฝโฒ). Note we add 399
primes to the target time-density combination to set them apart from ๐ท0, ๐ 0, and ๏ฟฝฬ๏ฟฝ in 400
equation [15] which will be gathered in the conjugation assay itself. 401
To determine the conjugation window and find a time-density combination, we mix 402
exponentially growing populations of donors and recipients in a large array of parallel 403
mating cultures for a full factorial treatment of initial densities and incubation times (Figure 404
4a and 4b). For ease of presentation, we will assume that donors and recipients start at 405
equal proportion and we will refer to the โinitial densityโ as the total cell density of these 406
two strains. In our schematic, columns of a 96 deep well microtiter plate are grouped by 407
the value of the initial density (3 columns per value; Figure 4a) and rows are grouped by 408
the value of the incubation time (2 rows per value; Figure 4b). At each time in the set of 409
incubation times being explored, growth medium selecting for transconjugants (see SI 410
section 12 for explanation) is added. This dilutes the mating cultures by ten-fold, which 411
hinders further conjugation and simultaneously permits the growth of any transconjugant 412
cells that previously formed (Figure 4b). After a longer incubation, we assess the mating 413
cultures (3 columns x 2 rows = 6 wells; Figure 4c) within each time-density treatment for 414
presence or absence of transconjugants (i.e., turbid culture or non-turbid culture). There 415
are three possible outcomes for each treatment: all matings have transconjugants, none 416
of the matings have transconjugants, or there is both transconjugant-containing and 417
transconjugant-free matings. We are interested in the last outcome (Figure 4c, gray dots). 418
These treatments meet the ๏ฟฝฬ๏ฟฝ0(๏ฟฝฬ๏ฟฝ) condition (i.e., 0 < ๏ฟฝฬ๏ฟฝ0(๏ฟฝฬ๏ฟฝ) < 1) thus identifying a valid 419
time-density combination for the LDM conjugation assay. As a general expectation, 420
mating cultures with high donor conjugation rates (๐พ๐ท) will require shorter incubation times 421
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than mating cultures with low donor conjugation rates for a given initial density. We note 422
that there will generally be multiple time-density treatments in the conjugation window 423
and the final treatment choice may be determined by experimental tractability. Thus, this 424
choice is not constrained and can be made based on an incubation time that is practically 425
feasible. In addition, the LDM does not require the initial density of donors and recipients 426
to be identical (as was assumed for this presentation). Indeed, in certain circumstances 427
it may be easier to have donor and recipient initial densities be unequal. 428
The chosen treatment yields the target time-density combination (๐ท0โฒ , ๐ 0
โฒ , ๏ฟฝฬ๏ฟฝโฒ), and 429
we proceed with executing the LDM conjugation assay to gather all components to 430
estimate the conjugation rate. We mix exponentially growing populations of donors and 431
recipients, inoculate many mating cultures at the target initial density (๐ท0โฒ+๐ 0
โฒ ) in a 96 deep 432
well plate, and incubate in non-selective growth medium for the target incubation time (๏ฟฝฬ๏ฟฝโฒ) 433
(Figure 4d). We note that the incubation time ๏ฟฝฬ๏ฟฝ may deviate slightly from the target 434
incubation time ๏ฟฝฬ๏ฟฝโฒ due to timing constraints in the laboratory, but small deviations are 435
generally permissible. Thus, the executed incubation time ๏ฟฝฬ๏ฟฝ should be used to calculate 436
the conjugation rate. To estimate the initial densities (๐ท0 and ๐ 0), three mating cultures at 437
the start of the assay are diluted and plated on donor-selecting and recipient-selecting 438
agar plates. We note that the initial densities (๐ท0 and ๐ 0) may deviate from the target 439
initial densities (๐ท0โฒ and ๐ 0
โฒ ), but small deviations are permissible. After the incubation time 440
๏ฟฝฬ๏ฟฝ, final densities (๐ท๐ก and ๐ ๐ก) are obtained by dilution-plating from the same mating 441
cultures. Growth medium selecting for transconjugants is subsequently added to the 442
remaining mating cultures. To obtain ๏ฟฝฬ๏ฟฝ0(๏ฟฝฬ๏ฟฝ), we assess the proportion of mating cultures 443
that are non-turbid after incubation in transconjugant-selecting medium, as these cultures 444
have no transconjugants (Figure 4d). This differs from the traditional LuriaโDelbrรผck 445
method since no plating is required to obtain ๏ฟฝฬ๏ฟฝ0(๏ฟฝฬ๏ฟฝ). The binary output (turbidity vs. no 446
turbidity) is assessed in a high throughput manner using a standard microtiter 447
spectrophotometer. Using the obtained densities (๐ท0, ๐ 0, ๐ท๐ก, and ๐ ๐ก), the incubation time 448
(๏ฟฝฬ๏ฟฝ), the proportion of transconjugant-free populations (๏ฟฝฬ๏ฟฝ0(๏ฟฝฬ๏ฟฝ)), and correcting for the 449
experimental culture volume (๐), the LDM estimate for donor conjugation rate (๐พ๐ท) can be 450
calculated via equation [15]. 451
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Figure 4 : Overview for finding the conjugation window (a-c) and executing the LDM conjugation assay (d). (a) Microtiter plate map designating placement of donors (red), recipients (blue), transconjugants (purple), and mating cultures over 10-fold dilutions (different shades of gray). For simplicity, donors and recipients are at the same proportion in each mating culture. The control wells are the last three columns. (b) Using the microtiter plate from part a, transconjugant-selecting medium (green well fill) is added at each time period designated by two rows in the microtiter plate. Two example wells from different time-density treatments are highlighted on the left where the gray background shading refers to the initial density treatment labeled in part a. In the top example well, transconjugant-selecting medium is added immediately, inhibiting growth of donor and recipient cells (grey dashed cells), and resulting in a non-turbid well as no transconjugants formed. In the bottom example well, the donor and recipient populations in the mating culture grow until transconjugant-selecting medium is added at 9-hours, inhibiting growth of donors and recipients, and permitting growth of the formed transconjugants. (c) After a lengthy incubation of the microtiter plate from part b, there are two well-types in the microtiter plate (bottom-left): transconjugant-containing (green well with purple cross) and transconjugant-free (green well). For each treatment (time-density pair), the 6 mating wells are considered as a group, resulting in one of three outcomes (top): all transconjugant-free wells (white dot), all transconjugant-containing wells (black dot), a proportion of both well-types (gray dot). Any treatment with a gray dot represents a viable combination of initial densities (๐ท0
โฒ + ๐ 0โฒ ) and incubation time (๏ฟฝฬ๏ฟฝโฒ). (d) The microtiter plate is set up with many matings (gray) at the
chosen target density (๐ท0โฒ +๐ 0
โฒ ) in addition to control wells. Three matings are sampled to determine the actual initial
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densities (๐ท0 and ๐ 0) and final densities (๐ท๏ฟฝฬ๏ฟฝ and ๐ ๏ฟฝฬ๏ฟฝ). After the incubation time (๏ฟฝฬ๏ฟฝ), which is as close to the target incubation time (๏ฟฝฬ๏ฟฝโฒ) as possible, transconjugant-selecting medium is added to the microtiter plate (excluding the matings used for density-plating). After a lengthy incubation, the proportion of transconjugant-free (i.e., non-turbid) wells is calculated, yielding ๏ฟฝฬ๏ฟฝ0(๏ฟฝฬ๏ฟฝ). Using the actual incubation time (๏ฟฝฬ๏ฟฝ), initial densities (๐ท0, ๐ 0), final densities (๐ท๏ฟฝฬ๏ฟฝ, ๐ ๏ฟฝฬ๏ฟฝ), and correcting for the experimental culture volume (๐), the LDM estimate is used to calculate donor conjugation rate (๐พ๐ท). Part d can be repeated to obtain replicate LDM estimates.
Application of the LDM 452
To explore the logistics of implementing the LDM, we measured the conjugation 453
rate of a conjugative, IncP-1 plasmid (pALTS29) encoding chloramphenicol resistance 454
in an Escherichia coli host4. Jordt et. al.4 used this plasmid-host pair as an ancestral strain 455
which was propagated in chloramphenicol antibiotic for 400 generations to explore 456
mechanisms of plasmid-host co-evolution. The evolved lineage acquired genetic 457
modifications in the host chromosome and the plasmid. The ancestral plasmid was 458
gradually lost from the ancestral host population when these plasmid-bearing cells were 459
propagated under conditions that did not select for the plasmid (Figure 5a, dashed line). 460
In contrast, the evolved plasmid was considerably more persistent in the absence of 461
selection in its co-evolved host (Figure 5a, solid line). 462
This pattern of increased plasmid persistence after plasmid-host co-evolution is 463
consistent with previous studies22,23. In most cases, the cost of plasmid carriage is 464
assumed to decrease with plasmid-host co-evolution; however, changes in other factors, 465
such as an increased rate of conjugation or a decreased rate of segregational loss, may 466
also result in increased plasmid persistence. Using the LDM method, we showed that the 467
observed increase in plasmid persistence in the Jordt. et. al.4 study is at least in part due 468
to a significant increase in the conjugation rate in the evolved lineage (Figure 5b, Mann-469
Whitney U test, p-value = 0.029). 470
Figure 5 : LDM conjugation rate estimates of an ancestral and evolved lineage of E. coli containing a
conjugative IncP-1 plasmid. (a) Plasmid persistence in the absence of chloramphenicol antibiotic was higher in the evolved plasmid-host pair (solid line) than the ancestral plasmid-host pair (dashed line). Data and figure adapted from Jordt et. al.4 (b) Conjugation rate significantly increased in the evolved (Evo) plasmid-host pair compared to the ancestral (Anc) plasmid-host pair. The points represent a conjugation rate estimate from an LDM conjugation assay. The black line is the mean of four replicates. The colored line is the median of the four replicates. The asterisk indicates a significant difference from a Mann-Whitney U test.
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Discussion 471
Here we have presented a new method for estimating the rate of plasmid 472
conjugative transfer from a donor cell to a recipient cell. We derived our LDM estimate 473
using a mathematical approach that captures the stochastic process of conjugation, which 474
was inspired by the method Luria and Delbrรผck applied to the process of mutation. This 475
departs from the mathematical approach for other conjugation rate estimates which 476
assume underlying deterministic frameworks guiding the dynamics of 477
transconjugants11,13,16. One noteworthy conjugation assay from Johnson & Kroer also 478
embraced the stochastic nature of conjugation21. Similar to one aspect of our approach, 479
they survey a series of mating cultures for the presence of transconjugants and use the 480
fraction of transconjugant-free cultures to derive a conjugation metric. However, their 481
metric yields the number of conjugation events and does not provide an estimate for the 482
conjugation rate. Thus, the population dynamics of donors and recipients, as well as the 483
incubation time, are not considered. The LDM captures positive features of both the 484
Johnson & Kroer approach and the deterministic-based approaches: specifically, it both 485
recognizes the stochastic nature of conjugation and provides an analytic estimate for the 486
conjugation rate. 487
Beyond the incorporation of stochasticity, the model and derivation behind the 488
LDM estimate relaxes assumptions that constrained former approaches. This makes 489
calculating conjugation rates accessible to a wide range of experimenters that use 490
different plasmid-donor-recipient combinations. The LDM estimate makes no restrictive 491
assumptions about growth rates or conjugation rates of different bacterial strains. Other 492
conjugation rate estimates assume little to no change in the density due to growth (TDR), 493
growth rates are identical across all strains (TDR and SIM), conjugation rates are identical 494
for donors and transconjugants (SIM), or conjugation events from transconjugants are 495
few (TDR and ASM)11,13,16. By relaxing these assumptions, we have increased the 496
applicability of this new estimate, and our simulation results confirmed the high accuracy 497
of the LDM under such scenarios. In contrast, the other conjugation methods (TDR, SIM, 498
and ASM) sometimes yielded inaccurate estimates when their underlying assumptions 499
were violated. Because violations of these assumptions are likely in natural plasmid 500
systems, these performance differences could be important. For instance, unequal growth 501
rates can occur among isogenic strains if the plasmid increases or decreases the growth 502
rate of a host5,24. Also, growth rates may be significantly different when the donor and 503
recipient are different strains or species. For example, some strains of Escherichia coli 504
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can divide every 20 minutes while some strains of Shewanella oneidensis only divide 505
every 45 minutes25,26. Consequently, the growth rates of all the cell types will differ due 506
to these combined effects in many plasmid transfer experiments. Such compounding 507
effects could increase the error in measuring conjugation rate with the TDR and SIM 508
estimates. Moreover, unequal growth rates are not likely to occur in isolation, but in 509
tandem with other modeling violations such as unequal plasmid transfer rates. The 510
consequences of such scenarios remain to be explored. 511
Perhaps the most significant advantage of our LDM method is the capability to 512
accurately estimate the donor conjugation rate when it deviates from the transconjugant 513
conjugation rate. For the previous methods, the error in estimating the donor conjugation 514
rate increases as the transconjugant conjugation rate gets larger than the donor 515
conjugation rate. These differences are likely to occur in plasmid transfer experiments. 516
For instance, Dimitriu et. al.27 used the TDR approach to measure donor conjugation rate 517
where the recipient was a clone-mate or a different species. The authors observed cross-518
species conjugation rates that were up to 4 orders of magnitude lower than the within-519
species conjugation rates. If this pattern held generally, then a cross-species mating 520
assay would violate a key assumption of previous methods, in that the transconjugant 521
conjugation rate would be much higher than the donor conjugation rate. Indeed, the 522
โunequal conjugation ratesโ simulations resulted in inflated conjugation estimates for the 523
other methods (Figure 2b). For these reasons, reported cross-species conjugation rates 524
may also be overestimates. In any case, unequal conjugation rates are likely, especially 525
in microbial communities with diverse species. In addition, unequal conjugation rates can 526
arise due to a molecular mechanism encoded on the conjugative plasmid, such as 527
transitory de-repression13,28. In this case, a conjugation event between donor and 528
recipient will temporarily elevate the conjugation rate of the newly formed transconjugant. 529
This phenomenon would lead to similar outcomes as described above for different 530
species. The LDM method can accurately estimate donor conjugation rates in systems 531
with unequal conjugation rates, whether the differences are taxonomic or molecular in 532
origin. 533
The LDM estimate also has advantages in terms of precision. Since conjugation is 534
a stochastic process, the number of transconjugants at any given time is a random 535
variable with a certain distribution. Therefore, estimates that rely on the number of 536
transconjugants (TDR, SIM and ASM) or the probability of their absence (LDM) will also 537
fall into distributions (Figure 2). Even in cases where the mean (first moment of the 538
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distribution) is close to the actual conjugation rate, the variance (second central moment) 539
may differ among estimates. In fact, the LDM estimate had smaller variance compared to 540
other estimates, even under parameter settings where different estimates shared similar 541
accuracy (Figure 2). This greater precision likely originates from the difference in the 542
distribution of the number of transconjugants (๐๏ฟฝฬ๏ฟฝ) and the distribution of the probability of 543
transconjugant absence (๐0(๏ฟฝฬ๏ฟฝ)), something we explore analytically in SI section 14. 544
Beyond the mean and variance, other features of these distributions (i.e., moments) may 545
also be important. For certain parameter settings, the estimates relying on transconjugant 546
numbers were asymmetric (the third moment was non-zero). In such cases, a small 547
number of replicate estimates could lead to bias. For example, when there is a strong 548
growth benefit to carrying the plasmid, the distribution for the ASM estimate is skewed 549
even though the mean of this distribution reflects the actual conjugation rate (see Figure 550
2a). Typically a small number of conjugation assays is often the standard in studies; thus, 551
the general position and shape of these estimate distributions may matter. Over the 552
portion of parameter space we explored, the LDM distribution facilitated accurate and 553
precise estimates through its position (a mean reflecting the true value) and its shape (a 554
small variance and a low skew). 555
For all the methods considered, the distribution of conjugation estimates changed 556
with time (Figure 3). Therefore, the incubation time becomes an important parameter. In 557
our simulations, the estimate distribution generally narrowed with longer incubation times 558
(Figure 3). This would suggest that one advantage of longer incubation is greater 559
precision. However, when underlying assumptions are violated, there can be an 560
interesting interaction between accuracy and precision over time. For instance, consider 561
the case where transconjugant conjugation rate exceeds donor conjugation rate (Figure 562
3b). For the SIM and ASM metrics, as incubation time rises, there is a shift from an 563
accurate imprecise estimate to an inaccurate precise estimate. For an assay with a limited 564
number of replicates in such a case, the optimal incubation time may be intermediate (a 565
topic worthy of formal investigation). Over a range of incubation times where ๐0(๏ฟฝฬ๏ฟฝ) is not 566
close to 0 or 1, the LDM distribution is generally immune to these problems, exhibiting 567
constant high accuracy and precision over time (Figure 3). 568
To execute a conjugation assay using any of the methods described, the 569
experimenter needs to make several decisions regarding various protocol features such 570
as incubation time and density of the mating culture. All of these choices need to conform 571
closely to the modeling assumptions of each method and ideally maximize the accuracy 572
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25
of the conjugation rate estimate. For instance, the TDR assay seems simpler than the 573
other methods, since it only requires measurement of the final densities of cells. However, 574
implementing the TDR assay can be difficult because it assumes little to no change in 575
density due to growth. For a growth-based TDR implementation, a very short incubation 576
time could limit cell growth, but the density of the culture may need to be very high so 577
transconjugants can form during the assay. Previous studies have shown that for many 578
plasmids the conjugation rate is maximal during exponential phase13,17,29; therefore, using 579
high density culture may underestimate the conjugation rate. A chemostat implementation 580
could circumvent some of these issues by allowing lower density culture, exponential 581
growth, and longer incubation times. However, unequal growth rates in a chemostat 582
would lead to changes in the density of the donors and recipients. For the SIM approach, 583
a standard growth-based 24-hour conjugation assay is often used5,30,31,32,33. This serves 584
as an easy default choice and has many laboratory benefits for efficiency. Even so, the 585
24-hour incubation involves more than initial and final sampling, as two additional 586
samplings are required to measure the maximum population growth rate during 587
exponential phase (SI section 2). If these additional samplings are forgotten for the SIM 588
approach, highly inaccurate estimates of the conjugation rate can result. In addition, the 589
underlying SIM assumptions of equal growth rates and conjugation rates are easy to 590
violate, and inaccuracies due to such violations can grow with incubation time (Figure 3b). 591
For the ASM, a standard incubation time is not recommended; rather, this time is left to 592
the experimenter to determine. However, the authors suggest short incubation times to 593
prevent biased estimates if modeling assumptions are violated (e.g., relatively high 594
transconjugant transfer rates). The authors mention that a follow-up assay to measure 595
the transconjugant transfer rate may be needed to determine if the incubation time was 596
short enough to prevent deviations. In contrast, the LDM protocol is designed to lead the 597
experimenter to valid incubation times by explicitly identifying the conjugation window 598
where ๐0(๏ฟฝฬ๏ฟฝ) can be estimated. Therefore, the experimenter can be sure the protocol 599
occurred during a usable incubation time if the assay results in a ๏ฟฝฬ๏ฟฝ0(๏ฟฝฬ๏ฟฝ) between 0 and 1. 600
We note that identifying the conjugation window does add an extra step in the LDM 601
approach; however, a similar assay could help an experimenter choose an optimal 602
incubation time for the other assays. Even so, the effects of these implementation choices 603
on the accuracy and precision of conjugation rate estimates are currently understudied. 604
The LDM protocol offers an additional advantage by eliminating plating of the 605
transconjugants. For the other methods, this is typically a source of error since the 606
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transconjugant density is often determined by plating the mating culture on 607
transconjugant selecting medium. With short culture incubation times before plating, 608
which will increase the accuracy of the other methods, the transconjugant density is 609
extremely low compared to the donor and recipient densities. Therefore, plating high 610
density cultures is needed to be able to detect the transconjugants. This can lead to 611
conjugation continuing on the agar plates, which produces extra transconjugant colonies 612
that did not result from conjugation events before plating,15,34. Such โmatings on 613
transconjugant-selecting agar platesโ can occur before the donors and recipients are 614
completely inhibited by the transconjugant-selecting agent (typically an antibiotic). Such 615
events will lead to an inflation of the transconjugant density and thus inaccuracy in the 616
conjugation rate estimate. We note that the LDM avoids this problem since it relies only 617
on recording transconjugant presence/absence in the liquid mating cultures. 618
We applied our high-throughput LDM protocol to our previous evolution experiment 619
to estimate the conjugation rate for an IncP-1 plasmid in an E. coli host. Interestingly, 620
we observed the evolution of conjugation rate. This comes as a bit of a surprise given 621
that the environment in which the E. coli evolved contained an antibiotic selecting for 622
plasmid carriage. Thus, plasmid-free cells would have been rare leading to few 623
opportunities for plasmid conjugation into such cells. Bacterial cells containing a 624
conjugative plasmid in a community with an abundance of plasmid-free cells is the 625
scenario in which conjugation rates are expected to and have been observed to evolve35. 626
Therefore, it is not immediately obvious what mechanism led to the increase in the 627
conjugation rate in our system. It is possible that increased conjugation rate could have 628
been a pleiotropic effect from a different trait under direct selection. Alternatively, higher 629
conjugation rate could evolve if (1) a mutation on the plasmid led to a higher transfer rate 630
and (2) if donor cells can conjugate with other donors in our systems36. In such a case, 631
the mutated plasmid would transfer quickly even under plasmid-selecting conditions and 632
it could spread through the population as long as any associated costs (e.g., lower growth 633
of its host or lower replication efficiency within mixed-plasmid hosts) were sufficiently low. 634
This hypothetical sequence of events could lead to an increased conjugation rate. 635
Regardless of the mechanism, it is interesting that bacterial populations in environments 636
containing an antibiotic could lead to the evolution of higher conjugation rates of plasmids 637
encoding resistance to that antibiotic. Importantly, such evolution could lead to higher 638
frequency of antibiotic resistance in microbial communities, even in the absence of the 639
antibiotic, due to more plasmid conjugation events. 640
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In conclusion, the LDM offers new possibilities for measuring the conjugation rate 641
for many types of plasmids and species. We have presented evidence that supports our 642
method being more accurate and precise than other widely used approaches. We note 643
that we did not fully review all the conjugation assays currently in use in the literature15,37, 644
focusing instead on the least laborious and most widely used methods. Importantly, the 645
LDM eliminates the possible bias caused by high transconjugant conjugation rates. Due 646
to the sensitivity of the LDM conjugation assay and the ease of implementation, we were 647
able to show the evolution of conjugation rate for an IncP-1 conjugative plasmid in an 648
Escherichia coli host. However, the LDM method can be applied even more broadly given 649
the flexibility of the underlying theoretical framework and the relative ease of assay 650
implementation. 651
652
Acknowledgements 653
This work is supported by the National Institute of Allergy and Infectious Diseases 654
Extramural Activities grant no. R01 AI084918 of the National Institutes of Health. O.K. is 655
supported by the NSF Graduate Research Fellowship grant no. DGE-1762114. C.E. is 656
supported by the NSF Graduate Research Fellowship. We thank Hannah Jordt, Simon 657
Snoeck, and members of the Kerr and Top laboratories for useful suggestions on the 658
manuscript. 659
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Supplemental Information660
SI section 1 : Historical theoretical perspective 661
In this section, we highlight three key methods for estimating conjugation rate. 662
While outlining the theoretical framework, we highlight the key distinctions and theoretical 663
assumptions of each approach. Levin et. al.13 introduced a simple mathematical model 664
for plasmid population dynamics to isolate the conjugation rate parameter (๐พ)13. In their 665
model, the donor population grows exponentially at a rate ๐ (equation [1]). The number 666
of transconjugants increases due to conjugation events (2nd term equation [3]) by donors 667
and by existing transconjugants as well as exponential growth of the existing 668
transconjugant pool (1st term equation [3]). The recipient population grows exponentially 669
at a rate ๐ and is depleted at the rate that new transconjugants form due to conjugation 670
events (2nd term equation [2]). For ease of reference, model variables and parameters 671
are in SI Table 1 (the LDM is included for a complete comparison). 672
SI Table 1: Variables and parameters used in plasmid dynamic models
Variable/Parameter Description Relevant Estimate Units
๐ท Donor density TDR, SIM, ASM, LDM
cfu
ml ๐ Recipient density TDR, SIM, ASM, LDM
๐ Transconjugant density TDR, SIM, ASM, LDM
๐ Growth rate (not population specific) TDR, SIM
1
h
๐๐ท Donor growth rate ASM, LDM
๐๐ Recipient growth rate ASM, LDM
๐๐ Transconjugant growth rate ASM, LDM
๐พ Conjugation rate (not population specific) TDR, SIM
ml
cfu โ h ๐พ๐ท Donor-recipient conjugation rate ASM, LDM
๐พ๐ Transconjugant-recipient conjugation rate ASM, LDM
Equations [1] - [3] contains four notable assumptions. First, conjugation is 673
described by mass-action kinetics, where conjugation events are proportional to the 674
product of donor and recipient cell densities, which is a reasonable assumption in well-675
mixed liquid cultures13. Second, the model assumes a negligible rate of segregation, a 676
process whereby a dividing plasmid-containing cell produces one plasmid-containing 677
daughter cell and one plasmid-free daughter. These first two assumptions also exist in all 678
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29
of the other conjugation estimate methods we discuss. Third, the growth rate is the same 679
for all cell types (donors, recipients, and transconjugants). Fourth, the plasmid 680
conjugation rate is the same from donors to recipients as from transconjugants to 681
recipients. 682
Levin et. al.13 used their plasmid dynamic model to derive an estimate for plasmid 683
conjugation rate (๐พ) by making three additional simplifying assumptions. First, the change 684
in cell density of donors due to growth is assumed to be negligible. Likewise, the change 685
in cell density of recipients due to growth and to conjugation (i.e., transformation into 686
transconjugants) is assumed to be negligible. Finally, transconjugants are assumed to be 687
rare in the population resulting in negligible growth of transconjugants and conjugation 688
from transconjugants to recipients. Thus, the increase in transconjugants cell density is 689
through plasmid conjugation from donors to recipients (i.e., ๐๐/๐๐ก = ๐พ๐ท๐ ). Using these 690
simplifying assumptions, Levin et. al.13 solved for an expression of the conjugation rate ๐พ 691
(equation [4]) in terms of the density of donors, recipients, and transconjugants (๐ท๐ก, ๐ ๐ก, 692
and ๐๐ก, respectively) after a period of incubation ๏ฟฝฬ๏ฟฝ (see SI section 3 for a few different 693
approaches to this derivation). We label the expression in equation [4] as the โTDRโ 694
estimate for the conjugation rate, where the letters in this acronym come from the dynamic 695
variables used in the estimate. TDR is a commonly used estimate14,13,15,17. However, the 696
simplifying assumptions used to derive TDR, such as no change in density due to growth, 697
can make implementation in the laboratory difficult. For ease of reference, all variables 698
used in the conjugation estimates are in SI Table 2. All assumptions underlying each 699
estimate are in SI Table 3. Both tables include LDM for a full comparison. 700
SI Table 2: Variables and parameters used to estimate* conjugation rate
Variable/Parameter Description Relevant Estimate Units
๏ฟฝฬ๏ฟฝ Incubation time (final sampling time) TDR, SIM**, ASM, LDM h
๐ท0, ๐ 0 Initial donor and recipient densities ASM, LDM
cfu
ml
๐ท๐ก , ๐ ๐ก Final donor and recipient densities TDR, SIM, ASM, LDM
๐๐ก Final transconjugant density TDR, SIM, ASM
๐0, ๐๐ก Initial and final total population density SIM
๐๐ Transconjugant growth rate ASM hโ1
๐0(๏ฟฝฬ๏ฟฝ) Probability a population has no transconjugants
LDM exp (cfu
ml)
* The laboratory estimates are used here (see SI section 6) ** If the SIM assay is conducted on exponentially growing cultures (see SI section 2), ๐กฬ along with ๐0 and ๐๏ฟฝฬ๏ฟฝ can be used to estimate ๐ (otherwise, an independent estimate of ๐ is needed).
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SI Table 3: Summary of modeling assumptions
Assumption TDR SIM ASM LDM
Conjugation events follow mass-action kinetics X X X X
There is no segregative loss of the plasmid X X X X
The cell populations do not change in size due to growth X
Processes of conjugation and growth are not resource dependent* X X X
The cell populations grow exponentially X X
The growth rate is identical for all cell types X X
The transconjugant conjugation rate is not high relative to the donor conjugation rate
X X X
* The SIM model is able to incorporate resource-dependent growth and conjugation because (1) growth and transfer rates are homogeneous and (2) the functional form for resource dependence is the same for growth and transfer.
In contrast to the assumption behind the TDR estimate, the model explored by 701
Simonsen et. al.16 allows for population growth. Indeed, they allowed for the rate of 702
population growth to change with the level of a resource in the environment, adding a 703
dynamic variable for the resource concentration. In addition, the conjugation rate can also 704
change with the resource concentration. The authors focus on a case where both growth 705
and conjugation rates vary with resource concentration according to the Monod function. 706
This choice was informed by experimental results showing that cells enter stationary 707
phase and conjugation ramps down to a negligible level as resources are depleted13. This 708
pattern occurs for various plasmid incompatibility groups, but not all29. Simonsen et. al.16 709
used this updated model to derive an estimate for plasmid conjugation rate (equation [5], 710
see SI section 4 for the derivation). We term equation [5] as the โSIMโ estimate for 711
conjugation rate, where SIM stands for โSimonsen et. al. Identicality Methodโ since the 712
underlying model assumes that all strains are identical with regards to growth, and donors 713
and transconjugants are identical with regards to conjugation rate (as in equations [1]-714
[3]). The SIM estimate involves measuring the density of the initial population (๐0), and 715
the final density of donors (๐ท๐ก), recipients (๐ ๐ก), transconjugants (๐๐ก), and the total 716
population (๐๐ก) after the incubation time ๏ฟฝฬ๏ฟฝ (e.g., the end of an assay). The SIM estimate 717
is a commonly used since it allows for the use of batch culture in the laboratory. Thus, it 718
circumvents the constraints of the laboratory implementation of TDR; however, the 719
underlying model (an enriched version of equations [1]-[3]) holds the same assumptions 720
as before: homogeneous growth rates and conjugation rates. 721
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Recently, Huisman et. al.11 updated the SIM model by introducing population 722
specific growth rates for donors, recipients, and transconjugants (๐๐ท, ๐๐ , and ๐๐, 723
respectively) and population specific conjugation rates for donors and transconjugants 724
(๐พ๐ท and ๐พ๐). Importantly, the updated model is an approximation of the SIM due to three 725
additional assumptions. First, conjugation and growth rates are assumed to be constant 726
until resources are depleted, eliminating the additional resource concentration equation. 727
Second, the increase in recipients due to growth greatly outpaces the decrease in 728
recipients due to conjugation (i.e., ๐๐ ๐ โซ ๐พ๐ท๐ท๐ + ๐พ๐๐๐ ). Third, the increase in 729
transconjugants due to growth or plasmid conjugation from donors to recipients greatly 730
outpaces the increase in transconjugants due to plasmid conjugation from 731
transconjugants to recipients (i.e., ๐๐๐ + ๐พ๐ท๐ท๐ โซ ๐พ๐๐๐ ). These model conditions are 732
reasonable if the system starts with donors and recipients present but transconjugants 733
are absent, the system is tracked for a short period of time ๏ฟฝฬ๏ฟฝ, conjugation rates are low 734
relative to growth rates, and the transconjugant conjugation rate (๐พ๐) is not much higher 735
than the donor conjugation rate (๐พ๐ท). With these added assumptions, equations [1]-[3] 736
can be reformulated as the following approximate system of equations: 737
๐๐ท
๐๐ก= ๐๐ท๐ท, [1.1]
๐๐
๐๐ก= ๐๐ ๐ , [1.2]
๐๐
๐๐ก= ๐๐๐ + ๐พ๐ท๐ท๐ , [1.3]
Huisman et. al. use this model to derive an estimate for conjugation rate (equation [6]) 738
where different cell types now can have different growth rates (see SI section 5 for the 739
derivation). We term equation [6] as the ASM estimate for donor conjugation rate, where 740
ASM stands for โApproximate Simonsen et. al. Methodโ. 741
742
SI section 2 : Comparison of laboratory implementations 743
In this section, we compare the laboratory implementations for various end-point 744
estimates: TDR, SIM, and ASM. Each method is explained either as recommended by its 745
authors or the most simplified protocol to acquire the information for the estimate. For 746
each, we describe proper laboratory implementation for the approaches based on the 747
model and derivation assumptions used to acquire the estimate. Note in this section, we 748
do not explore the assumptions that are violated due to the biological samples being 749
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32
tested (i.e., specific species or plasmids) which can result in violations such as unequal 750
conjugation rates or growth rates. These are explored in the main text via stochastic 751
simulations. Thus, we focus solely on the parameters under the experimenterโs control. 752
For ease of reference, key implementation differences are highlighted in SI Table 4 (LDM 753
is included for a full comparison). 754
SI Table 4: Comparison of implementations
Summary TDR SIM ASM LDM
Assay conditions minimizing the change in density due to growth X
Minimize incubation time necessary for producing transconjugants X
An incubation time results in a subset of parallel populations having no transconjugants
X
Assay occurs over a period of exponential cell growth X* X X
Assay requires multiple parallel mating cultures per replicate X
Assay requires a measurement of transconjugant density X X X
Assay requires a measurement of population growth rate X*
Assay requires a measurement of transconjugant growth rate X
* For the SIM assay, either the entire assay is conducted over exponentially growing cultures or an independent estimate for (maximum) population growth rate is needed.
The TDR estimate has a simple form (equation [4]). Donors and recipients are 755
mixed in non-selective growth medium and incubated for a specified time ๏ฟฝฬ๏ฟฝ. Typically, 756
densities after the incubation time are determined using selective plating. The derivation 757
assumes the density in donors and recipients does not change which sets specific 758
constraints on incubation. In the original study, Levin et. al.9, used a chemostat to keep 759
the population constant. Other studies shorten the incubation time ๏ฟฝฬ๏ฟฝ such that population 760
growth is negligible and use various laboratory tools to detect the small number of 761
transconjugants15,17. 762
SIM circumvents the incubation limitations. Under conditions where the population 763
is growing exponentially over the entire assay, we can rework the SIM estimate (equation 764
[5]) into an alternative form for the laboratory (see SI section 6 for details). 765
๐พ =1
๏ฟฝฬ๏ฟฝ[ln (1 +
๐๐ก๐ ๐ก
๐๐ก๐ท๐ก)]ln๐๐ก โ ln๐0๐๐ก โ๐0
[2.1]
Donor and recipient populations in exponential phase are mixed in non-selective growth 766
medium. The initial population density (๐0) is determined by dilution plating on non-767
selective medium. After the mating mixture is incubated (for a period of ๏ฟฝฬ๏ฟฝ), the final 768
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densities (๐ท๐ก, ๐ ๐ก, ๐๐ก, and ๐๐ก) are determined by dilution plating on selective and non-769
selective media. To implement SIM as written in equation [2.1], the specified incubation 770
time ๏ฟฝฬ๏ฟฝ must occur well before stationary phase is reached to collect proper data for 771
estimating the population growth rate (๐ = (ln๐๐ก โ ln๐0)/๏ฟฝฬ๏ฟฝ). There is an alternative 772
option for implementing the SIM using equation [5]. The donor and recipient populations 773
are mixed and incubated under batch culture conditions (including lag, exponential, and 774
stationary phase). However, the (maximum) population growth rate (๐) needs to be 775
determined with two additional samplings from the mixed population at times ๐ก๐ and ๐ก๐, 776
both occurring within exponential phase: 777
๐ =ln (๐๐ก๐/๐๐ก๐)
๐ก๐ โ ๐ก๐ [2.2]
The population densities ๐๐ก๐ and ๐๐ก๐ can be estimated either through colony counts from 778
plating or optical density from a spectrophotometer. Either way, the timing of exponential 779
phase is important for this approach and at least some analysis during this phase is 780
required regardless of the implementation strategy. 781
For the ASM estimate, we can rework equation [6] into an alternate form for the 782
laboratory (see SI section 6 for details). 783
๐พ๐ท = {1
๏ฟฝฬ๏ฟฝ(ln๐ท๐ก๐ ๐ก โ ln๐ท0๐ 0) โ ๐๐}
๐๏ฟฝฬ๏ฟฝ
(๐ท๐ก๐ ๏ฟฝฬ๏ฟฝ โ ๐ท0๐ 0๐๐๐๐ก) [2.3]
Donor and recipient populations in exponential phase are mixed in non-selective medium. 784
Initial densities (๐ท0 and ๐ 0) are determined by plating dilutions on the appropriate 785
selective media. After donor and recipient incubate (๏ฟฝฬ๏ฟฝ), final densities (๐ท๐ก, ๐ ๐ก, and ๐๏ฟฝฬ๏ฟฝ) are 786
determined by plating dilutions on the appropriate selective media. From the 787
transconjugant-selecting agar plates, a transconjugant clone is incubated in monoculture 788
then sampled twice (at times ๐ก๐ and ๐ก๐) in exponential phase to measure the 789
transconjugant growth rate (๐๐ = ln(๐๐ก๐/๐๐ก๐)/(๐ก๐ โ ๐ก๐)). The authors point out a critical 790
consideration for proper implementation of the ASM is the incubation time ๏ฟฝฬ๏ฟฝ. Not only is 791
sampling in exponential phase important, but if the incubation time ๏ฟฝฬ๏ฟฝ is too long and 792
passes a critical time (๐ก๐๐๐๐ก) the approximations used to derive the ASM break down. To 793
avoid overshooting ๐ก๐๐๐๐ก, the authors recommend sampling as soon as measurable 794
transconjugants arise. To determine that the incubation time used was below the critical 795
time (๐ก๐๐๐๐ก), a second assay is recommended by the authors to measure the 796
transconjugant conjugation rate ๐พ๐, which will determine if the original incubation time ๏ฟฝฬ๏ฟฝ 797
was below ๐ก๐๐๐๐ก for measuring the donor conjugation rate. This second assay would have 798
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34
the transconjugant clone become the donor in the mixture, while a newly marked recipient 799
must be used so that donors and recipients can be distinguished using selective plating. 800
Each method has aspects of implementation in common. Each one shares the 801
basic approach of mixing donors and recipients over some incubation time ๏ฟฝฬ๏ฟฝ. Each 802
estimate requires reliable selectable markers to differentiate donors, recipients, and 803
transconjugants. However, all estimates have some constraints on initial densities and 804
time of measurement. This can occur because the experimenter needs to capture 805
conjugation events (all estimates require this), avoid population growth (TDR), or keep 806
growth exponential (ASM, and at least parts of SIM). Even so, each method has clear 807
distinctions. The most notable is the incubation time ๏ฟฝฬ๏ฟฝ (i.e., the end of the assay). The 808
TDR method is constrained to conditions where no change in population size due to 809
growth can occur. For SIM, initial and final sampling are not constrained to a particular 810
phase of growth; however, measurement of the growth rate must occur during the 811
exponential growth phase. For ASM, initial sampling is in early exponential phase, and 812
the final sampling needs to occur during a specific time window. In other words, the assay 813
needs to be long enough that measurable transconjugants appear, but short enough so 814
that assumptions arenโt violated (which can occur if transconjugant density becomes too 815
large). 816
817
SI section 3 : TDR derivation 818
We present a few derivations for the TDR method. The derivations differ by the 819
assumptions used to solve for TDR. We use this section to show all of the conditions 820
where the TDR method can be used. In addition, we use this section to walk through all 821
of the mathematical steps and provide an easy reference for comparing the derivations 822
of all four methods (TDR, SIM, ASM, and LDM). The original condition presented by Levin 823
et. al.13 is the last derivation described in this section. 824
Here we assume that we are considering conditions where population growth is 825
not occurring. These conditions could involve an environment hindering cell division. 826
Alternatively, the environment could promote growth, but we restrict the time period 827
sufficiently so that population change is negligible. Or finally, if all strains possess the 828
same growth rate, the population could be growing over longer periods of time but doing 829
so in a continuous flow-through device (e.g., a chemostat) such that population density 830
remains constant (see Levin et. al.13). Because we assume no segregative loss of the 831
plasmid, the donor population must remain constant 832
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๐ท๐ก = ๐ท0, [3.1]
for any time ๐ก under consideration. Importantly, even though growth is not occurring, 833
conjugation can proceed. We assume ๐0 = 0, but the population of transconjugants can 834
increase over time. Because every transconjugant was formerly a recipient cell, it must 835
be the case that 836
๐ ๐ก + ๐๐ก = ๐ 0, 837
for any time ๐ก under consideration. Therefore, 838
๐ ๐ก = ๐ 0 โ ๐๐ก. [3.2]
The dynamics of transconjugants is given by 839
๐๐๐ก๐๐ก
= ๐พ๐ท๐ท๐ก๐ ๐ก + ๐พ๐๐๐ก๐ ๐ก [3.3]
By substituting terms from equations [3.1] and [3.2] 840
๐๐๐ก๐๐ก
= ๐พ๐ท๐ท0(๐ 0 โ ๐๐ก) + ๐พ๐๐๐ก(๐ 0 โ ๐๐ก), 841
๐๐๐ก๐๐ก
= (๐พ๐ท๐ท0 + ๐พ๐๐๐ก)(๐ 0 โ ๐๐ก), 842
๐๐๐ก๐๐ก
= โ๐พ๐ (๐๐ก +๐พ๐ท๐ท0๐พ๐
) (๐๐ก โ ๐ 0), 843
We can solve this differential equation by a separation of variables: 844
โซ๐๐๐ก
โ๐พ๐ (๐๐ก +๐พ๐ท๐ท0๐พ๐
) (๐๐ก โ ๐ 0)
๐ก
0
= โซ ๐๐ก๐ก
0
. 845
The following identity is relevant here: 846
๐ {1
๐(๐ โ ๐)ln๐ฅ โ ๐๐ฅ โ ๐}
๐๐ฅ=
1
๐(๐ฅ โ ๐)(๐ฅ โ ๐) . 847
Letting ๐ฅ = ๐๐ก, ๐ = โ๐พ๐, ๐ = โ๐พ๐ท๐ท0
๐พ๐, and ๐ = ๐ 0, we can proceed as follows: 848
{1
๐พ๐ (๐พ๐ท๐ท0๐พ๐
+ ๐ 0)ln๐๐ก +
๐พ๐ท๐ท0๐พ๐
๐๐ก โ ๐ 0}|
0
๐ก
= (๐ก)|0๐ก , 849
1
๐พ๐ (๐พ๐ท๐ท0๐พ๐
+ ๐ 0)(ln
๐๐ก +๐พ๐ท๐ท0๐พ๐
๐๏ฟฝฬ๏ฟฝ โ ๐ 0โ ln
๐0 +๐พ๐ท๐ท0๐พ๐
๐0 โ ๐ 0) = ๏ฟฝฬ๏ฟฝ. 850
Because ๐0 = 0, 851
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1
๐พ๐ (๐พ๐ท๐ท0๐พ๐
+ ๐ 0)(ln
๐๐ก +๐พ๐ท๐ท0๐พ๐
๐๐ก โ ๐ 0โ ln
๐พ๐ท๐ท0๐พ๐โ๐ 0
) = ๏ฟฝฬ๏ฟฝ, 852
1
๐พ๐ (๐พ๐ท๐ท0๐พ๐
+ ๐ 0)(ln
โ๐ 0 (๐๏ฟฝฬ๏ฟฝ +๐พ๐ท๐ท0๐พ๐
)
๐พ๐ท๐ท0๐พ๐
(๐๐ก โ ๐ 0)) = ๏ฟฝฬ๏ฟฝ, 853
ln1 +
๐พ๐๐๐ก๐พ๐ท๐ท0
1 โ๐๐ก๐ 0
= (๐พ๐ท๐ท0 + ๐พ๐๐ 0)๏ฟฝฬ๏ฟฝ, [3.4]
1 +๐พ๐๐๐ก๐พ๐ท๐ท0
1 โ๐๐ก๐ 0
= exp{(๐พ๐ท๐ท0 + ๐พ๐๐ 0)๏ฟฝฬ๏ฟฝ}, 854
1 +๐พ๐๐๐ก๐พ๐ท๐ท0
= (1 โ๐๐ก๐ 0) exp{(๐พ๐ท๐ท0 + ๐พ๐๐ 0)๏ฟฝฬ๏ฟฝ}, 855
๐พ๐๐๐ก๐พ๐ท๐ท0
+๐๐ก๐ 0exp{(๐พ๐ท๐ท0 + ๐พ๐๐ 0)๏ฟฝฬ๏ฟฝ} = exp{(๐พ๐ท๐ท0 + ๐พ๐๐ 0)๏ฟฝฬ๏ฟฝ} โ 1, 856
๐๐ก [๐พ๐๐พ๐ท๐ท0
+1
๐ 0exp{(๐พ๐ท๐ท0 + ๐พ๐๐ 0)๏ฟฝฬ๏ฟฝ}] = exp{(๐พ๐ท๐ท0 + ๐พ๐๐ 0)๏ฟฝฬ๏ฟฝ} โ 1, 857
๐๐ก =๐ 0(exp{(๐พ๐ท๐ท0 + ๐พ๐๐ 0)๏ฟฝฬ๏ฟฝ} โ 1)
๐พ๐๐ 0๐พ๐ท๐ท0
+ exp{(๐พ๐ท๐ท0 + ๐พ๐๐ 0)๏ฟฝฬ๏ฟฝ}. [3.5]
Thus, equation [3.5] is a general solution for the number of transconjugants at any time. 858
If we assume that ๐พ๐ = 0, then we can rewrite [3.4] as 859
ln1
1 โ๐๐ก๐ 0
= ๐พ๐ท๐ท0๏ฟฝฬ๏ฟฝ, 860
โ ln (1 โ๐๐ก๐ 0) = ๐พ๐ท๐ท0๏ฟฝฬ๏ฟฝ, 861
๐พ๐ท =โ ln (1 โ
๐๐ก๐ 0)
๐ท0๏ฟฝฬ๏ฟฝ. [3.6]
When ๐ 0 โซ ๐๏ฟฝฬ๏ฟฝ, a first-order Taylor approximation ensures โ ln (1 โ๐๏ฟฝฬ๏ฟฝ
๐ 0) โ
๐๏ฟฝฬ๏ฟฝ
๐ 0, and 862
therefore 863
๐พ๐ท โ
๐๏ฟฝฬ๏ฟฝ๐ 0๐ท0๏ฟฝฬ๏ฟฝ
, 864
๐พ๐ท โ๐๏ฟฝฬ๏ฟฝ
๐ท0๐ 0๏ฟฝฬ๏ฟฝ . 865
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Because ๐ท๐ก = ๐ท0 (see [3.1]) and ๐ ๐ก โ ๐ 0 (when ๐ 0 โซ ๐๏ฟฝฬ๏ฟฝ by [3.2]), we have 866
๐พ๐ท โ๐๏ฟฝฬ๏ฟฝ
๐ท๐ก๐ ๐ก ๏ฟฝฬ๏ฟฝ . 867
On the other hand, if we assume ๐พ๐ท = ๐พ๐ = ๐พ, then we can rewrite [3.4] as 868
ln1 +
๐๐ก๐ท0
1 โ๐๐ก๐ 0
= ๐พ(๐ท0 + ๐ 0)๏ฟฝฬ๏ฟฝ, 869
๐พ =1
๏ฟฝฬ๏ฟฝ(๐ท0 + ๐ 0)ln1 +
๐๐ก๐ท0
1 โ๐๐ก๐ 0
, 870
๐พ =1
๏ฟฝฬ๏ฟฝ(๐ท0 + ๐ 0){ln (1 +
๐๐ก๐ท0) โ ln (1 โ
๐๐ก๐ 0)}. [3.7]
When ๐ท0 โซ ๐๐ก and ๐ 0 โซ ๐๐ก, first-order Taylor approximations ensure 871
๐พ โ1
๏ฟฝฬ๏ฟฝ(๐ท0 + ๐ 0)(๐๐ก๐ท0+๐๐ก๐ 0), 872
๐พ โ๐๏ฟฝฬ๏ฟฝ
๏ฟฝฬ๏ฟฝ(๐ท0 + ๐ 0)(1
๐ท0+1
๐ 0), 873
๐พ โ๐๏ฟฝฬ๏ฟฝ
๏ฟฝฬ๏ฟฝ(๐ท0 + ๐ 0)(๐ 0+๐ท0๐ท0๐ 0
), 874
๐พ โ๐๐ก
๐ท0๐ 0๏ฟฝฬ๏ฟฝ . 875
Because ๐ท๐ก = ๐ท0 and ๐ ๐ก โ ๐ 0 (when ๐ 0 โซ ๐๐ก), we have 876
๐พ โ๐๐ก
๐ท๐ก๐ ๐ก ๏ฟฝฬ๏ฟฝ . 877
So we have general expressions for donor conjugation rate when ๐พ๐ = 0 (equation [3.6]) 878
or when ๐พ๐ท = ๐พ๐ (equation [3.7]). However, when ๐ท0 โซ ๐๐ก and ๐ 0 โซ ๐๐ก, for all ๐ก under 879
consideration, both of these measures are well approximated by equation [4]. Here we 880
extend the application of equation [4] even further. When ๐ 0 โซ ๐๐ก, then ๐ ๐ก โ ๐ 0. Let us 881
assume ๐ ๐ก = ๐ 0. We will also assume 882
๐พ๐ท๐ท0๐ 0 โซ ๐พ๐๐๐ก๐ 0, [3.8]
namely, the rate of formation of transconjugants by donors is much greater than the 883
formation by transconjugants. Of course, if ๐พ๐ = 0 or ๐พ๐ท = ๐พ๐ , then ๐ท0 โซ ๐๐ก ensures 884
assumption [3.8]. However, assumption [3.8] does not require ๐พ๐ = 0 or ๐พ๐ท = ๐พ๐ (indeed, 885
assumption [3.8] is satisfied when ๐พ๐ท โฅ ๐พ๐ and ๐ท0 โซ ๐๐ก, and ๐ 0 > 0). In general, this 886
inequality is satisfied when ๐0 = 0, ๐ท0 and ๐ 0 are is large, ๐พ๐ is not dramatically higher 887
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than ๐พ๐ท, and the period is small. Under assumption [3.8], the dynamics can be well 888
approximated by a simplified version of the differential equation [3.3] (where the 889
transconjugant conjugation term is gone). This is the differential equation originally solved 890
by Levin et. al., 891
๐๐๐ก๐๐ก
= ๐พ๐ท๐ท0๐ 0. 892
Because everything on the right-hand-side of the equation is a constant, the solution is 893
straightforward: 894
โซ ๐๐๐ก
๐ก
0
= โซ ๐พ๐ท๐ท0๐ 0๐๐ก๐ก
0
, 895
(๐๐ก)|0๐ก = (๐พ๐ท๐ท0๐ 0๐ก)|0
๐ก , 896
๐๐ก โ ๐0 = ๐พ๐ท๐ท0๐ 0๏ฟฝฬ๏ฟฝ. 897
Because ๐0 = 0, 898
๐๐ก = ๐พ๐ท๐ท0๐ 0๏ฟฝฬ๏ฟฝ, 899
๐พ๐ท =๐๐ก
๐ท0๐ 0๏ฟฝฬ๏ฟฝ . 900
Because ๐ท๐ก = ๐ท0 and ๐ ๐ก = ๐ 0 (by assumption), we have recovered equation [4]. 901
902
SI section 4 : SIM derivation 903
In this section, we walk through all of the mathematical steps in the Simonsen et. 904
al. derivation. We include this derivation as a quick reference to help the reader compare 905
the derivations of all four methods (TDR, SIM, ASM, and LDM). Simonsen et. al. modified 906
the model (equations [1]-[3]) by adding a dynamic variable for resource concentration (๐ถ). 907
The dynamics of this resource incorporate an additional parameter for the amount of 908
resource needed to produce a new cell (๐). Both the growth rate and conjugation rate are 909
taken to be functions of the resource concentration. 910
๐๐ท
๐๐ก= ๐(๐ถ)๐ท, [4.1]
๐๐
๐๐ก= ๐(๐ถ)๐ โ ๐พ(๐ถ)๐ (๐ท + ๐), [4.2]
๐๐
๐๐ก= ๐(๐ถ)๐ + ๐พ(๐ถ)๐ (๐ท + ๐), [4.3]
๐๐ถ
๐๐ก= โ๐(๐ถ)(๐ + ๐ท + ๐)๐. [4.4]
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The other variables are consistent with their use in equations [1]-[3]. In Simonsen et. al.16, 911
when resources are depleted, growth and conjugation stop. A Monod function introduces 912
batch culture dynamics (i.e., exponential and stationary phase) making growth and 913
conjugation both increase in a saturated manner with resource concentration, where the 914
resource concentration yielding ยฝ the maximal rate is given by the parameter ๐. 915
Importantly, conjugation and growth are assumed to have the same functional form: 916
๐(๐ถ) =๐๐๐๐ฅ๐ถ
๐ + ๐ถ, 917
๐พ(๐ถ) =๐พ๐๐๐ฅ๐ถ
๐ + ๐ถ, 918
where ๐ is the half saturation constant. 919
Here we derive the estimation of conjugation rate from Simonsen et. al.16 focusing 920
on the exponential phase of growth where resources are common (๐ถ is large) thus the 921
following are reasonable assumptions: 922
๐(๐ถ) = ๐๐๐๐ฅ 923
๐พ(๐ถ) = ๐พ๐๐๐ฅ 924
Under these assumptions, conjugation rate (๐พ) and growth rate (๐) are constant. As 925
resources are assumed to remain abundant such that there are no changes to the rates 926
of growth and conjugation, equation [4.4] can be dropped. Simonsen et. al.16 assume that 927
all strains have the same growth rate (๐๐ท = ๐๐ = ๐๐ = ๐๐๐๐ฅ) and conjugation rates from 928
donors and transconjugants are the same (๐พ๐ท = ๐พ๐ = ๐พ๐๐๐ฅ). Additionally, Simonsen et. 929
al.16 assume no segregative loss. Thus, with our assumption of exponential growth, we 930
recover equations [1]-[3]. We define ๐ = ๐ท + ๐ + ๐. Therefore, using equations [1]-[3] 931
๐๐
๐๐ก=๐๐ท
๐๐ก+๐๐
๐๐ก+๐๐
๐๐ก, 932
๐๐
๐๐ก= ๐๐๐๐ฅ๐ท + ๐๐๐๐ฅ๐ โ ๐พ๐๐๐ฅ๐ (๐ท + ๐) + ๐๐๐๐ฅ๐ + ๐พ๐๐๐ฅ๐ (๐ท + ๐), 933
๐๐
๐๐ก= ๐๐๐๐ฅ(๐ท + ๐ + ๐), 934
๐๐
๐๐ก= ๐๐๐๐ฅ๐. [4.5]
The general solutions for equation [1] and [4.5] are found by (1) separating variables, (2) 935
integrating, and (3) solving for ๐ท๐ก or ๐๐ก, respectively. 936
๐ท๐ก = ๐ท0๐๐๐๐๐ฅ๐ก , 937
๐๐ก = ๐0๐๐๐๐๐ฅ๐ก . 938
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40
Define the fraction of donors in the population to be ๐๐ก = ๐ท๐ก/๐๐ก. We note 939
๐๐ก =๐ท๐ก๐๐ก=๐ท0๐
๐๐๐๐ฅ๐ก
๐0๐๐๐๐๐ฅ๐ก=๐ท0๐0
= ๐0, 940
Thus, this fraction does not change over time (i.e., ๐๐ก is a constant). Lastly, we define a 941
fraction ๐๐ก = ๐๐ก/๐ ๐ก. Using equations [2] and [3], 942
๐๐
๐๐ก=๐(๐/๐ )
๐๐ก=
๐๐๐๐ก๐ โ
๐๐ ๐๐ก๐
๐ 2, 943
๐๐
๐๐ก={๐๐๐๐ฅ๐ + ๐พ๐๐๐ฅ๐ (๐ท + ๐)}๐ โ {๐๐๐๐ฅ๐ โ ๐พ๐๐๐ฅ๐ (๐ท + ๐)}๐
๐ 2, 944
๐๐
๐๐ก=๐๐๐๐ฅ๐๐ + ๐พ๐๐๐ฅ(๐ท + ๐)๐
2 โ๐๐๐๐ฅ๐๐ + ๐พ๐๐๐ฅ(๐ท + ๐)๐๐
๐ 2, 945
๐๐
๐๐ก=๐พ๐๐๐ฅ(๐ท + ๐)๐
2 + ๐พ๐๐๐ฅ(๐ท + ๐)๐๐
๐ 2, 946
๐๐
๐๐ก=๐พ๐๐๐ฅ(๐ท + ๐)๐ + ๐พ๐๐๐ฅ(๐ท + ๐)๐
๐ , 947
๐๐
๐๐ก=๐พ๐๐๐ฅ(๐ท๐ + ๐๐ + ๐ท๐ + ๐
2)
๐ , 948
๐๐
๐๐ก= ๐พ๐๐๐ฅ
๐ท๐ + ๐ ๐ + ๐2 + ๐ท๐
๐ , 949
๐๐
๐๐ก= ๐พ๐๐๐ฅ (
๐(๐ท + ๐ + ๐)
๐ + ๐ท). 950
Because ๐ = ๐ท + ๐ + ๐, 951
๐๐
๐๐ก= ๐พ๐๐๐ฅ (
๐๐
๐ + ๐ท), 952
๐๐
๐๐ก= ๐พ๐๐๐ฅ๐ (
๐
๐ +๐ท
๐). 953
Because ๐ = ๐ท/๐ and ๐ = ๐/๐ , 954
๐๐
๐๐ก= ๐พ๐๐๐ฅ๐(๐ + ๐), 955
(1
๐ + ๐)๐๐
๐๐ก= ๐พ๐๐๐ฅ๐. 956
Equation [4.5] ensures ๐ = (1
๐๐๐๐ฅ)๐๐
๐๐ก, which yields 957
(1
๐ + ๐)๐๐
๐๐ก= ๐พ๐๐๐ฅ (
1
๐๐๐๐ฅ)๐๐
๐๐ก, 958
๐๐๐๐ฅ (1
๐ + ๐)๐๐
๐๐ก= ๐พ๐๐๐ฅ
๐๐
๐๐ก, 959
๐พ๐๐๐ฅ(๐๐) = ๐๐๐๐ฅ (๐๐
๐ + ๐). 960
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41
To emphasize that ๐ is a constant, we write this as ๐0 and then integrate both sides, 961
๐พ๐๐๐ฅ โซ ๐๐๐ก
0
= ๐๐๐๐ฅโซ (๐๐
๐ + ๐0) ,
๐ก
0
962
Making the time dependence of the variables explicit via subscripts 963
๐พ๐๐๐ฅ๐๐ก|0๐ก = ๐๐๐๐ฅln(๐๐ก + ๐0)|0
๐ก , 964
๐พ๐๐๐ฅ(๐๐ก โ๐0) = ๐๐๐๐ฅ{ln(๐๐ก + ๐0) โ ln(๐0 + ๐0)}. 965
Because ๐0 = ๐๐ก 966
๐พ๐๐๐ฅ(๐๐ก โ๐0) = ๐๐๐๐ฅ{ln(๐๐ก + ๐๐ก) โ ln(๐0 + ๐๐ก)}, 967
๐พ๐๐๐ฅ(๐๐ก โ ๐0) = ๐๐๐๐ฅ ln (๐๐ก + ๐๐ก๐0 + ๐๐ก
). 968
Because ๐๐ก = ๐ท๐ก/๐๐ก and ๐๐ก = ๐๐ก/๐ ๏ฟฝฬ๏ฟฝ, 969
๐พ๐๐๐ฅ(๐๐ก โ ๐0) = ๐๐๐๐ฅ ln (
๐๐ก๐ ๐ก+๐ท๐ก๐๐ก
๐0๐ 0+๐ท๐ก๐๐ก
). 970
Because ๐0 = 0, 971
๐พ๐๐๐ฅ(๐๐ก โ ๐0) = ๐๐๐๐ฅ ln (
๐๐ก๐ ๐ก+๐ท๐ก๐๐ก
๐ท๐ก๐๐ก
), 972
๐พ๐๐๐ฅ(๐๐ก โ ๐0) = ๐๐๐๐ฅ ln (๐๐ก๐ ๏ฟฝฬ๏ฟฝ
๐๐ก๐ท๐ก+ 1) , 973
๐พ๐๐๐ฅ = ๐๐๐๐ฅ ln (1 +๐๐ก๐ ๐ก
๐๐ก๐ท๐ก)
1
๐๐ก โ ๐0. 974
Dropping the โmaxโ subscripts, we have recovered equation [5]. 975
976
SI section 5 : ASM derivation 977
In this section, we walk through all of the mathematical steps in the Huisman et. 978
al.11 derivation. We include this derivation as a quick reference to help the reader compare 979
the derivations of all four methods (TDR, SIM, ASM, and LDM). We start with a system of 980
ordinary differential equations described by Huisman et. al.11, the Extended Simonsen 981
Model (ESM): 982
๐๐ท
๐๐ก= ๐๐ท(๐ถ)๐ท, [5.1]
๐๐
๐๐ก= ๐๐ (๐ถ)๐ โ (๐พ๐ท(๐ถ)๐ท + ๐พ๐(๐ถ)๐)๐ , [5.2]
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๐๐
๐๐ก= ๐๐(๐ถ)๐ + (๐พ๐ท(๐ถ)๐ท + ๐พ๐(๐ถ)๐)๐ , [5.3]
๐๐ถ
๐๐ก= โ(๐๐ท(๐ถ)๐ท + ๐๐ (๐ถ)๐ + ๐๐(๐ถ)๐)๐, [5.4]
Where ๐ subscripts specify population specific growth rates, ๐พ subscripts specify donor 983
and transconjugant specific conjugation rates, and other variables are consistent with 984
equations [1]-[3] or equations [4.1]-[4.4]. As with the SIM, growth and conjugation rate 985
are both dependent on resource concentration. When resources are depleted, growth 986
and conjugation stop. 987
๐๐ด(๐ถ) =๐๐ด,๐๐๐ฅ๐ถ
๐ + ๐ถ, 988
๐พ๐ต(๐ถ) =๐พ๐ต,๐๐๐ฅ๐ถ
๐ + ๐ถ, 989
where ๐ด โ {๐ท, ๐ , ๐} and ๐ต โ {๐ท, ๐}. 990
Under the assumption that growth rate and conjugation rate are constant (where 991
๐๐ด(๐ถ) = ๐๐ด,๐๐๐ฅ and ๐พ๐ต(๐ถ) = ๐พ๐ต,๐๐๐ฅ), equation [5.4] can be dropped. Although the 992
maximal rates of growth and conjugation are assumed, in what follows, we drop the โmaxโ 993
subscript for notational convenience. This new set of simplified ordinary differential 994
equations is termed the Approximate Simonsen et. al. Method (โASMโ). 995
๐๐ท
๐๐ก= ๐๐ท๐ท, [5.5]
๐๐
๐๐ก= ๐๐ ๐ โ (๐พ๐ท๐ท + ๐พ๐๐)๐ , [5.6]
๐๐
๐๐ก= ๐๐๐ + (๐พ๐ท๐ท + ๐พ๐๐)๐ , [5.7]
Under initial culture conditions, the final simple set of equations [1.1] - [1.3] arise 996
by assuming the recipient population is dominated by growth ๐๐ ๐ โซ (๐พ๐ท๐ท + ๐พ๐๐)๐ and 997
the transconjugant population is dominated by growth and conjugation from donors ๐๐๐ +998
๐พ๐ท๐ท๐ โซ ๐พ๐๐๐ . The solutions to differential equations [1.1] and [1.2] are: 999
๐ท๐ก = ๐ท0๐๐๐ท๐ก [5.8]
๐ ๐ก = ๐ 0๐๐๐ ๐ก [5.9]
Here we will derive the solution for the differential equation [1.3] for transconjugant density 1000
๐. We know that ๐ท๐ก = ๐ท0๐๐๐ท๐ก and ๐ ๐ก = ๐ 0๐
๐๐ ๐ก, therefore 1001
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๐๐
๐๐ก= ๐๐๐ + ๐พ๐ท๐ท0๐ 0๐
(๐๐ท+๐๐ )๐ก [5.10]
We propose that the transconjugant density can be written as a product of time dependent 1002
functions: 1003
๐๐ก = ๐ข๐ก๐ฃ๐ก [5.11]
Weโll drop the ๐ก subscripts for notational ease. By the product rule, 1004
๐๐
๐๐ก= ๐ข
๐๐ฃ
๐๐ก+ ๐ฃ
๐๐ข
๐๐ก [5.12]
We can rewrite equation [5.10] by plugging in equations [5.11] and [5.12] as follows: 1005
๐ข๐๐ฃ
๐๐ก+ ๐ฃ
๐๐ข
๐๐ก= ๐๐๐ข๐ฃ + ๐พ๐ท๐ท0๐ 0๐
(๐๐ท+๐๐ )๐ก 1006
๐ข๐๐ฃ
๐๐ก+ ๐ฃ (
๐๐ข
๐๐กโ ๐๐๐ข) = ๐พ๐ท๐ท0๐ 0๐
(๐๐ท+๐๐ )๐ก [5.13]
We have some freedom to pick ๐ข๐ก as we please. So, letโs choose a function such that the 1007
second term of the left-hand side of equation [5.13] is zero: 1008
๐๐ข
๐๐กโ ๐๐๐ข = 0 1009
๐๐ข
๐๐ก= ๐๐๐ข 1010
The solution to this differential equation is: 1011
๐ข๐ก = ๐ข0๐๐๐๐ก [5.14]
where ๐ข0 is a constant. 1012
Thus, we can rewrite equation [5.13] as: 1013
๐ข0๐๐๐๐ก
๐๐ฃ
๐๐ก= ๐พ๐ท๐ท0๐ 0๐
(๐๐ท+๐๐ )๐ก 1014
๐ข0๐๐ฃ
๐๐ก=๐พ๐ท๐ท0๐ 0๐
(๐๐ท+๐๐ )๐ก
๐๐๐๐ก 1015
๐ข0๐๐ฃ
๐๐ก= ๐พ๐ท๐ท0๐ 0๐
(๐๐ท+๐๐ )๐ก๐โ๐๐๐ก 1016
๐ข0๐๐ฃ
๐๐ก= ๐พ๐ท๐ท0๐ 0๐
(๐๐ท+๐๐ โ๐๐)๐ก 1017
To solve this equation, we integrate 1018
๐ข0โซ๐๐ฃ = ๐พ๐ท๐ท0๐ 0โซ๐(๐๐ท+๐๐ โ๐๐)๐ก๐๐ก 1019
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๐ข0๐ฃ๐ก =๐พ๐ท๐ท0๐ 0
๐๐ท + ๐๐ โ ๐๐๐(๐๐ท+๐๐ โ๐๐)๐ก + ๐ [5.15]
where ๐ is a constant of the integration. To solve for ๐, plug in ๐ก = 0, 1020
๐ข0๐ฃ0 =๐พ๐ท๐ท0๐ 0
๐๐ท +๐๐ โ๐๐๐(๐๐ท+๐๐ โ๐๐)0 + ๐ 1021
๐ข0๐ฃ0 =๐พ๐ท๐ท0๐ 0
๐๐ท +๐๐ โ ๐๐+ ๐ 1022
๐ = ๐ข0๐ฃ0 โ๐พ๐ท๐ท0๐ 0
๐๐ท +๐๐ โ๐๐ 1023
Because ๐0 = ๐ข0๐ฃ0, 1024
๐ = ๐0 โ๐พ๐ท๐ท0๐ 0
๐๐ท +๐๐ โ ๐๐ 1025
So, we now can find the solution for ๐ฃ๐ก by plugging in our solution for ๐ into equation 1026
[5.15]: 1027
๐ข0๐ฃ๐ก =๐พ๐ท๐ท0๐ 0
๐๐ท +๐๐ โ๐๐๐(๐๐ท+๐๐ โ๐๐)๐ก + ๐0 โ
๐พ๐ท๐ท0๐ 0๐๐ท +๐๐ โ๐๐
1028
๐ข0๐ฃ๐ก = ๐0 +๐พ๐ท๐ท0๐ 0
๐๐ท +๐๐ โ ๐๐{๐(๐๐ท+๐๐ โ๐๐)๐ก โ 1} 1029
๐ฃ๐ก =1
๐ข0(๐0 +
๐พ๐ท๐ท0๐ 0๐๐ท +๐๐ โ๐๐
{๐(๐๐ท+๐๐ โ๐๐)๐ก โ 1}) [5.16]
Because ๐๐ก = ๐ข๐ก๐ฃ๐ก through substitution of equations [5.14] and [5.16], we have 1030
๐๐ก = [๐ข0๐๐๐๐ก] [
1
๐ข0(๐0 +
๐พ๐ท๐ท0๐ 0๐๐ท + ๐๐ โ ๐๐
{๐(๐๐ท+๐๐ โ๐๐)๐ก โ 1})] 1031
๐๐ก = ๐๐๐๐ก (๐0 +
๐พ๐ท๐ท0๐ 0๐๐ท +๐๐ โ ๐๐
{๐(๐๐ท+๐๐ โ๐๐)๐ก โ 1}) 1032
๐๐ก = ๐0๐๐๐๐ก +
๐พ๐ท๐ท0๐ 0๐๐ท + ๐๐ โ๐๐
{๐(๐๐ท+๐๐ )๐ก โ ๐๐๐๐ก} 1033
Because ๐0 = 0, we arrive at the result in Huisman et al.11 1034
๐๐ก =๐พ๐ท๐ท0๐ 0
๐๐ท + ๐๐ โ๐๐{๐(๐๐ท+๐๐ )๐ก โ ๐๐๐๐ก} 1035
Given that ๐ท๐ก = ๐ท0๐๐๐ท๐ก and ๐ ๐ก = ๐ 0๐
๐๐ ๐ก, this can be rewritten as 1036
๐๏ฟฝฬ๏ฟฝ =๐พ๐ท
๐๐ท + ๐๐ โ ๐๐{๐ท๐ก๐ ๐ก โ ๐ท0๐ 0๐
๐๐๐ก} 1037
Solving for ๐พ๐ท gives equation [6]. 1038
1039
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SI section 6 : Alternative laboratory forms for conjugation estimates 1040
To rearrange the SIM estimate, we start with equation [5]. If the entire period from 1041
๐ก = 0 to ๐ก = ๏ฟฝฬ๏ฟฝ involves exponential growth, then ๐๐ก = ๐0๐๐๐ก . In such a case, ๐ =1042
(1/๏ฟฝฬ๏ฟฝ) ln(๐๐ก/๐0). We arrive at the alternative laboratory form for SIM which is equation 1043
[2.1]. 1044
๐พ =1
๏ฟฝฬ๏ฟฝ[ln (1 +
๐๐ก๐ ๐ก
๐๐ก๐ท๐ก)]ln๐๐ก โ ln๐0๐๐ก โ๐0
. 1045
To rearrange the ASM estimate, we start with equation [6]. While the equations 1046
๐๐ท = (1/๏ฟฝฬ๏ฟฝ) ln(๐ท๐ก/๐ท0) and ๐๐ = (1/๏ฟฝฬ๏ฟฝ) ln(๐ ๐ก/๐ 0) again provide estimates on donor and 1047
recipient growth rates, we cannot express the transconjugant growth rate (๐๐) as a simple 1048
expression of time and initial/final densities of members of the mating culture. However, 1049
data from a transconjugant monoculture supply an estimate on this parameter. Thus, we 1050
arrive at the laboratory form for ASM which is equation [2.3]. 1051
๐พ๐ท = {1
๏ฟฝฬ๏ฟฝ(ln๐ท๐ก๐ ๐ก โ ln๐ท0๐ 0) โ ๐๐}
๐๐ก
(๐ท๐ก๐ ๐ก โ ๐ท0๐ 0๐๐๐๐ก). 1052
To rearrange the LDM estimate, we start with equation [13]. Since ๐ท๐ก = ๐ท0๐๐๐ท๐ก 1053
and ๐ ๐ก = ๐ 0๐๐๐ ๐ก, it is the case that ๐๐ท = (1/๏ฟฝฬ๏ฟฝ) ln(๐ท๐ก/๐ท0) and ๐๐ = (1/๏ฟฝฬ๏ฟฝ) ln(๐ ๐ก/๐ 0). So, 1054
we have 1055
๐พ๐ท =1
๏ฟฝฬ๏ฟฝln ๐0(๏ฟฝฬ๏ฟฝ)
ln (๐ท๐ก๐ท0) + ln (
๐ ๐ก๐ 0)
๐ท0๐ 0 โ ๐ท๐ก๐ ๐ก 1056
After rearrangement, we have 1057
๐พ๐ท =1
๏ฟฝฬ๏ฟฝ{โ ln ๐0(๏ฟฝฬ๏ฟฝ)}
ln(๐ท๐ก๐ ๐ก) โ ln(๐ท0๐ 0)
๐ท๐ก๐ ๐ก โ ๐ท0๐ 0. 1058
In the laboratory, we measure an estimate (๏ฟฝฬ๏ฟฝ0(๏ฟฝฬ๏ฟฝ)) of the probability that a population has 1059
no transconjugants (๐0(๏ฟฝฬ๏ฟฝ)) which is simply the fraction of the populations (i.e., parallel 1060
cultures) that have no transconjugants at the incubation time ๏ฟฝฬ๏ฟฝ (see SI section 9 for 1061
details). In addition, if a 1 ml volume is not used for each mating culture (assuming that 1062
all cell densities are measured in cfu
ml units), then we must add a correction factor ๐ which 1063
is the number of experimental volume units (evu) per ml (see SI section 10 for details and 1064
an example). Thus, we arrive at the laboratory form for the LDM which is equation [15]. 1065
๐พ๐ท = ๐ {1
๏ฟฝฬ๏ฟฝ[โ ln ๏ฟฝฬ๏ฟฝ0(๏ฟฝฬ๏ฟฝ)]
ln๐ท๐ก๐ ๐ก โ ln๐ท0๐ 0๐ท๐ก๐ ๐ก โ ๐ท0๐ 0
} 1066
1067
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SI section 7 : Derivation of ๐๐(๏ฟฝฬ๏ฟฝ) for the LDM estimate 1068
We define ๐๐(๐ก) to be the probability that there are ๐ transconjugants at time ๐ก, 1069
where ๐ is a non-negative integer (i.e., ๐๐(๐ก) = Pr{๐๐ก = ๐}). We focus here on the 1070
probability that transconjugants are absent (namely, where ๐ = 0) and derive an 1071
expression for ๐0(๐ก). By definition ๐0(๐ก + ฮ๐ก) = Pr{๐๐ก+ฮ๐ก = 0}. However, ๐๐ก+ฮ๐ก = 0 implies 1072
๐๐ก = 0, so we can write 1073
๐0(๐ก + ฮ๐ก) = Pr{(๐๐ก+ฮ๐ก = 0) โฉ (๐๐ก = 0)} = Pr{๐๐ก+ฮ๐ก = 0 | ๐๐ก = 0}Pr{๐๐ก = 0}, 1074
Given that ๐0(๐ก) = Pr{๐๐ก = 0}, we can use equation [11] to write the following time-1075
increment recursion for ๐0(๐ก): 1076
๐0(๐ก + ฮ๐ก) = (1 โ ๐พ๐ท๐ท๐ก๐ ๐กฮ๐ก)๐0(๐ก). 1077
This can be simplified as follows 1078
๐0(๐ก + ฮ๐ก) โ ๐0(๐ก)
ฮ๐ก= โ๐พ๐ท๐ท๐ก๐ ๐ก๐0(๐ก). 1079
Taking the limit as ฮ๐ก โ 0 gives 1080
limฮ๐กโ0
๐0(๐ก + ฮ๐ก) โ ๐0(๐ก)
ฮ๐ก=๐๐0(๐ก)
๐๐ก. 1081
Therefore, we have the following differential equation: 1082
๐๐0(๐ก)
๐๐ก= โ๐พ๐ท๐ท๐ก๐ ๐ก๐0(๐ก). 1083
We are assuming ๐ท๐ก = ๐ท0๐๐๐ท๐ก and ๐ ๐ก = ๐ 0๐
๐๐ ๐ก. We note that these assumptions are 1084
reasonable if the densities of donors and recipients are reasonably large and the rate of 1085
transconjugant generation per recipient cell (๐พ๐ท๐ท๐ก + ๐พ๐๐๐ก , or if ๐๐ก = 0, simply ๐พ๐ท๐ท๐ก) 1086
remains very small relative to per capita recipient growth rate. Under these assumptions, 1087
our differential equation becomes: 1088
๐๐0(๐ก)
๐๐ก= โ๐พ๐ท๐ท0๐ 0๐
(๐๐ท+๐๐ )๐ก๐0(๐ก). 1089
We solve this differential equation via separation of variables, integrating from 0 to our 1090
incubation time of interest ๏ฟฝฬ๏ฟฝ: 1091
โซ๐๐0(๐ก)
๐0(๐ก)
๐ก
0
= โซโ๐พ๐ท๐ท0๐ 0๐(๐๐ท+๐๐ )๐ก๐๐ก
๐ก
0
, 1092
ln ๐0(๐ก)|0๐ก =
โ๐พ๐ท๐ท0๐ 0๐๐ท +๐๐
๐(๐๐ท+๐๐ )๐ก|0
๐ก
, 1093
ln ๐0(๏ฟฝฬ๏ฟฝ) โ ln ๐0(0) =โ๐พ๐ท๐ท0๐ 0๐๐ท +๐๐
๐(๐๐ท+๐๐ )๐ก โโ๐พ๐ท๐ท0๐ 0๐๐ท +๐๐
. 1094
Given that ๐0(0) = 1, 1095
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ln ๐0(๏ฟฝฬ๏ฟฝ) =โ๐พ๐ท๐ท0๐ 0๐๐ท +๐๐
(๐(๐๐ท+๐๐ )๐ก โ 1), 1096
๐0(๏ฟฝฬ๏ฟฝ) = exp {โ๐พ๐ท๐ท0๐ 0๐๐ท +๐๐
(๐(๐๐ท+๐๐ )๐ก โ 1)}, 1097
which is equation [12]. 1098
1099
SI section 8 : Derivation of mutation rate from the Luria-Delbrรผck experiment 1100
Here we derive the classic estimate of mutation rate from Luria and Delbrรผck. We 1101
assume that there is a population of wild-type cells that grow according to the following 1102
equation: 1103
๐๐ก = ๐0๐๐๐๐ก , [8.1]
where ๐๐ก is the number of wild type cells at time ๐ก and ๐๐ is the per capita growth rate. 1104
The wild-type population dynamics are assumed to be deterministic (a reasonable 1105
assumption if the initial population size is reasonably large, i.e., ๐0 โซ 0). We are also 1106
ignoring the loss of wild-type cells to mutational transformation, but this omission is 1107
reasonable if the mutation rate is very small relative to per capita wild-type growth rate. 1108
Let the number of mutants be given by a random variable ๐๐ก. This variable takes 1109
on non-negative integer values. For a very small interval of time, ฮ๐ก, the current number 1110
of mutants will either increase by one or remain constant. The probabilities of each 1111
possibility are given as follows: 1112
Pr{๐๐ก+ฮ๐ก = ๐๐ก + 1} = ๐๐๐กฮ๐ก + ๐๐๐๐กฮ๐ก, [8.2]
Pr{๐๐ก+ฮ๐ก = ๐๐ก} = 1 โ (๐๐๐ก + ๐๐๐๐ก)ฮ๐ก. [8.3]
The two terms on the right-hand side of equation [8.2] give the ways that a mutant can 1113
be generated. The first term measures the probability that a wild-type cell undergoes a 1114
mutation (๐ is the mutation rate). The second term gives the probability that a mutant cell 1115
divides and produces two mutant cells (๐๐ is the mutant growth rate). Equation [8.3] is 1116
the probability that neither of these processes occur. 1117
Analogous to the procedure in SI section 7 (with ๐0(๐ก) = Pr {๐๐ก = 0}): 1118
๐0(๐ก + ฮ๐ก) = (1 โ ๐๐๐กฮ๐ก)๐0(๐ก). 1119
By rearranging, taking the limit as ฮ๐ก โ 0, and utilizing equation [8.1], we have 1120
๐๐0(๐ก)
๐๐ก= โ๐๐0๐
๐๐๐ก๐0(๐ก). 1121
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This differential equation can be solved in an analogous way as well 1122
โซ๐๐0(๐ก)
๐0(๐ก)
๐ก
0
= โซโ๐๐0๐๐๐๐ก๐๐ก
๐ก
0
, 1123
ln ๐0(๐ก)|0๐ก =
โ๐๐0๐๐
๐๐๐๐ก|0
๐ก
, 1124
ln ๐0(๏ฟฝฬ๏ฟฝ) โ ln ๐0(0) =โ๐๐0๐๐
๐๐๐๐ก โโ๐๐0๐๐
. 1125
Because we assume ๐0 = 0, we must have ๐0(0) = 1, and 1126
ln ๐0(๏ฟฝฬ๏ฟฝ) =โ๐๐0๐๐
(๐๐๐๐ก โ 1). 1127
Solving for the mutation rate ๐ 1128
๐ = โ ln ๐0(๏ฟฝฬ๏ฟฝ)๐๐
๐0(๐๐๐๐ก โ 1), 1129
which is equation [14]. This equation can also be expressed as: 1130
๐ = โ ln ๐0(๏ฟฝฬ๏ฟฝ)๐๐
๐๐ก โ๐0. 1131
In Luria and Delbrรผckโs original formulation, they measured time in units 1132
proportional to the bacterial division cycles. Specifically, a new time variable ๐ง is defined: 1133
๐ง =๐ก
๐ก๐ ln 2โ, 1134
where ๐ก๐ is the period required for population doubling during exponential growth. 1135
Because 1136
๐0๐๐๐๐ก = ๐02
๐ก/๐ก๐ , 1137
it is the case that 1138
๐๐ =1
๐ก๐ ln 2โ. 1139
Therefore, ๐ง = ๐๐๐ก and the equation ๐๐ก = ๐0๐๐๐๐ก can be expressed as 1140
๐๐ง = ๐0๐๐ง 1141
Performing the same analysis on this nondimensionalized equation gives the original 1142
formulation (their equations [4] and [5], where our ๐ is given by their โaโ and our ๏ฟฝฬ๏ฟฝ is given 1143
by their โtโ): 1144
๐ =โ ln ๐0(๏ฟฝฬ๏ฟฝ)
๐๐ง โ๐0. 1145
1146
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SI section 9 : The LDM MLE derivation 1147
We start by focusing on ๐0(๏ฟฝฬ๏ฟฝ), the probability that a population will have no 1148
transconjugants at time ๏ฟฝฬ๏ฟฝ (for notational convenience weโll drop the time variable). What 1149
we actually measure is the number of independent populations (or wells) that have no 1150
transconjugants (call this ๐ค) out of the total number of populations (or wells) tracked (call 1151
this W). What is our best estimate of ๐0, given our data ๐ค? That is, what value ๐0 1152
maximizes the likelihood function: 1153
โ(๐0) = Pr {๐ค|๐0} = (W๐ค) (๐0)
๐ค(1 โ ๐0)Wโ๐ค 1154
Because โln(๐ฅ) is a monotonic decreasing function, the value ๐0 that maximizes โ(๐0) 1155
will be the same value that minimizes: 1156
L(๐0) = โln{โ(๐0)} 1157
This can be rewritten as follows: 1158
L(๐0) = โln {(W๐ค) (๐0)
๐ค(1 โ ๐0)Wโ๐ค} 1159
L(๐0) = โln(W๐ค) โ ๐ค ln ๐0 โ (W โ๐ค) ln(1 โ ๐0) 1160
To find the maximum likelihood estimate, we find the critical points of L by setting its 1161
derivative to zero, or: 1162
๐L
๐๐0= โ
๐ค
๐0+Wโ๐ค
1 โ ๐0= 0 1163
The maximum likelihood estimate for ๐0 (which weโll denote ๏ฟฝฬ๏ฟฝ0) solves the above 1164
equation, or 1165
Wโ๐ค
1 โ ๏ฟฝฬ๏ฟฝ0=๐ค
๏ฟฝฬ๏ฟฝ0 1166
(Wโ ๐ค)๏ฟฝฬ๏ฟฝ0 = ๐ค(1 โ ๏ฟฝฬ๏ฟฝ0) 1167
W๏ฟฝฬ๏ฟฝ0 โ๐ค๏ฟฝฬ๏ฟฝ0 = ๐ค โ๐ค๏ฟฝฬ๏ฟฝ0 1168
W๏ฟฝฬ๏ฟฝ0 = ๐ค 1169
๏ฟฝฬ๏ฟฝ0 =๐ค
W 1170
This answer makes intuitive sense: the most likely estimate for ๐0 (the probability that a 1171
population has no transconjugants) is simply the fraction of the populations that have no 1172
transconjugants. 1173
To double check that this estimate actually corresponds to a minimum of L(๐0), 1174
consider the second derivative: 1175
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๐2L
๐๐02=๐ค
๐02+
Wโ๐ค
(1 โ ๐0)2 1176
Evaluating this derivative at the critical point yields: 1177
๐2L
๐๐02|๐0=๏ฟฝฬ๏ฟฝ0
=๐ค
(๐คW)
2 +Wโ๐ค
(Wโ๐คW )
2 1178
๐2L
๐๐02|๐0=๏ฟฝฬ๏ฟฝ0
= W2 (1
๐ค+
1
Wโ๐ค) 1179
As long as 0 < ๐ค < W, 1180
๐2L
๐๐02|๐0=๏ฟฝฬ๏ฟฝ0
> 0, 1181
which means that L is convex at ๏ฟฝฬ๏ฟฝ0 and therefore, this value is a local minimum. In turn, 1182
this means that ๏ฟฝฬ๏ฟฝ0 is a local maximum for โ. 1183
1184
SI section 10 : Experimental volume unit conversion using ๐ 1185
In this section, we walk through the addition of ๐ to the LDM estimate. This is 1186
important to maintain the typical units ml
h โ cfu used for reporting the conjugation rates. In the 1187
original differential equations [1]-[3], the units of the dynamic variables were cfu
ml. If we want 1188
to deal with numbers instead of density, the let us define a new volume unit termed the 1189
โevuโ standing for โexperimental volume unitโ where we will assume there are ๐ evuโs per 1190
ml. Focusing on the number of donors in the experiment (which we label ๏ฟฝฬ๏ฟฝ), we have the 1191
following conversion: 1192
๏ฟฝฬ๏ฟฝcfu
evu=(๐ท
cfuml)
(๐evuml), 1193
Equivalently, 1194
๏ฟฝฬ๏ฟฝ =๐
๐ 1195
๏ฟฝฬ๏ฟฝ =๐
๐ 1196
In our original differential equations, let us multiply both sides of all the differential 1197
equations by 1/๐, yielding: 1198
1
๐
๐๐ท
๐๐ก= ๐
1
๐๐ท, 1199
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1
๐
๐๐
๐๐ก= ๐
1
๐๐ โ ๐พ
1
๐๐ (๐ท + ๐), 1200
1
๐
๐๐
๐๐ก= ๐
1
๐๐ + ๐พ
1
๐๐ (๐ท + ๐). 1201
This can be reworked as 1202
๐๏ฟฝฬ๏ฟฝ
๐๐ก= ๐๏ฟฝฬ๏ฟฝ, 1203
๐๏ฟฝฬ๏ฟฝ
๐๐ก= ๐๏ฟฝฬ๏ฟฝ โ ๐พ๏ฟฝฬ๏ฟฝ(๐ท + ๐), 1204
๐๏ฟฝฬ๏ฟฝ
๐๐ก= ๐๏ฟฝฬ๏ฟฝ + ๐พ๏ฟฝฬ๏ฟฝ(๐ท + ๐). 1205
We have a mix of densities (cfu
ml) and numbers (
cfu
evu) here, so we do the following: 1206
๐๏ฟฝฬ๏ฟฝ
๐๐ก= ๐๏ฟฝฬ๏ฟฝ, 1207
๐๏ฟฝฬ๏ฟฝ
๐๐ก= ๐๏ฟฝฬ๏ฟฝ โ ๐๐พ๏ฟฝฬ๏ฟฝ (
๐ท + ๐
๐), 1208
๐๏ฟฝฬ๏ฟฝ
๐๐ก= ๐๏ฟฝฬ๏ฟฝ + ๐๐พ๏ฟฝฬ๏ฟฝ (
๐ท + ๐
๐). 1209
Again, making substitutions: 1210
๐๏ฟฝฬ๏ฟฝ
๐๐ก= ๐๏ฟฝฬ๏ฟฝ, 1211
๐๏ฟฝฬ๏ฟฝ
๐๐ก= ๐๏ฟฝฬ๏ฟฝ โ { ๐๐พ}๏ฟฝฬ๏ฟฝ(๏ฟฝฬ๏ฟฝ + ๏ฟฝฬ๏ฟฝ), 1212
๐๏ฟฝฬ๏ฟฝ
๐๐ก= ๐๏ฟฝฬ๏ฟฝ + {๐๐พ}๏ฟฝฬ๏ฟฝ(๏ฟฝฬ๏ฟฝ + ๏ฟฝฬ๏ฟฝ). 1213
If we let 1214
๏ฟฝฬ๏ฟฝ = ๐๐พ 1215
we note the units on ๏ฟฝฬ๏ฟฝ are evu
h โ cfu. Incorporating this relation, we have 1216
๐๏ฟฝฬ๏ฟฝ
๐๐ก= ๐๏ฟฝฬ๏ฟฝ, 1217
๐๏ฟฝฬ๏ฟฝ
๐๐ก= ๐๏ฟฝฬ๏ฟฝ โ ๏ฟฝฬ๏ฟฝ๏ฟฝฬ๏ฟฝ(๏ฟฝฬ๏ฟฝ + ๏ฟฝฬ๏ฟฝ), 1218
๐๏ฟฝฬ๏ฟฝ
๐๐ก= ๐๏ฟฝฬ๏ฟฝ + ๏ฟฝฬ๏ฟฝ๏ฟฝฬ๏ฟฝ(๏ฟฝฬ๏ฟฝ + ๏ฟฝฬ๏ฟฝ). 1219
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This set of equations tracks the number of cells (per evu). Thus, if the above equations 1220
were used, then the derivations of the LDM estimate could flow exactly like we show in 1221
the โTheoretical Frameworkโ section and SI section 7. That is, the following will be correct: 1222
๏ฟฝฬ๏ฟฝ = โ ln ๐0(๏ฟฝฬ๏ฟฝ) (๐๐ท +๐๐
๏ฟฝฬ๏ฟฝ0๏ฟฝฬ๏ฟฝ0(๐(๐๐ท+๐๐ )๐ก โ 1)
) 1223
Note, no correction is needed on ๐0(๏ฟฝฬ๏ฟฝ) as everything is in terms of numbers, which was 1224
how this quantity was derived. Because ๏ฟฝฬ๏ฟฝ =๐ท
๐ and ๏ฟฝฬ๏ฟฝ =
๐
๐, we can rewrite the above as 1225
๏ฟฝฬ๏ฟฝ = โ ln ๐0(๏ฟฝฬ๏ฟฝ) (๐๐ท +๐๐
๐ท0๐๐ 0๐(๐(๐๐ท+๐๐ )๐ก โ 1)
) 1226
Or: 1227
๏ฟฝฬ๏ฟฝ
๐= ๐ {โ ln๐0(๏ฟฝฬ๏ฟฝ) (
๐๐ท + ๐๐
๐ท0๐ 0(๐(๐๐ท+๐๐ )๐ก โ 1)
)} 1228
Because ๐พ =๐พ
๐, we have 1229
๐พ = ๐ {โ ln ๐0(๏ฟฝฬ๏ฟฝ) (๐๐ท +๐๐
๐ท0๐ 0(๐(๐๐ท+๐๐ )๐ก โ 1)
)} 1230
Note that if our evu was 1 ml, then ๐ = 1 and we could use our estimate exactly as written. 1231
Generally, we have to correct our original metric by multiplying by ๐. 1232
Now just as a check, letโs see what happens when we flip the unit of volume for a 1233
different metric, say SIM. Suppose that we use the cfu
evu units for cell densities, which 1234
means we are calculating 1235
๏ฟฝฬ๏ฟฝ = ๐ ln(1 +๏ฟฝฬ๏ฟฝ๐ก
๏ฟฝฬ๏ฟฝ๐ก
๏ฟฝฬ๏ฟฝ๐ก
๏ฟฝฬ๏ฟฝ๐ก)
1
(๏ฟฝฬ๏ฟฝ๐ก โ ๏ฟฝฬ๏ฟฝ0) 1236
where ๏ฟฝฬ๏ฟฝ has units evu
h โ cfu. Using our conversion, we can rewrite the right-hand side as 1237
follows: 1238
๏ฟฝฬ๏ฟฝ = ๐ ln (1 +๐๐ก/๐
๐ ๐ก/๐
๐๐ก/๐
๐ท๐ก/๐)
1
(๐๐ก โ๐0)/๐ 1239
Canceling and rearranging yields 1240
1241
๏ฟฝฬ๏ฟฝ
๐= ๐ ln (1 +
๐๏ฟฝฬ๏ฟฝ๐ ๐ก
๐๐ก๐ท๐ก)
1
(๐๐ก โ๐0) 1242
However, ๐พ =๐พ
๐, so we have 1243
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๐พ = ๐ ln (1 +๐๐ก๐ ๐ก
๐๐ก๐ท๐ก)
1
(๐๐ก โ ๐0) 1244
Therefore, if we flip to cell densities in the new unit cfu
ml, we will automatically get the 1245
conjugation rate in the new unit ml
h โ cfu. The same applies to the TDR and ASM estimates. 1246
Therefore, there are no correction factors needed for these metrics as volume units 1247
change. Note, in these cases, the numerical value of the transfer rate in the new units is 1248
not generally the same as the rate in the old units (๐พ โ ๏ฟฝฬ๏ฟฝ when ๐ โ 1). 1249
1250
SI section 11 : Stochastic simulation framework 1251
We developed a stochastic simulation framework using the Gillespie algorithm. We 1252
extended the base model (equations [1] - [3]) to include plasmid loss due to segregation 1253
at rate ๐. Thus, transconjugants are transformed into plasmid-free recipients due to 1254
plasmid loss via segregation at rate ๐๐. The donors are transformed into plasmid-free 1255
cells due to plasmid loss via segregation at rate ๐๐ท. Therefore, the extended model 1256
(equations [11.1] - [11.4]) tracks the change in density of a new population type, plasmid-1257
free former donors (๐น). In total, the extended model describes the change in density of 1258
four populations (๐ท, ๐ , ๐, and ๐น) due to various biological parameters: growth rates (๐๐ท,1259
๐๐ , ๐๐, and ๐๐น), conjugation rates (๐พ๐ท and ๐พ๐) and segregation rates (๐๐ท and ๐๐). For 1260
simplicity, here we assume donor-to-recipient conjugation rate is equal to donor-to-1261
plasmid-free-former-donor conjugation rate and represent both with ๐พ๐ท. In addition, we 1262
assume transconjugant-to-recipient conjugation rate is equal to transconjugant-to-1263
plasmid-free-donor conjugation rate and represent both with ๐พ๐. 1264
๐๐ท
๐๐ก= ๐๐ท๐ท + (๐พ๐ท๐ท + ๐พ๐๐)๐น โ ๐๐ท๐ท, [11.1]
๐๐
๐๐ก= ๐๐ ๐ โ (๐พ๐ท๐ท + ๐พ๐๐)๐ + ๐๐๐, [11.2]
๐๐
๐๐ก= ๐๐๐ + (๐พ๐ท๐ท + ๐พ๐๐)๐ โ ๐๐๐. [11.3]
๐๐น
๐๐ก= ๐๐น๐น โ (๐พ๐ท๐ท + ๐พ๐๐)๐น + ๐๐ท๐ท. [11.4]
Each simulation examined a biological process (i.e., growth, conjugation, or 1265
segregation) in isolation by manipulating one or two of the relevant parameters in order 1266
to determine the effect on accuracy and precision of various estimates. A simulation was 1267
initiated with a positive density of donors (๐ท0), a positive density of recipients (๐ 0), 1268
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biological parameters for each population type, and then run for a specified incubation 1269
time ๏ฟฝฬ๏ฟฝ. For each initial parameter setting, we simulated 10,000 parallel populations and 1270
calculated the conjugation rate using each method (TDR, SIM, ASM, and LDM). Each 1271
estimate has different requirements for calculating the conjugation rate. The LDM 1272
estimate needs multiple populations to calculate ๏ฟฝฬ๏ฟฝ0(๏ฟฝฬ๏ฟฝ). Thus, for a single estimate via 1273
LDM, we reserved 100 parallel populations, where 99 were used to calculate ๏ฟฝฬ๏ฟฝ0(๏ฟฝฬ๏ฟฝ) and 1274
the remaining population was used to calculate initial and final cell densities. In other 1275
words, the 10,000 parallel populations yielded 100 replicate LDM estimates for each 1276
parameter setting. For the other methods, we calculated a conjugation rate for each 1277
simulated population. In other words, the 10,000 simulated populations yielded 10,000 1278
replicate estimates for each initial parameter setting. 1279
The window of time for population growth (i.e., incubation time ๏ฟฝฬ๏ฟฝ) is different for the 1280
LDM vs. the other methods. A non-zero conjugation rate can be calculated for TDR, SIM, 1281
and ASM whenever there is a non-zero number of transconjugants. This contrasts with 1282
the LDM since conjugation rate is calculated when a proportion of populations have zero 1283
transconjugants. Given different incubation time requirements for each method, we use 1284
a deterministic simulation for each parameter setting to find the incubation time (๏ฟฝฬ๏ฟฝ) where 1285
we calculated conjugation rate. For the LDM, the incubation time was chosen to occur 1286
where on average a single transconjugant was present in the population. For the other 1287
methods, the incubation time occurred where on average ten transconjugant cells were 1288
present in the population. For Figure 2, we used a โbaselineโ set of parameters (๐๐ท =1289
๐๐ = ๐๐ = ๐๐น = 1, ๐พ๐ท = ๐พ๐ = 1 ร 10โ6, ๐๐ท = ๐๐ = 0) and initial densities (๐ท0 = ๐ 0 =1290
1 ร 102) unless otherwise indicated. Given that the Gillespie algorithm is computationally 1291
expensive, we chose low initial densities and an unusually high conjugation rate. See 1292
below and Figure 3 for more realistic rates. 1293
For Figure 3, the conjugation rate was estimated for each method at 30-minute 1294
intervals. The 30-minute time intervals occur over an earlier time interval for the LDM vs. 1295
the other methods for reasons described above. For the 30-minute time intervals, we 1296
applied estimate-specific filters. For the TDR, SIM, and ASM estimates, the range for 1297
these intervals was determined by the condition that at least 90 percent of the simulated 1298
populations contained transconjugants at the incubation time ๏ฟฝฬ๏ฟฝ. For the LDM estimate, 1299
the range for these intervals was determined by the condition that at least one parallel 1300
population had zero transconjugants. Therefore, estimates for the LDM are shown at 1301
earlier incubation times than those for the other methods (Figure 3). Given that Figure 3 1302
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55
explores only three parameter combinations (i.e., one for each panel), the computational 1303
expense of the Gillespie algorithm was of less concern than in Figure 2 where there are 1304
many more parameter settings per figure panel. Thus, for Figure 3, we set a conjugation 1305
rate that is often reported in the literature (๐พ๐ท = ๐พ๐ = 1 ร 10โ14) and more reasonable 1306
initial densities (๐ท0 = ๐ 0 = 1 ร 105 ) unless otherwise indicated. We note that the 1307
Gillespie algorithm is computationally expensive when the densities are very large. 1308
Therefore, due to the longer incubation times needed for TDR, SIM, and ASM, only 100 1309
populations of the 10,000 were incubated through the later time intervals (i.e., in Figure 1310
3, each time point has 100 conjugation rate estimates). 1311
SI Figure 1: The effect of different host growth rates on conjugation rate estimation. The parameter values were
๐๐ท = ๐๐ = ๐๐ = 1, ๐พ๐ท = ๐พ๐ = 1 ร 10โ6, and ๐๐ท = ๐๐ = 0 unless otherwise indicated. The dynamic variables were initialized with ๐ท0 = ๐ 0 = 1 ร 102 and ๐0 = ๐น0 = 0. All incubation times are short but are specific to each parameter setting. Conjugation rate was estimated using 10,000 populations and summarized using boxplots. The gray dashed line indicates the โtrueโ value for the donor conjugation rate. The box contains the 25th to 75th percentile. The vertical lines connected to the box contain 1.5 times the interquartile range above the 75th percentile and below the 25th percentile with the caveat that the whiskers were always constrained to the range of the data. The colored line in the box indicates the median. The solid black line indicates the mean. The boxplots in gray indicate the baseline parameter setting, and all colored plots represent deviation of one or two parameters from baseline. In this plot, we explored unequal host growth rates by tracking simulated populations over a range ๐๐ = ๐๐ โ {0.0625, 0.125, 0.24, 0.5, 1, 2, 4, 8}. This captures the situation in which the recipient (and therefore transconjugant) is a different strain or species from the donor and differs in growth rate.
SI section 12 : The LDM protocol 1312
Here we provide a general protocol for implementing the LDM in the laboratory. 1313
Some aspects need to be tailored to the specific experimental system, such as selection 1314
markers and incubation times. At the end of the protocol, we summarize some protocol 1315
additions that may be useful if the experimental system allows. For the main body of this 1316
section, we left out these โadditionsโ and only included the minimum supplies, strains, 1317
and selection conditions needed to estimate conjugation rate using the LDM. For a more 1318
general overview of the LDM see the โThe LDM Implementationโ section. 1319
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To start, identify the donor and recipient strains to be used (SI Table 5). The 1320
recipients must contain a selectable marker unique to the recipient host strain (๐๐ ๐ป). The 1321
donors must also contain a unique selectable marker, but it will be encoded on the donorโs 1322
conjugative plasmid (๐๐). The combination of the unique selectable markers allows for 1323
selection of the transconjugant cell type, a recipient strain containing the conjugative 1324
plasmid. 1325
SI Table 5: Overview of cell types in the conjugation assay.
Cell type Cell type description Selectable marker (๐)
๐ Plasmid-free recipients Recipient host marker (๐๐ ๐ป)
๐ท Plasmid donors Donor plasmid marker (๐๐)
๐ Transconjugants Recipient host marker (๐๐ ๐ป) Donor plasmid marker (๐๐)
Part I. Characterize cell types (estimated time = 4 days) 1326
Four key pieces of information necessary to proceed with Part II are determined here. 1327
(1) The approximate cell density for donors and recipients after overnight growth. (2) The 1328
minimum inhibitory concentrations (MIC) for the donors, recipients, and transconjugants. 1329
If the selectable markers involve antibiotic resistance, this will identify the antibiotic 1330
concentration (selection for ๐๐ ๐ป + ๐๐) to select for transconjugants and inhibit donors and 1331
recipients. (3) Verifying the conjugative plasmid will transfer horizontally between cells in 1332
liquid culture. Note that some plasmids transfer inefficiently in liquid culture. (4) The length 1333
of lag phase for donor and recipient cells. This information is needed because the donors 1334
and recipients are mixed in the exponential phase when determining โthe conjugation 1335
windowโ (Part II) and executing the LDM conjugation assay (Part III). 1336
1337
Day I.1 1338
Step I.1.1 Start three replicate overnight cultures from freezer stocks of the donors and 1339
recipients (SI Figure 2a). Include selection for ๐๐ in the donor cultures to 1340
ensure plasmid maintenance. 1341
Step I.1.2 Incubate cultures. 1342
Day I.2 1343
Step I.2.1 Transfer the donor and recipient cultures created in Step I.1.1 using a 1:100 1344
dilution (SI Figure 2b). Include selection for ๐๐ in the donor cultures to 1345
ensure plasmid maintenance. 1346
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Step I.2.2 To start three replicate mating cultures, dilute the donor and recipient 1347
cultures created in Step I.1.1 using a 1:100 dilution. Mix the diluted donors 1348
and recipients at a 1:1 ratio (SI Figure 2b). 1349
Note: These mating cultures allow for plasmid conjugation between donors 1350
and recipients to create transconjugants. This will (1) verify that 1351
transconjugants can form in the liquid condition, (2) generate the 1352
transconjugant strain that will verify the transconjugant-selecting conditions 1353
during the MIC assay. 1354
Note: If antibiotic selection is used to maintain the donor plasmid, wash 1355
donor cells (through three repeated rounds of centrifugation, removal of the 1356
supernatant, addition of an equal volume of saline, and vortexing) before 1357
this step. 1358
Step I.2.3 Incubate cultures. 1359
Day I.3 1360
Step I.3.1 Determine the overnight culture density for the donor and recipient cultures 1361
created in Step I.2.1, for example by creating a dilution series and 1362
separately plating each culture on non-selective medium. 1363
Note: Gathering density information for donors and recipients can help the 1364
experimenter tailor Part II most effectively (see Figure 4a) 1365
Step I.3.2 Determine the minimum inhibitory concentration (MIC) for all cell types 1366
(donors, recipients and transconjugants) using transconjugant selection 1367
(selection for ๐๐ ๐ป + ๐๐). 1368
Note: The MIC needs to be in the growth format planned for the remaining 1369
phases (container, temperature, growth medium type, etc.). Since drug 1370
interaction effects can occur, it is not appropriate to assume that the MIC 1371
values established for two strains with a single selectable marker can be 1372
combined to yield conditions that will inhibit a strain with both selectable 1373
markers (i.e., the MIC for the donor and the recipient are not necessarily 1374
concentrations that will inhibit the transconjugant). 1375
I.3.2.1 Create the MIC microtiter plate with a twofold gradient of 1376
transconjugant-selecting medium across the rows. See SI Figure 2c, 1377
for example. 1378
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I.3.2.2 Dilute the donor, recipient and mating cultures created on Day I.2 using 1379
a 1:100 dilution. Inoculate the MIC plate with the diluted donors, 1380
recipients, and mating cultures in the appropriate columns. 1381
Note: If transconjugants are at too of low of a percentage in the mating 1382
cultures, a transconjugant may not get into the MIC microtiter plate. 1383
Thus, the MIC for the โtransconjugantโ may appear to be equal to the 1384
maximum MIC of the donors and recipients. In this case, a mating 1385
culture should be plated on transconjugant-selecting agar plates to 1386
isolate a transconjugant and then determine the MIC. 1387
I.3.2.3 Incubate the MIC microtiter plate. 1388
Step I.3.3 Characterize strainโs growth using a spectrophotometer. 1389
I.3.3.1 Dilute the donor and recipient cultures created on Day I.2. using a 1390
1:100 dilution and inoculate three replicate wells for both the diluted 1391
donors and recipients. 1392
I.3.3.2 Incubate the microtiter plate in a spectrophotometer using the 1393
discontinuous kinetic reading setting which will measure the optical 1394
density (O.D.) of each growing culture at frequent intervals. 1395
Note: The spectrophotometerโs setting for the kinetic read should 1396
include shaking and temperature control which matches the growth 1397
conditions used for culturing the strains. 1398
Day I.4 1399
Step I.4.1 Calculate the overnight culture density by counting the cfu on the agar plates 1400
from Step I.3.1. 1401
Step I.4.2 Determine the transconjugant selection MIC for each cell type by analyzing 1402
the MIC microtiter plate created in Step I.3.2. 1403
Note: The transconjugant-selecting concentration to be chosen for Part II 1404
should be below the MIC for transconjugants and above the MIC for donors 1405
and recipients. All the tested concentrations in SI Figure 2d are usable for 1406
Part II. 1407
Step I.4.3 Analyze the growth kinetic readings from Step I.3.3 to determine the length 1408
of lag phase for the donor and recipient strains. 1409
Note: In Part II and Part III, the donors and recipient strains need to grow 1410
through lag phase and be mixed while in exponential phase. The length of 1411
lag phase may differ between donors and recipients. 1412
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1413
Part II. Determine the conjugation window (estimated time = 4 days) 1414
Day II.1 1415
Step II.1.1 Start overnight cultures by repeating Day I.1 and adding in three replicate 1416
wells for the transconjugant strain isolated in Part I. 1417
Day II.2 1418
Step II.2.1 Transfer cultures by repeating Step I.2.1 and Step I.2.3 from Day I.2 1419
including the transconjugant cultures. 1420
Day II.3 1421
Step II.3.1 Transfer the donor and recipient cultures created in Step II.2.1 using a 1422
1:100 dilution. Add selection for ๐๐ to the donor cultures to ensure plasmid 1423
maintenance. 1424
Step II.3.2 Incubate the donor and recipient cultures created in Step II.3.1 until both 1425
are in exponential phase. The incubation time for each strain was 1426
determined in Step I.4.3. 1427
Step II.3.3 Make a 10-fold dilution series of the donor and recipient cultures after the 1428
incubation in Step II.3.2 and plate appropriate dilutions to determine 1429
densities. Also, dilute the transconjugant cultures created in Step II.2.1. 1430
Note: Reserve a portion of these dilutions to create control wells in a future 1431
step. 1432
Step II.3.4 Mix a portion of the corresponding donor and recipient dilutions created in 1433
Step II.3.3. 1434
Note: If antibiotic selection is used to maintain the donor plasmid, wash 1435
donor cells as described previously. 1436
Step II.3.5 Add the diluted mixed cultures created in Step II.3.4 into the corresponding 1437
columns in a deep-well microtiter plate (Figure 4a gray columns). Create 1438
multiple replicate wells for each density-time treatment (see Figure 4a and 1439
4b for an example microtiter plate map). 1440
Note: The culture will be diluted 1:10 with transconjugant-selecting medium 1441
in a future step. Thus, add a culture volume which allows for a 1:10 dilution 1442
in the deep-well. This will vary depending on the volume capacity of the 1443
deep-well plate used. 1444
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Step II.3.6 Add the diluted donor, recipient, and transconjugant cultures created in 1445
Step II.3.3 into the separate corresponding columns (Figure 4a red, blue, 1446
and purple column). 1447
Note: Multiple dilutions were created for the donors, recipients, and 1448
transconjugants in Step II.3.3. All of the dilutions can be used for controls 1449
by using an additional deep-well plate. Figure 4a shows a simple scenario 1450
where the densest dilution is used for the control column. 1451
Step II.3.7 Add the diluted cultures for the donors and recipients created in Step II.3.3 1452
into additional separate columns (not shown in Figure 4) to be used as 1453
mating controls at each time point. 1454
Note: The mating controls are created at each time point by mixing donors 1455
and recipients and immediately adding transconjugant-selecting media. The 1456
mating controls confirm that plasmid conjugation is not occurring after the 1457
transconjugant-selecting medium is added. 1458
Step II.3.8 At each time point, 1459
II.3.8.1 Mix the donor and recipient cultures created for the mating controls in 1460
Step II.3.7 into an empty well in the corresponding row (i.e., the mating 1461
control column). 1462
II.3.8.2 Add transconjugant-selecting medium to all wells in the designated 1463
row(s) which dilutes the cultures ten-fold (see Figure 4b). 1464
Step II.3.9 Incubate the microtiter plate until turbidity is stable. 1465
Note: Since transconjugants are rare, it will take longer for cultures to reach 1466
stationary phase. Thus, even for fast-growing bacteria this incubation may 1467
be longer than one day. 1468
Note: The donor and recipient control wells should stay clear throughout 1469
incubation. If turbidity starts to occur, antibiotic may be degrading. 1470
Day II.4 1471
Step II.4.1 Calculate the culture density from Step II.3.1 by counting the cfu on the agar 1472
plates created in Step II.3.3. 1473
Step II.4.2 Record the turbidity results from the deep-well plate(s) (Figure 4c). 1474
Step II.4.3 Analyze the data from Step II.4.1 and Step II.4.2 then select a target 1475
incubation time (๏ฟฝฬ๏ฟฝโฒ) and target initial density for the donors (๐ท0โฒ ) and 1476
recipients (๐ 0โฒ ) to use in Part III (see Figure 4c for an example). 1477
1478
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Part III. Executing the actual LDM conjugation assay (estimated time = 4 days) 1479
Day III.1 1480
Step III.1.1 Start overnight cultures by repeating Step II.1.1. 1481
Day III.2 1482
Step III.2.1 Transfer cultures by repeating Step II.2.1. 1483
Day III.3 1484
Step III.3.1 Create donor and recipient cultures in exponential phase by repeating Step 1485
II.3.1 and Step II.3.2. 1486
Step III.3.2 Dilute the donor and recipient cultures from Step III.3.1 to the target initial 1487
densities (๐ท0โฒ , ๐ 0
โฒ ). Dilute the transconjugant cultures created in Step III.2.1. 1488
Note: Reserve a portion of the dilutions to create control wells. 1489
Step III.3.3 Mix a portion of the diluted donor and recipient cultures created in Step 1490
III.3.2. 1491
Step III.3.4 Add the diluted mixed culture created in Step III.3.3 into the corresponding 1492
wells in a deep-well microtiter plate (Figure 4d gray wells). 1493
Note: Add additional culture volume to the mating wells to be used to 1494
determine the initial and final densities. Thus, after the additional volume is 1495
removed for the initial density plating (๐ท0, ๐ 0) the remaining culture volume 1496
will match the other mating wells during the incubation time. 1497
Step III.3.5 Add the diluted donor, recipient, and transconjugant cultures created in 1498
Step III.3.2 into the corresponding separate wells (Figure 4d red, blue, and 1499
purple wells). 1500
Step III.3.6 Add the diluted donor and recipient cultures created in Step III.3.2 into 1501
additional wells (not shown in Figure 4) to be used as mating controls. 1502
Note: The mating controls are created after the incubation time (๏ฟฝฬ๏ฟฝ) by mixing 1503
donor and recipient cultures and immediately adding transconjugant-1504
selecting medium. The mating controls confirm that the donors and 1505
recipients cannot grow in the transconjugant-selecting medium and that 1506
plasmid conjugation is not occurring after transconjugant-selecting medium 1507
is added. One deep-well plate can be used for Part III by rearranging the 1508
microtiter plate map shown in Figure 4d to include the extra wells needed 1509
for the mating controls). 1510
Step III.3.7 Determine initial densities of donor and recipients (๐ท0, ๐ 0) by dilution-1511
plating from three designated mating wells. 1512
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Step III.3.8 Incubate the deep well plate for the chosen time (๏ฟฝฬ๏ฟฝ). 1513
Step III.3.9 Determine final densities of donor and recipients (๐ท๐ก, ๐ ๐ก) by dilution-plating 1514
from the three designated mating wells used in Step III.3.7. 1515
Step III.3.10 Add transconjugant-selecting medium to all wells using a 1:10 dilution 1516
(see Figure 4d). 1517
Step III.3.11 Incubate the microtiter plate. 1518
Note: Since transconjugants are rare, it will take longer for cultures to reach 1519
stationary phase. Thus, even for fast-growing bacteria this incubation may 1520
be longer than one day. 1521
1522
Day III.4 1523
Step III.4.1 Calculate the initial (๐ท0, ๐ 0) and final densities (๐ท๐ก, ๐ ๐ก) by counting the cfu 1524
on the agar plates from Step III.3.7 and Step III.3.9. 1525
Step III.4.2 Record the turbidity results from the deep-well plate (Figure 4d). 1526
Step III.4.3 Analyze the data from Step III.4.1 and Step III.4.2 to calculate the donor 1527
conjugation rate using the LDM estimate (equation [15]). 1528
1529
Protocol โadditionsโ to consider: 1530
- Add an additional selectable marker to either the donor host strain (๐๐ท๐ป) or the 1531
recipient host strain (๐๐ ๐ป2). Since the selectable marker in the recipient host strain 1532
(๐๐ ๐ป) is often chromosomally encoded through a single mutation, the donor may 1533
easily become resistant to the selecting agent (e.g., corresponding antibiotic) 1534
through a similar or different mutation in the donor chromosome. Thus, it becomes 1535
valuable to be able to distinguish transconjugant-containing populations from 1536
mutant-donor-containing populations through an additional marker. This type of 1537
mutant arose in our experiments. However, our donor strain had an additional 1538
selectable marker (๐๐ท๐ป), thus the final microtiter plate in the LDM assay was pin 1539
replicated into medium selecting for such donors (selection for ๐๐ท๐ป + ๐๐ + ๐๐ ๐ป). All 1540
mutant-donor-containing populations were identified and eliminated from the LDM 1541
calculation. If an additional selectable marker was available for the recipient host 1542
strain (๐๐ ๐ป2), the transconjugant-selecting medium could have all three selective 1543
agents (selection for ๐๐ + ๐๐ ๐ป + ๐๐ ๐ป2); here, a spurious result would require two 1544
mutations in the same donor lineage, which occurs with a very low probability. 1545
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63
- Adding additional densities and time intervals. In Figure 4, we have shown a 1546
version of Part II. This portion of the assay is highly customizable depending on 1547
the strains that are used. If the experimenter has an idea of the conjugation 1548
window, then Part II could be simplified. If the experimenter is working with strains 1549
that have large unknowns, then adding additional densities or time intervals is 1550
advisable. 1551
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SI Figure 2: Protocol overview for determining the minimum inhibitory concentration in Part I. (a) On day 1, the donors (red) and recipients (blue) are inoculated into three replicate wells in the standard 96-well microtiter plate. (b) On day 2 after incubation of the microtiter plate in a, the grown donor (red crosses) and recipient (blue crosses) replicate cultures are transferred (and diluted) into new wells. In addition, the grown donor a nd recipient are mixed to inoculate three replicate mating wells (gray) to allow for plasmid transfer between donors and recipients to create transconjugants. (c) On day 3 after incubation, the drug microtiter plate is created using transconjugant-selecting medium (selection for ๐๐ ๐ป + ๐๐) over a two-fold gradient. The bottom row of the microtiter plate contains non-selective medium thus serves as a positive control for bacterial growth. The corresponding grown culture from part (b) is diluted and inoculated into the designated columns in the drug microtiter plate. (d) After incubation of the microtiter plate in (c), MIC is determined for the donors, recipients, and for transconjugants (purple) that arose in the mating cultures. For recipients (blue cross) and donors (red cross), the MIC for transconjugant selecting medium is below the 1X drug concentration. For transconjugants (purple cross), the MIC for transconjugant-selecting medium is above the 64X drug concentration. In the bottom row with non-selective medium, all cell types can be present (mixed cross with blue, red and purple).
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65
1552
SI section 13 : Probability generating function and low-order moments 1553 1554 Keller and Antal (2015)38 studied a generalization of the process explored by Luria and 1555
Delbrรผck (1943)20. To start, Keller and Antal consider a wildtype population expanding 1556
from a single cell as follows: 1557
๐๐ก = ๐(๐ก) = ๐๐ฟ๐ก . 1558
Each wildtype cell generates a mutant cell at a rate ๐โฒ, which grows as a stochastic birth 1559
process with rate ๐ผ (Keller and Antal studied a supercritical birth-death process, but we 1560
will focus on the special case of a pure birth process). In this case, mutants form at a 1561
rate ๐โฒ๐(๐ก), such that the times of mutant arrival conform to a non-homogeneous 1562
Poisson process. We note that if we start with ๐0 cells, then mutants form at a rate 1563
๐0๐โฒ๐(๐ก). Alternatively, we can set ๐ = ๐0๐โฒ, such that mutants form at a rate ๐๐(๐ก), 1564
which is the case explored by Keller and Antal. 1565
1566
Keller and Antal derive the probability generating function for the total number of 1567
mutants at an arbitrary time: 1568
๐บ(๐ง, ๐ก) = exp {๐
๐ฟ(๐น (1, ๐ ; 1 + ๐ ;
๐ง
๐ง โ 1๐โ๐ผ๐ก) โ ๐๐ฟ๐ก๐น (1, ๐ ; 1 + ๐ ;
๐ง
๐ง โ 1))}, 1569
where ๐น is the Gaussian hypergeometric function and ๐ =๐ฟ
๐ผ. 1570
1571
Our process of interest (the formation and growth of transconjugants) can be seen as 1572
an instance of their formulation by making the following substitutions: 1573
๐ฟ = ๐๐ท + ๐๐ , 1574
๐ = ๐พ๐ท๐ท0๐ 0, 1575
๐ผ = ๐๐ . 1576
With these substitutions, the generating function becomes: 1577
๐บ(๐ง, ๐ก) = exp {๐พ๐ท๐ท0๐ 0๐๐ท + ๐๐
(๐น (1,๐๐ท + ๐๐ ๐๐
; 1 +๐๐ท +๐๐ ๐๐
;๐ง
๐ง โ 1๐โ๐๐๐ก)1578
โ ๐(๐๐ท+๐๐ )๐ก๐น (1,๐๐ท +๐๐ ๐๐
; 1 +๐๐ท +๐๐ ๐๐
;๐ง
๐ง โ 1))}. 1579
Because 1580
๐บ(๐ง, ๐ก) = โ๐๐(๐ก)๐ง๐
โ
๐=0
, 1581
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66
the probability of zero transconjugants now becomes straightforward (given 1582
๐น (1,๐๐ท+๐๐
๐๐; 1 +
๐๐ท+๐๐
๐๐; 0) = 1): 1583
๐0(๐ก) = ๐บ(0, ๐ก) = exp {โ๐พ๐ท๐ท0๐ 0๐๐ท + ๐๐
(๐(๐๐ท+๐๐ )๐ก โ 1)}, 1584
which agrees with the result from SI section 7. 1585
1586
Making the appropriate substitutions, we can also write the mean and variance (eqs. 8 1587
and 9 from Keller and Antal) for the transconjugants: 1588
1589
๐ธ[๐๐ก] = {
๐พ๐ท๐ท0๐ 0๐(๐๐ท+๐๐ )๐ก๐ก if ๐๐ท + ๐๐ = ๐๐
๐พ๐ท๐ท0๐ 0(๐(๐๐ท+๐๐ )๐ก โ ๐๐๐๐ก)
๐๐ท + ๐๐ โ๐๐ if ๐๐ท + ๐๐ โ ๐๐
1590
Var[๐๐ก] =
{
2๐พ๐ท๐ท0๐ 0(๐
2(๐๐ท+๐๐ )๐ก โ ๐(๐๐ท+๐๐ )๐ก)
๐๐ท + ๐๐ โ ๐พ๐ท๐ท0๐ 0๐
(๐๐ท+๐๐ )๐ก๐ก if ๐๐ท +๐๐ = ๐๐
2๐พ๐ท๐ท0๐ 0(๐(๐๐ท+๐๐ )๐ก/2 โ ๐(๐๐ท+๐๐ )๐ก)
๐๐ท + ๐๐ + 2๐พ๐ท๐ท0๐ 0๐
(๐๐ท+๐๐ )๐ก๐ก if ๐๐ท + ๐๐ = 2๐๐
๐พ๐ท๐ท0๐ 0 {2๐2๐๐๐ก(๐๐ โ (๐๐ท + ๐๐ )) โ ๐
๐๐๐ก(2๐๐ โ (๐๐ท + ๐๐ )) + (๐๐ท + ๐๐ )๐(๐๐ท+๐๐ )๐ก
(2๐๐ โ (๐๐ท + ๐๐ ))(๐๐ โ (๐๐ท +๐๐ ))} otherwise
1591
We provide derivations for these expressions in Box 1. In all cases, the variance grows 1592
relative to the mean over time (see Box 2 below for the derivations). 1593
1594
Box 1 : Derivations of the first and second central moments 1595
In this Box, we derive the equations for the moments using terminology close to what 1596
Keller and Antal used (focusing on the case where ๐ โ {1,2}). From above, ๐ = ๐๐ฟ๐ก and 1597
๐โ1/๐ = ๐๐ผ๐ก. Keller and Antal denote the number of mutants (analogous to our 1598
transconjugants) as ๐ต. Also, letting ๐ =๐ง
๐งโ1 and ๐ =
๐
๐ผ and dropping the time argument for 1599
notational convenience, the probability generating function (PGF) can be written as: 1600
1601
๐บ๐ต(๐ง) = exp {๐๐
๐ [1
๐๐น (1, ๐ ; 1 + ๐ ; ๐๐โ
1๐ ) โ ๐น(1, ๐ ; 1 + ๐ ; ๐)]}. 1602
The expected value for the number of mutants can be obtained from the PGF in the usual 1603
way: 1604
๐ธ[๐ต] = ๐บ๐ตโฒ (1โ) = lim
๐งโ1โ๐บ๐ต(๐ง)
๐๐
1 + ๐
1
(๐ง โ 1)2ฮฃ, 1605
where 1606
ฮฃ = โ๐โ1โ1๐ ๐น (2,1 + ๐ ; 2 + ๐ ; ๐๐โ
1๐ ) + ๐น(2,1 + ๐ ; 2 + ๐ ; ๐). 1607
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We use the following identity 1608
๐น(๐, ๐; ๐; ๐ง) = (1 โ ๐ง)๐โ๐โ๐๐น(๐ โ ๐, ๐ โ ๐; ๐; ๐ง), 1609
to obtain 1610
๐น(2,1 + ๐ ; 2 + ๐ ; ๐) = (1 โ ๐)โ1๐น(๐ , 1,2 + ๐ , ๐) 1611
= (1 โ ๐ง)๐น (๐ , 1,2 + ๐ ,๐ง
๐ง โ 1) 1612
Using a related identity 1613
๐น(๐, ๐; ๐; ๐ง) = (1 โ ๐ง)โ๐๐น (๐ โ ๐, ๐; ๐;๐ง
๐ง โ 1), 1614
we get: 1615
๐น(2,1 + ๐ ; 2 + ๐ ; ๐) = (1 โ ๐ง)๐น (๐ , 1,2 + ๐ ,๐ง
๐ง โ 1) 1616
= (1 โ ๐ง)2๐น(2,1,2 + ๐ , ๐ง) 1617
We will use a similar procedure for the first term in ฮฃ, obtaining: 1618
๐น(2,1 + ๐ ; 2 + ๐ ; ๐๐โ1/๐ ) = (1 โ ๐๐โ1/๐ )โ1๐น(๐ , 1,2 + ๐ ; ๐๐โ1/๐ ) 1619
= (1 โ ๐๐โ1/๐ )โ1๐น (๐ , 1,2 + ๐ ;๐ฅ
๐ฅ โ 1) 1620
= (1 โ ๐๐โ1/๐ )โ1(1 โ ๐ฅ)๐น(2,1,2 + ๐ ; ๐ฅ) 1621
=(๐ง โ 1)2
(๐ง๐โ1/๐ โ ๐ง + 1)2๐น(2,1; 2 + ๐ ; ๐ฅ) 1622
where ๐ฅ =๐ง๐โ1/๐
๐ง๐โ1/๐ โ๐ง+1. 1623
In sum, we have shown that the first derivative of the PGF can be expressed in the 1624
following way: 1625
๐บ๐ตโฒ (๐ง) = ๐บ๐ต(๐ง)
๐๐
1 + ๐ (โ
๐โ1โ1๐
(๐ง๐โ1๐ โ ๐ง + 1)2
๐น(2,1;2 + ๐ ; ๐ฅ) + ๐น(2,1,2 + ๐ , ๐ง)). [B1.1] 1626
Using the fact that lim๐งโ1โ
๐บ๐ต(๐ง) = 1 and the Gauss identity ๐น(๐, ๐; ๐; 1) =ฮ(๐)ฮ(๐โ๐โ๐)
ฮ(๐โ๐)ฮ(๐โ๐) (which 1627
holds if ๐ > ๐ + ๐) we obtain 1628
๐ธ[๐ต] = ๐บ๐ตโฒ (1โ) =
๐๐
๐ โ 1(โ๐โ1+1/๐ + 1). 1629
To calculate the variance for the number of mutants, we note that 1630
๐บ๐ตโณ(1โ) = ๐ธ[๐ต2] โ ๐ธ[๐ต], 1631
and thus 1632
Var[๐ต] = ๐บ๐ตโณ(1โ) + ๐ธ[๐ต] โ (๐ธ[๐ต])2 1633
= ๐บ๐ตโณ(1โ) + ๐บ๐ต
โฒ (1โ) โ (๐บ๐ตโฒ (1โ))2 1634
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From equation [B1.1] we can express the first derivative of the PGF as: 1635
๐บ๐ตโฒ (๐ง) = ๐บ๐ต(๐ง)
๐๐
1 + ๐ ฮฃ1. 1636
It follows that 1637
Var[๐ต] = ๐บ๐ตโณ(1โ)+ ๐บ๐ต
โฒ (1โ) โ (๐บ๐ตโฒ (1โ))2 1638
=๐๐
1 + ๐ lim๐งโ1โ
(ฮฃ1 + ฮฃ1โฒ ) 1639
= ๐ธ[๐ต] +๐๐
1 + ๐ lim๐งโ1โ
ฮฃ1โฒ 1640
To calculate the derivative of ฮฃ1, we note that 1641
ฮฃ1 = โ๐โ1โ
1๐
(๐ง๐โ1๐ โ ๐ง + 1)2
๐น(2,1; 2 + ๐ ; ๐ฅ) + ๐น(2,1,2 + ๐ , ๐ง). 1642
We can calculate the derivative of the second term: 1643
โ๐น(2,1,2 + ๐ , ๐ง)
โ๐ง=
2
2 + ๐ ๐น(3,2; 3 + ๐ ; ๐ง), 1644
and of the first term: 1645
โ[โ๐โ1โ1/๐
(๐ง๐โ1/๐ โ ๐ง + 1)2๐น(2,1; 2 + ๐ ; ๐ฅ)]
โ๐ง1646
= โ๐โ1โ1/๐ (โ2(๐โ1/๐ โ 1)
(๐โ1/๐ ๐ง โ ๐ง + 1)3๐น(2,1; 2 + ๐ ; ๐ฅ)1647
+2๐โ1/๐
(๐โ1/๐ ๐ง โ ๐ง + 1)41
2 + ๐ ๐น(3,2;3 + ๐ ; ๐ฅ)) 1648
Taken together, and using Gauss' equality, 1649
lim๐งโ1โ
ฮฃ1โฒ = 2(๐โ1+1/๐ โ ๐โ1+2/๐ )
๐ + 1
๐ โ 1+
2
2 + ๐
(๐ + 2)(๐ + 1)
(๐ โ 1)(๐ โ 2)(1 โ ๐โ1+2/๐ ) 1650
= 2๐โ1+2/๐ ๐ + 1
๐ โ 1(โ1 โ
1
๐ โ 2) + 2๐โ1+1/๐
๐ + 1
๐ โ 1+
2(๐ + 1)
(๐ โ 1)(๐ โ 2) 1651
= 2๐โ1+2/๐ ๐ + 1
2 โ ๐ + 2๐โ1+1/๐
๐ + 1
๐ โ 1+
2(๐ + 1)
(๐ โ 1)(๐ โ 2) 1652
Coming back to 1653
Var[๐ต] = ๐ธ[๐ต] +๐๐
1 + ๐ lim๐งโ1โ
ฮฃ1โฒ , 1654
we obtain 1655
Var[๐ต] = ๐๐ [2
2 โ ๐ ๐โ1+
2๐ +
1
๐ โ 1๐โ1+
1๐ +
๐
(๐ โ 1)(๐ โ 2)]. 1656
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Making the appropriate substitutions yields our expressions for the mean and variance 1657
for our transconjugants. 1658
1659
Box 2 : Behavior of the variance relative to the mean over time 1660
In this Box, we will show that the variance in transconjugant numbers grows relative to 1661
the mean over time. To do so, we will need to consider several cases. 1662
1663
Letโs consider the ratio of the variance to the mean, which we denote ๐๐ก =Var[๐๐ก]
๐ธ[๐๐ก]. We will 1664
start by focusing on the case where ๐๐ท + ๐๐ โ ๐๐ and ๐๐ท +๐๐ โ 2๐๐: 1665
๐๐ก =๐พ๐ท๐ท0๐ 0 {
2๐2๐๐๐ก(๐๐ โ (๐๐ท + ๐๐ )) โ ๐๐๐๐ก(2๐๐ โ (๐๐ท + ๐๐ )) + (๐๐ท +๐๐ )๐
(๐๐ท+๐๐ )๐ก
(2๐๐ โ (๐๐ท + ๐๐ ))(๐๐ โ (๐๐ท + ๐๐ ))}
๐พ๐ท๐ท0๐ 0(๐(๐๐ท+๐๐ )๐ก โ ๐๐๐๐ก)๐๐ท +๐๐ โ ๐๐
. 1666
This can be simplified as follows: 1667
๐๐ก =2๐2๐๐๐ก(๐๐ โ (๐๐ท + ๐๐ )) โ ๐
๐๐๐ก(2๐๐ โ (๐๐ท + ๐๐ )) + (๐๐ท +๐๐ )๐(๐๐ท+๐๐ )๐ก
(๐๐ท +๐๐ โ 2๐๐)(๐(๐๐ท+๐๐ )๐ก โ ๐๐๐๐ก). 1668
Letting ๐ =๐๐ท+๐๐
๐๐, 1669
๐๐ก =2๐2๐๐๐ก(1 โ ๐ ) โ ๐๐๐๐ก(2 โ ๐ ) + ๐ ๐(๐๐ท+๐๐ )๐ก
(๐ โ 2)(๐(๐๐ท+๐๐ )๐ก โ ๐๐๐๐ก), 1670
๐๐ก =๐ ๐(๐๐ท+๐๐ )๐ก โ 2(๐ โ 1)๐2๐๐๐ก + (๐ โ 2)๐๐๐๐ก
(๐ โ 2)(๐(๐๐ท+๐๐ )๐ก โ ๐๐๐๐ก). 1671
To determine the behavior of this ratio over time we take the derivative with respect to 1672
time: 1673
๐๐๐ก๐๐ก
1674
=
{(๐ (๐๐ท +๐๐ )๐
(๐๐ท+๐๐ )๐ก โ 4๐๐(๐ โ 1)๐2๐๐๐ก + (๐ โ 2)๐๐๐
๐๐๐ก)(๐ โ 2)(๐(๐๐ท+๐๐ )๐ก โ ๐๐๐๐ก)
โ(๐ ๐(๐๐ท+๐๐ )๐ก โ 2(๐ โ 1)๐2๐๐๐ก + (๐ โ 2)๐๐๐๐ก)(๐ โ 2) ((๐๐ท +๐๐ )๐(๐๐ท+๐๐ )๐ก โ ๐๐๐
๐๐๐ก)}
((๐ โ 2)(๐(๐๐ท+๐๐ )๐ก โ ๐๐๐๐ก))2 1675
๐๐๐ก๐๐ก
={
(๐ (๐๐ท + ๐๐ )๐
(๐๐ท+๐๐ )๐ก โ 4๐๐(๐ โ 1)๐2๐๐๐ก + (๐ โ 2)๐๐๐
๐๐๐ก)(๐ โ 2)(๐(๐๐ท+๐๐ )๐ก)
โ(๐ (๐๐ท +๐๐ )๐(๐๐ท+๐๐ )๐ก โ 4๐๐(๐ โ 1)๐
2๐๐๐ก + (๐ โ 2)๐๐๐๐๐๐ก)(๐ โ 2)(๐๐๐๐ก)
โ(๐ ๐(๐๐ท+๐๐ )๐ก โ 2(๐ โ 1)๐2๐๐๐ก + (๐ โ 2)๐๐๐๐ก)(๐ โ 2) ((๐๐ท + ๐๐ )๐(๐๐ท+๐๐ )๐ก)
+(๐ ๐(๐๐ท+๐๐ )๐ก โ 2(๐ โ 1)๐2๐๐๐ก + (๐ โ 2)๐๐๐๐ก)(๐ โ 2)(๐๐๐๐๐๐ก) }
((๐ โ 2)(๐(๐๐ท+๐๐ )๐ก โ ๐๐๐๐ก))2 , 1676
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๐๐๐ก๐๐ก
={
(โ4๐๐(๐ โ 1)๐2๐๐๐ก + (๐ โ 2)๐๐๐
๐๐๐ก)(๐ โ 2)(๐(๐๐ท+๐๐ )๐ก)
โ(๐ (๐๐ท + ๐๐ )๐(๐๐ท+๐๐ )๐ก โ 4๐๐(๐ โ 1)๐
2๐๐๐ก)(๐ โ 2)(๐๐๐๐ก)
โ(โ2(๐ โ 1)๐2๐๐๐ก + (๐ โ 2)๐๐๐๐ก)(๐ โ 2)((๐๐ท +๐๐ )๐(๐๐ท+๐๐ )๐ก)
+(๐ ๐(๐๐ท+๐๐ )๐ก โ 2(๐ โ 1)๐2๐๐๐ก)(๐ โ 2)(๐๐๐๐๐๐ก) }
((๐ โ 2)(๐(๐๐ท+๐๐ )๐ก โ ๐๐๐๐ก))2 , 1677
๐๐๐ก๐๐ก
={
(๐๐ท + ๐๐ โ 2๐๐)2(๐ โ 1)(๐ โ 2)๐
2๐๐๐ก(๐(๐๐ท+๐๐ )๐ก)
โ(๐๐ท + ๐๐ โ๐๐)(๐ โ 2)2๐๐๐๐ก(๐(๐๐ท+๐๐ )๐ก)
โ(๐๐ท +๐๐ โ๐๐)๐ (๐ โ 2)๐๐๐๐ก(๐(๐๐ท+๐๐ )๐ก)
+2๐๐(๐ โ 1)(๐ โ 2)๐3๐๐๐ก }
((๐ โ 2)(๐(๐๐ท+๐๐ )๐ก โ ๐๐๐๐ก))2 , 1678
๐๐๐ก๐๐ก
=
{
(๐๐ท + ๐๐ โ 2๐๐)2(๐ โ 1)(๐ โ 2)๐2๐๐๐ก(๐(๐๐ท+๐๐ )๐ก)
โ(๐๐ท + ๐๐ โ๐๐)2(๐ โ 1)(๐ โ 2)๐๐๐๐ก(๐(๐๐ท+๐๐ )๐ก)
+2๐๐(๐ โ 1)(๐ โ 2)๐3๐๐๐ก
}
((๐ โ 2)(๐(๐๐ท+๐๐ )๐ก โ ๐๐๐๐ก))2 , 1679
๐๐๐ก๐๐ก
=(๐ โ 1)(๐ โ 2){(๐ โ 2)๐(๐๐ท+๐๐ +2๐๐)๐ก โ (๐ โ 1)๐(๐๐ท+๐๐ +๐๐)๐ก + ๐3๐๐๐ก}
(12๐๐
) ((๐ โ 2)(๐(๐๐ท+๐๐ )๐ก โ ๐๐๐๐ก))2
, 1680
๐๐๐ก๐๐ก
=(๐ โ 1)(๐ โ 2)๐๐๐๐ก{(๐ โ 2)๐(๐๐ท+๐๐ +๐๐)๐ก โ (๐ โ 1)๐(๐๐ท+๐๐ )๐ก + ๐2๐๐๐ก}
(12๐๐
) ((๐ โ 2)(๐(๐๐ท+๐๐ )๐ก โ ๐๐๐๐ก))2
, 1681
๐๐๐ก๐๐ก
=(๐ โ 1)(๐ โ 2){(๐ โ 2)๐(๐ +1)๐๐๐ก โ (๐ โ 1)๐๐ ๐๐๐ก + ๐2๐๐๐ก}
(๐โ๐๐๐ก
2๐๐) ((๐ โ 2)(๐(๐๐ท+๐๐ )๐ก โ ๐๐๐๐ก))
2. 1682
The denominator is always positive, so the sign of this derivative is governed by the 1683
numerator 1684
sign [๐๐๐ก๐๐ก] = sign[(๐ โ 1)(๐ โ 2){(๐ โ 2)๐(๐ +1)๐๐๐ก โ (๐ โ 1)๐๐ ๐๐๐ก + ๐2๐๐๐ก}]. 1685
For ๐ > 2, which is when the growth rate of the transconjugant is less than the average 1686
of the donor and recipient growth rates, we have 1687
sign [๐๐๐ก๐๐ก] = sign[(๐ โ 2)๐(๐ +1)๐๐๐ก โ (๐ โ 1)๐๐ ๐๐๐ก + ๐2๐๐๐ก]. 1688
For any ๐ก > 0, for ๐ = 2 1689
sign [๐๐๐ก๐๐ก] = sign[0 โ ๐2๐๐๐ก + ๐2๐๐๐ก] = 0. 1690
Now letting ๐(๐ ) = (๐ โ 2)๐(๐ +1)๐๐๐ก โ (๐ โ 1)๐๐ ๐๐๐ก + ๐2๐๐๐ก, we have 1691
๐โฒ(๐ ) = ๐(๐ +1)๐๐๐ก + (๐ โ 2)๐๐๐ก๐(๐ +1)๐๐๐ก โ ๐๐ ๐๐๐ก โ (๐ โ 1)๐๐๐ก๐
๐ ๐๐๐ก , 1692
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๐โฒ(๐ ) = {1 + (๐ โ 2)๐๐๐ก}๐(๐ +1)๐๐๐ก โ {1 + (๐ โ 1)๐๐๐ก}๐
๐ ๐๐๐ก . 1693
Letting โ(๐ฅ) = {1 + (๐ โ 1 โ ๐ฅ)๐๐๐ก}๐(๐ +๐ฅ)๐๐๐ก, we see 1694
๐โฒ(๐ ) = โ(1) โ โ(0). 1695
But we see that 1696
โโฒ(๐ฅ) = โ๐๐๐ก๐(๐ +๐ฅ)๐๐๐ก + {1 + (๐ โ 1 โ ๐ฅ)๐๐๐ก}๐๐๐ก๐
(๐ +๐ฅ)๐๐๐ก , 1697
โโฒ(๐ฅ) = ๐๐๐ก๐(๐ +๐ฅ)๐๐๐ก(โ1 + {1 + (๐ โ 1 โ ๐ฅ)๐๐๐ก}), 1698
โโฒ(๐ฅ) = ๐๐๐ก๐(๐ +๐ฅ)๐๐๐ก(๐ โ 1 โ ๐ฅ)๐๐๐ก. 1699
If ๐ > 2 and ๐ก > 0, then โโฒ(๐ฅ) > 0 for 0 โค ๐ฅ โค 1. Since โ(๐ฅ) is a continuous function, we 1700
must have 1701
โ(1) > โ(0). 1702
This implies that for ๐ > 2 and ๐ก > 0, 1703
๐โฒ(๐ ) > 0. 1704
Since ๐(๐ ) is also continuous, and since ๐(2) = 0, we now know that for ๐ > 2 and ๐ก >1705
0, 1706
๐(๐ ) > 0. 1707
Therefore sign [๐๐๐ก
๐๐ก] = 1, or the derivative is positive for ๐ก > 0 and ๐ > 2, which means 1708
the variance grows relative to the mean over time. 1709
For 1 < ๐ < 2, 1710
sign [๐๐๐ก๐๐ก] = sign[โ(๐ โ 2)๐(๐ +1)๐๐๐ก + (๐ โ 1)๐๐ ๐๐๐ก โ ๐2๐๐๐ก]. 1711
Letting ๐ = โ(๐ โ 2), we know that 0 < ๐ < 1, and we can rewrite the above 1712
sign [๐๐๐ก๐๐ก] = sign[๐๐(3โ๐)๐๐๐ก + (1 โ ๐)๐(3โ(1+๐))๐๐๐ก โ ๐2๐๐๐ก]. 1713
Because ๐(3โ๐ฅ)๐๐๐ก is convex (๐ก > 0 โน๐2๐(3โ๐ฅ)๐๐๐ก
๐๐ฅ2> 0), Jensenโs inequality ensures 1714
๐๐(3โ๐)๐๐๐ก + (1 โ ๐)๐(3โ(1+๐))๐๐๐ก โ ๐2๐๐๐ก > ๐(3โ(๐2+(1โ๐)(1+๐)))๐๐๐ก โ ๐2๐๐๐ก , 1715
๐๐(3โ๐)๐๐๐ก + (1 โ ๐)๐(3โ(1+๐))๐๐๐ก โ ๐2๐๐๐ก > ๐(3โ(๐2+1โ๐2))๐๐๐ก โ ๐2๐๐๐ก , 1716
๐๐(3โ๐)๐๐๐ก + (1 โ ๐)๐(3โ(1+๐))๐๐๐ก โ ๐2๐๐๐ก > ๐2๐๐๐ก โ ๐2๐๐๐ก , 1717
๐๐(3โ๐)๐๐๐ก + (1 โ ๐)๐(3โ(1+๐))๐๐๐ก โ ๐2๐๐๐ก > 0. 1718
Therefore sign [๐๐๐ก
๐๐ก] = 1, or the derivative is positive for ๐ก > 0 and 1 < ๐ < 2, which 1719
means the variance grows relative to the mean over time. 1720
1721
Next, we turn to 0 < ๐ < 1, where 1722
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sign [๐๐๐ก๐๐ก] = sign[(๐ โ 2)๐(๐ +1)๐๐๐ก โ (๐ โ 1)๐๐ ๐๐๐ก + ๐2๐๐๐ก]. 1723
For any ๐ก > 0, for ๐ = 1 1724
sign [๐๐๐ก๐๐ก] = sign[โ๐2๐๐๐ก โ 0 + ๐2๐๐๐ก] = 0. 1725
Now letting ๐(๐ ) = (๐ โ 2)๐(๐ +1)๐๐๐ก โ (๐ โ 1)๐๐ ๐๐๐ก + ๐2๐๐๐ก, we have from above 1726
๐โฒ(๐ ) = {1 + (๐ โ 2)๐๐๐ก}๐(๐ +1)๐๐๐ก โ {1 + (๐ โ 1)๐๐๐ก}๐
๐ ๐๐๐ก . 1727
Letting โ(๐ฅ) = {1 + (๐ โ 1 โ ๐ฅ)๐๐๐ก}๐(๐ +๐ฅ)๐๐๐ก, we see 1728
๐โฒ(๐ ) = โ(1) โ โ(0). 1729
But we see from above that 1730
โโฒ(๐ฅ) = ๐๐๐ก๐(๐ +๐ฅ)๐๐๐ก(๐ โ 1 โ ๐ฅ)๐๐๐ก. 1731
If 0 < ๐ < 1 and ๐ก > 0, then โโฒ(๐ฅ) < 0 for 0 โค ๐ฅ โค 1. Since โ(๐ฅ) is a continuous function, 1732
we must have 1733
โ(1) < โ(0). 1734
This implies that for 0 < ๐ < 1 and ๐ก > 0, 1735
๐โฒ(๐ ) < 0. 1736
Since ๐(๐ ) is continuous, and since ๐(1) = 0, we now know that for 0 < ๐ < 1 and ๐ก >1737
0, ๐(๐ ) > 0. Therefore sign [๐๐๐ก
๐๐ก] = 1, or the derivative is positive for ๐ก > 0 and 0 < ๐ < 1, 1738
which means the variance grows relative to the mean over time. 1739
1740
Now consider the special case of ๐ = 2 1741
๐๐ก =
2๐พ๐ท๐ท0๐ 0(๐๐๐๐ก โ ๐2๐๐๐ก)2๐๐
+ 2๐พ๐ท๐ท0๐ 0๐2๐๐๐ก๐ก
๐พ๐ท๐ท0๐ 0(๐2๐๐๐ก โ ๐๐๐๐ก)
2๐๐ โ ๐๐
, 1742
๐๐ก =(๐๐๐๐ก โ ๐2๐๐๐ก) + 2๐๐๐ก๐
2๐๐๐ก
(๐2๐๐๐ก โ ๐๐๐๐ก), 1743
๐๐ก =(2๐๐๐ก โ 1)๐
2๐๐๐ก + ๐๐๐๐ก
(๐2๐๐๐ก โ ๐๐๐๐ก). 1744
Taking the derivative with respect to time yields 1745
๐๐๐ก๐๐ก
=
{(2๐๐๐
2๐๐๐ก + 2๐๐(2๐๐๐ก โ 1)๐2๐๐๐ก + ๐๐๐
๐๐๐ก)(๐2๐๐๐ก โ ๐๐๐๐ก)
โ ((2๐๐๐ก โ 1)๐2๐๐๐ก + ๐๐๐๐ก) (2๐๐๐
2๐๐๐ก โ ๐๐๐๐๐๐ก)
}
(๐2๐๐๐ก โ ๐๐๐๐ก)2, 1746
๐๐๐ก๐๐ก
= ๐๐(4๐๐๐ก๐
2๐๐๐ก + ๐๐๐๐ก)(๐2๐๐๐ก โ ๐๐๐๐ก) โ ((2๐๐๐ก โ 1)๐2๐๐๐ก + ๐๐๐๐ก) (2๐2๐๐๐ก โ ๐๐๐๐ก)
(๐2๐๐๐ก โ ๐๐๐๐ก)2, 1747
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๐๐๐ก๐๐ก
= ๐๐
{4๐๐๐ก๐
4๐๐๐ก + ๐3๐๐๐ก โ 4๐๐๐ก๐3๐๐๐ก โ ๐2๐๐๐ก
โ2(2๐๐๐ก โ 1)๐4๐๐๐ก โ 2๐3๐๐๐ก + (2๐๐๐ก โ 1)๐
3๐๐๐ก + ๐2๐๐๐ก}
(๐2๐๐๐ก โ ๐๐๐๐ก)2, 1748
๐๐๐ก๐๐ก
= ๐๐{4๐๐๐ก โ 2(2๐๐๐ก โ 1)}๐
4๐๐๐ก โ {4๐๐๐ก + 1 โ (2๐๐๐ก โ 1)}๐3๐๐๐ก
(๐2๐๐๐ก โ ๐๐๐๐ก)2, 1749
๐๐๐ก๐๐ก
= ๐๐2๐4๐๐๐ก โ {2๐๐๐ก + 2}๐
3๐๐๐ก
(๐2๐๐๐ก โ ๐๐๐๐ก)2, 1750
๐๐๐ก๐๐ก
= 2๐๐๐3๐๐๐ก
๐๐๐๐ก โ๐๐๐ก โ 1
(๐2๐๐๐ก โ ๐๐๐๐ก)2. 1751
Using the Taylor expansion 1752
๐๐๐ก๐๐ก
= 2๐๐๐3๐๐๐ก
(1 + ๐๐๐ก +(๐๐๐ก)
2
2! +(๐๐๐ก)
3
3! +(๐๐๐ก)
4
4! โฆ ) โ ๐๐๐ก โ 1
(๐2๐๐๐ก โ ๐๐๐๐ก)2, 1753
๐๐๐ก๐๐ก
= 2๐๐๐3๐๐๐ก
((๐๐๐ก)
2
2! +(๐๐๐ก)
3
3! +(๐๐๐ก)
4
4! โฆ )
(๐2๐๐๐ก โ ๐๐๐๐ก)2. 1754
Therefore, ๐๐๐ก
๐๐ก> 0 for ๐ก > 0 and ๐ = 2, which means the variance grows relative to the 1755
mean over time. 1756
1757
Finally, consider the special case of ๐ = 1 1758
๐๐ก =
2๐พ๐ท๐ท0๐ 0(๐2๐๐๐ก โ ๐๐๐๐ก)๐๐
โ ๐พ๐ท๐ท0๐ 0๐๐๐๐ก๐ก
๐พ๐ท๐ท0๐ 0๐๐๐๐ก๐ก , 1759
๐๐ก =2๐2๐๐๐ก โ (2 + ๐๐๐ก)๐
๐๐๐ก
๐๐๐ก๐๐๐๐ก , 1760
๐๐ก =2๐๐๐๐ก โ (2 + ๐๐๐ก)
๐๐๐ก . 1761
Using the Taylor expansion 1762
๐๐ก =2 (1 + ๐๐๐ก +
(๐๐๐ก)2
2! +(๐๐๐ก)
3
3! +(๐๐๐ก)
4
4! โฆ ) โ (2 + ๐๐๐ก)
๐๐๐ก , 1763
๐๐ก =๐๐๐ก + 2(
(๐๐๐ก)2
2! +(๐๐๐ก)
3
3! +(๐๐๐ก)
4
4! โฆ )
๐๐๐ก , 1764
๐๐ก = 1 + 2(๐๐๐ก
2!+(๐๐๐ก)
2
3!+(๐๐๐ก)
3
4!โฆ). 1765
Once again, the variance grows relative to the mean over time. 1766
1767
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Overall, we have shown that for all ๐ > 0 and ๐ก > 0, the ratio of the transconjugant 1768
variance to the mean number of transconjugants is amplified over time. 1769
1770
SI section 14 : Variance in Estimates 1771
Here we will focus on two estimates, ASM and LDM, and ask about their variance 1772
(enabling us to compare precision). We will focus exclusively on the contributions to this 1773
variance coming from the stochasticity in the transconjugant numbers (i.e., ignoring 1774
contributions coming from assessment of initial and final donor and recipient populations). 1775
1776
We start with the ASM estimate (here we express the estimate in terms of growth rate 1777
parameters): 1778
๐พ๐ท =๐๐ท +๐๐ โ ๐๐
๐ท0๐ 0(๐(๐๐ท+๐๐ )๐ก โ ๐๐๐๐ก)๐๐ก . 1779
Because we are only focusing on the contribution of the transconjugant variation, all 1780
parameters (initial densities and growth rates will be taken to be fixed). Thus, we can think 1781
about the ASM estimate as a random variable ฮASM, where 1782
ฮASM = ๐1๐๏ฟฝฬ๏ฟฝ , 1783
where the constant ๐1 is 1784
๐1 =๐๐ท +๐๐ โ ๐๐
๐ท0๐ 0(๐(๐๐ท+๐๐ )๐ก โ ๐๐๐๐ก)
. 1785
The variance of the ASM estimate is then 1786
var(ฮASM) = ๐12{var(๐๏ฟฝฬ๏ฟฝ)}. 1787
But we have a closed form expression for var(๐๐ก). If ๐๐ โ {๐๐ท + ๐๐ , (๐๐ท +๐๐ )/2}, we 1788
have: 1789
var(ฮASM) = (๐๐ท+๐๐ โ๐๐
๐ท0๐ 0(๐(๐๐ท+๐๐ )๏ฟฝฬ๏ฟฝโ๐๐๐๏ฟฝฬ๏ฟฝ)
)
2
๐พ๐ท๐ท0๐ 0 {
(๐๐ท+๐๐ )๐(๐๐ท+๐๐ )๏ฟฝฬ๏ฟฝ
(๐๐ท+๐๐ โ๐๐)(๐๐ท+๐๐ โ2๐๐)+
๐๐๐๏ฟฝฬ๏ฟฝ
๐๐ท+๐๐ โ๐๐โ
2๐2๐๐๏ฟฝฬ๏ฟฝ
๐๐ท+๐๐ โ2๐๐}, 1790
var(ฮASM) =๐พ๐ท
๐ท0๐ 0(
๐๐ท+๐๐ โ๐๐
๐(๐๐ท+๐๐ )๏ฟฝฬ๏ฟฝโ๐๐๐๏ฟฝฬ๏ฟฝ)2{(๐๐ท+๐๐ )๐
(๐๐ท+๐๐ )๏ฟฝฬ๏ฟฝ+(๐๐ท+๐๐ โ2๐๐)๐๐๐๏ฟฝฬ๏ฟฝโ(๐๐ท+๐๐ โ๐๐)2๐
2๐๐๏ฟฝฬ๏ฟฝ
(๐๐ท+๐๐ โ๐๐)(๐๐ท+๐๐ โ2๐๐)}, 1791
var(ฮASM) =๐พ๐ท(๐๐ท+๐๐ โ๐๐)
๐ท0๐ 0{(๐๐ท+๐๐ )๐
(๐๐ท+๐๐ )๏ฟฝฬ๏ฟฝ+(๐๐ท+๐๐ โ2๐๐)๐๐๐๏ฟฝฬ๏ฟฝโ(๐๐ท+๐๐ โ๐๐)2๐
2๐๐๏ฟฝฬ๏ฟฝ
(๐๐ท+๐๐ โ2๐๐)(๐(๐๐ท+๐๐ )๏ฟฝฬ๏ฟฝโ๐๐๐๏ฟฝฬ๏ฟฝ)
2 }. 1792
The formulas for ๐๐ = ๐๐ท + ๐๐ and 2๐๐ = ๐๐ท +๐๐ could also be derived via simple 1793
substitution (note, lim๐๐โ๐๐ท+๐๐
๐1 = 1/(๐ท0๐ 0๐ก๐(๐๐ท+๐๐ )๐ก). These formulas allow us to project 1794
variance in the ASM estimate over time due to transconjugant variation if all parameters 1795
are known. 1796
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1797
We now turn to the LDM estimate: 1798
๐พ๐ท = โ ln ๏ฟฝฬ๏ฟฝ0(๏ฟฝฬ๏ฟฝ) (๐๐ท +๐๐
๐ท0๐ 0(๐(๐๐ท+๐๐ )๐ก โ 1)
). 1799
What we actually measure is the number of populations (or wells) that have no 1800
transconjugants (call this ๐ค) out of the total number of populations (or wells) tracked (call 1801
this W). As we show in SI section 9, the maximum likelihood estimate of ๐0(๏ฟฝฬ๏ฟฝ) is 1802
๏ฟฝฬ๏ฟฝ0(๏ฟฝฬ๏ฟฝ) =๐ค
W. 1803
Of course, from experiment to experiment, there will be variance in the number of 1804
populations with no transconjugants. Let us consider a random variable ๐น, which 1805
represents the fraction of total populations that have no transconjugants. The expectation 1806
of ๐น is (we drop the time argument for notational convenience): 1807
E[๐น] = โ (W๐ค) (๐0)
๐ค(1 โ ๐0)Wโ๐ค (
๐ค
W)
W
๐ค=0
, 1808
E[๐น] =1
Wโ
W!
๐ค! (W โ๐ค)!(๐0)
๐ค(1 โ ๐0)Wโ๐ค
W
๐ค=1
๐ค, 1809
E[๐น] =1
Wโ
W!
(w โ 1)! (W โ ๐ค)!(๐0)
๐ค(1 โ ๐0)Wโ๐ค ,
W
๐ค=1
1810
E[๐น] = ๐0 โ(Wโ 1)!
(w โ 1)! ((Wโ 1) โ (w โ 1))!(๐0)
๐คโ1(1 โ ๐0)(Wโ1)โ(๐คโ1)
W
๐ค=1
. 1811
Letting ๐ = ๐ค โ 1, we have 1812
E[๐น] = ๐0 โ(Wโ 1)!
๐! ((W โ 1) โ ๐)!(๐0)
๐(1 โ ๐0)(Wโ1)โ๐
Wโ1
๐=0
. 1813
However, because โ (W โ 1๐
) (๐0)๐(1 โ ๐0)
(Wโ1)โ๐Wโ1๐=0 = 1, we have 1814
E[๐น] = ๐0, 1815
which makes sense. The second central moment of ๐น is 1816
var[๐น] = โ (W๐ค) (๐0)
๐ค(1 โ ๐0)Wโ๐ค (
๐ค
Wโ ๐0)
2W
๐ค=0
, 1817
var[๐น] = โ (W๐ค) (๐0)
๐ค(1 โ ๐0)Wโ๐ค ((
๐ค
W)2
โ 2๐0 (๐ค
W) + (๐0)
2) ,
W
๐ค=0
1818
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var[๐น] = โ (W๐ค) (๐0)
๐ค(1 โ ๐0)Wโ๐ค (
๐ค
W)2
W
๐ค=0
โ 2๐0 โ (W๐ค) (๐0)
๐ค(1 โ ๐0)Wโ๐ค (
๐ค
W)
W
๐ค=0
1819
+ (๐0)2 โ (
W๐ค) (๐0)
๐ค(1 โ ๐0)Wโ๐ค
W
๐ค=0
, 1820
var[๐น] = {(1
W2)โ (
W๐ค) (๐0)
๐ค(1 โ ๐0)Wโ๐ค๐ค2
W
๐ค=0
} โ 2๐0E[๐น] + (๐0)2, 1821
var[๐น] = {(1
W2)โ (
W๐ค) (๐0)
๐ค(1 โ ๐0)Wโ๐ค๐ค(๐ค โ 1 + 1)
W
๐ค=0
} โ 2(๐0)2 + (๐0)
2, 1822
var[๐น] = {(1
W2)โ
W!
๐ค! (Wโ ๐ค)!(๐0)
๐ค(1 โ ๐0)Wโ๐ค๐ค(๐ค โ 1)
W
๐ค=2
1823
+ (1
W2)โ (
W๐ค)(๐0)
๐ค(1 โ ๐0)Wโ๐ค๐ค
W
๐ค=0
} โ (๐0)2, 1824
var[๐น] = {(1
W2)โ
W!
(๐ค โ 2)! (W โ๐ค)!(๐0)
๐ค(1 โ ๐0)Wโ๐ค
W
๐ค=2
1825
+ (1
W)โ (
W๐ค)(๐0)
๐ค(1 โ ๐0)Wโ๐ค (
๐ค
W)
W
๐ค=0
} โ (๐0)2, 1826
var[๐น] = (W(Wโ 1)(๐0)
2
W2){โ
(Wโ 2)!
(๐ค โ 2)! ((Wโ 2) โ (๐ค โ 2))!(๐0)
๐คโ2(1
W
๐ค=2
1827
โ ๐0)(Wโ2)โ(๐คโ2)} +
๐0Wโ (๐0)
2. 1828
Letting ๐ = ๐ค โ 2, we have 1829
var[๐น] = ((Wโ 1)(๐0)
2
W){โ (
Wโ 2๐
) (๐0)๐(1 โ ๐0)
(Wโ2)โ๐
Wโ2
๐=0
}+๐0Wโ (๐0)
2, 1830
var[๐น] = ((Wโ 1)(๐0)
2
W)+
๐0Wโ (๐0)
2, 1831
var[๐น] =(W โ 1)(๐0)
2 + ๐0 โW(๐0)2
W, 1832
var[๐น] =โ(๐0)
2 + ๐0W
, 1833
var[๐น] =๐0(1 โ ๐0)
W. 1834
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1835
Because we are only focusing on the contribution of the transconjugant variation, all 1836
parameters (initial densities and growth rates will be taken to be fixed). Thus, we can think 1837
about the LDM estimate as a random variable ฮLDM, 1838
ฮLDM = ๐2 ln ๐น, 1839
where the constant ๐2 is 1840
๐2 = โ(๐๐ท +๐๐
๐ท0๐ 0(๐(๐๐ท+๐๐ )๐ก โ 1)
). 1841
1842
The variance of the LDM estimate is then 1843
var(ฮLDM) = ๐22{var(ln๐น)}. 1844
Here we use a first-order Taylor series approximation for ln ๐น centered at E[๐น] in the 1845
following way: 1846
ln ๐น โ ln(E[๐น]) +1
E[๐น](๐น โ E[๐น]), 1847
ln ๐น โ๐น
E[๐น]+ ln(E[๐น]) โ 1. 1848
1849
1850
So, because ln(E[๐น]) โ 1 is constant, we have 1851
var[ln ๐น] โ var {๐น
E[๐น]}, 1852
var[ln๐น] โ (1
E[๐น])2
var[๐น], 1853
var[ln ๐น] โ (1
๐0)2
(๐0(1 โ ๐0)
W), 1854
var[ln ๐น] โ1
W(1
๐0โ 1). 1855
This approximation will be accurate when the deviation between ๐น and E[๐น] is very small 1856
(i.e., |๐นโE[๐น]|
E[๐น]โช 1). As W (the number of replicate populations in the experiment) gets large, 1857
the distribution of ๐น will tighten around E[๐น], making the approximation more reasonable. 1858
1859
Now, we have the following expression for ๐0 (reintroducing the time argument): 1860
๐0(๏ฟฝฬ๏ฟฝ) = exp {โ๐พ๐ท๐ท0๐ 0๐๐ท +๐๐
(๐(๐๐ท+๐๐ )๐ก โ 1)} . 1861
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Therefore, 1862
var[ln ๐น๐ก] โ1
W(exp {
๐พ๐ท๐ท0๐ 0๐๐ท + ๐๐
(๐(๐๐ท+๐๐ )๐ก โ 1)} โ 1),1863
where we make the time dependence of ๐น clear. Returning to the variance for the LDM 1864
estimate, 1865
var(ฮLDM) โ1
W(
๐๐ท +๐๐
๐ท0๐ 0(๐(๐๐ท+๐๐ )๐ก โ 1)
)
2
(exp {๐พ๐ท๐ท0๐ 0๐๐ท +๐๐
(๐(๐๐ท+๐๐ )๐ก โ 1)} โ 1).1866
1867
If we define 1868
๐๐ก =๐๐ท +๐๐
๐ท0๐ 0(๐(๐๐ท+๐๐ )๐ก โ 1)
, 1869
then we have 1870
var(ฮLDM) โ๐๐ก2
W(๐
(๐พ๐ท๐๏ฟฝฬ๏ฟฝ)โ 1).1871
1872
In SI Figure 3, we explore the variances (approximate in the case of LDM) as a function 1873
of time. The LDM estimates (for two different numbers of populations) are more precise 1874
(lower variance) for much of the time range. However, if the time gets too high (๏ฟฝฬ๏ฟฝ โ 5 for 1875
the parameter set shown in SI Figure 3), then the LDM variance blows up (while the ASM 1876
variance remains very low). In a case like this, the LDM is predicted to be more precise 1877
when the time of the assay is sufficiently low. 1878
SI Figure 3 : The variance of the ASM (green) and LDM (pink) estimates. Different numbers of populations (W) are
used for the LDM estimates, as indicated. The parameters used here are ๐พ๐ท = 10โ12, ๐ท0 = ๐ 0 = 104, ๐๐ท = 1, and๐๐ = ๐๐ = 1.5.
ASMLDM
W=10
W=100
1 2 3 4 5 6time
1.ร10-21
2.ร10-21
3.ร10-21
4.ร10-21
5.ร10-21
6.ร10-21
variance
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79
To show mathematically that the LDM estimate is more precise for short times, we will 1879
approximate the expressions for the variance when ๏ฟฝฬ๏ฟฝ is very small. This enables us to 1880
use a first-order Maclaurin approximation ๐๐๐ก โ 1 + ๐๏ฟฝฬ๏ฟฝ. This allows the following 1881
approximation of the variance for the ASM estimate: 1882
var(ฮASM)1883
โ๐พ๐ท(๐๐ท +๐๐ โ ๐๐)
๐ท0๐ 0{(๐๐ท +๐๐ )(1 + (๐๐ท + ๐๐ )๐กฬ) + (๐๐ท + ๐๐ โ 2๐๐)(1 + ๐๐๐กฬ) โ 2(๐๐ท + ๐๐ โ ๐๐)(1 + 2๐๐๐กฬ)
(๐๐ท + ๐๐ โ 2๐๐)((๐๐ท + ๐๐ )๐กฬ โ ๐๐๐กฬ)2}. 1884
We can then simplify this expression: 1885
var(ฮASM) โ๐พ๐ท(๐๐ท + ๐๐ โ ๐๐)
๐ท0๐ 0{(๐๐ท + ๐๐ )(๐๐ท +๐๐ )๐กฬ + (๐๐ท + ๐๐ โ 2๐๐)๐๐๐กฬ โ 2(๐๐ท +๐๐ โ ๐๐)2๐๐๐กฬ
(๐๐ท + ๐๐ โ 2๐๐)(๐๐ท + ๐๐ โ ๐๐)2๐กฬ2}, 1886
var(ฮASM) โ๐พ๐ท
๐ท0๐ 0๐กฬ{(๐๐ท + ๐๐ )(๐๐ท + ๐๐ ) + (๐๐ท + ๐๐ โ 2๐๐)๐๐ โ 2(๐๐ท + ๐๐ โ๐๐)2๐๐
(๐๐ท + ๐๐ โ 2๐๐)(๐๐ท + ๐๐ โ ๐๐)}, 1887
var(ฮASM)1888
โ๐พ๐ท
๐ท0๐ 0๐กฬ{(๐๐ท +๐๐ โ 2๐๐)(๐๐ท + ๐๐ ) + 2๐๐(๐๐ท +๐๐ ) + (๐๐ท + ๐๐ โ 2๐๐)๐๐ โ 2(๐๐ท + ๐๐ โ ๐๐)2๐๐
(๐๐ท + ๐๐ โ 2๐๐)(๐๐ท + ๐๐ โ๐๐)}, 1889
var(ฮASM)1890
โ๐พ๐ท
๐ท0๐ 0๐กฬ{(๐๐ท + ๐๐ โ 2๐๐)(๐๐ท + ๐๐ ) + 2๐๐๐๐ท + 2๐๐๐๐ + ๐๐๐๐ท +๐๐๐๐ โ 2๐๐
2 โ 4๐๐๐๐ท โ 4๐๐๐๐ + 4๐๐2
(๐๐ท +๐๐ โ 2๐๐)(๐๐ท + ๐๐ โ ๐๐)}, 1891
var(ฮASM) โ๐พ๐ท
๐ท0๐ 0๏ฟฝฬ๏ฟฝ{(๐๐ท + ๐๐ โ 2๐๐)(๐๐ท +๐๐ ) โ ๐๐๐๐ท โ ๐๐๐๐ + 2๐๐
2
(๐๐ท + ๐๐ โ 2๐๐)(๐๐ท + ๐๐ โ๐๐)}, 1892
1893
var(ฮASM) โ๐พ๐ท
๐ท0๐ 0๏ฟฝฬ๏ฟฝ{(๐๐ท +๐๐ โ 2๐๐)(๐๐ท + ๐๐ ) โ (๐๐ท + ๐๐ โ 2๐๐)๐๐
(๐๐ท + ๐๐ โ 2๐๐)(๐๐ท + ๐๐ โ๐๐)}, 1894
var(ฮASM) โ๐พ๐ท
๐ท0๐ 0๏ฟฝฬ๏ฟฝ{(๐๐ท + ๐๐ โ 2๐๐)(๐๐ท + ๐๐ โ๐๐)
(๐๐ท + ๐๐ โ 2๐๐)(๐๐ท + ๐๐ โ๐๐)}, 1895
var(ฮASM) โ๐พ๐ท
๐ท0๐ 0๏ฟฝฬ๏ฟฝ . 1896
We now turn to the variance for the LDM (which we note is already an approximation). 1897
For very small ๏ฟฝฬ๏ฟฝ, 1898
๐๐ก โ๐๐ท +๐๐
๐ท0๐ 0(1 + (๐๐ท + ๐๐ )๏ฟฝฬ๏ฟฝ โ 1), 1899
๐๐ก โ1
๐ท0๐ 0๏ฟฝฬ๏ฟฝ . 1900
Thus, we have 1901
var(ฮLDM) โ(
1๐ท0๐ 0๏ฟฝฬ๏ฟฝ
)2
W(๐(๐พ๐ท๐ท0๐ 0๐ก) โ 1). 1902
Using the Maclaurin approximation again yields 1903
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var(ฮLDM) โ(
1๐ท0๐ 0๏ฟฝฬ๏ฟฝ
)2
W(1 + ๐พ๐ท๐ท0๐ 0๏ฟฝฬ๏ฟฝ โ 1), 1904
var(ฮLDM) โ๐พ๐ท
W๐ท0๐ 0๏ฟฝฬ๏ฟฝ . 1905
Our LDM assay requires W > 1. Therefore when ๏ฟฝฬ๏ฟฝ is very small 1906
var(ฮLDM) < var(ฮASM). 1907
Again, we note the caveat that our estimate for the variance of the LDM estimate was 1908
already an approximation. 1909
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