Volume 4 Number 6 June 1977 Nucleic Acids Research

A program for least squares analysis of reassociation and hybridization data

*

W.R. Pearson, E.H. Davidson and R. J. Britten

Division of Biology, California Institute of Technology, Pasadena, CA 91125, USA

Received 21 December 1976

ABSTRACTA computer program is described for the rapid calculation of least

squares solutions for data fitted to different functions normally used inreassociation and hybridization kinetic measurements. The equations for thefraction not reacted as a function of Cot follow: First order, exp(-kCot);second order, (1+kCot)Y1; variable order, (1+kCot)-n; apRoximate fractionof DNA sequence remaining single stranded, (1+kCot)F ; and a functiondescribing the pairing of tracer when the rate constant for the tracer (k) isdistinct frym the driver rate constant (kd):exptkL1-(1+kdCot) -nJ/Lkd(1-n)J}. Several components may be used for most ofthese functional forms. The standard deviations of the individual parametersat the solutions are calculated.

INTRODUCTIONThe quantitative examination of reassociation and hybridization kinetic

measurements has become increasingly important as the sophistication of the

measurements has grown. In this paper we describe a general computer program

which can conveniently apply the variety of functions now used to interpret

kinetic measurements. We have chosen to use a non-linear least squares method

so that the solutions give equal weight to all of the individual measurements

and no preliminary assumptions need be made about the initial or terminal

values of the reaction.

Least squares computer programs have been applied to the problem of

resolving repetitive and single copy kinetic components in DNA renaturation

experiments carried out on many organisms (e.g. 1,2,3). They are also used

to determine rate constants in RNA excess hybridization reactions (e.g. 4,5).The use of cDNA orobes to determine the complexity of RNA Populations by

kinetic rather than saturation measurements (e.g. 6,7) also relies heavily on

the resolution of abundance classes by accurate determination of their rate

constants.

The five functions listed in Table 1 are used for the examination of the

following kinds of measurements. The second order equation (FINGER) describes

© Information Retrieval Limited 1 Falconberg Court London Wl V 5FG England 1727

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TABLE 1 Functional forms used by the program

Number Name Form Use

1 FINGER fi(l + kiCot)f1 second order reaction: DNArenaturation measured byhydroxyapatite chromatography

2 WHATOR f(1 + kCot)-n variable form reaction: to determinevalues of n when the apparent order ofthe reaction is unknown

3 NUFORM fiexDtkiL1-(kdCot)1nJ/Lk (1-n)J1 describes rate oftgacer reaction when tracer rateconstant k differs from driver rateconstant k

-2d4 EXCESS fiexp(-kiCot) first order function: for RNA excess

experiments

5 WHTCMP f.(1 + k.Cot) 0 44 modified second order function: S-1nuclease assay of hybridization

accurately the form of DNA reassociation kinetics assayed by hydroxyapatite

chromatography (8,9) though for fairly comDlex reasons (10,11). The

pseudo-first- order equation (EXCESS) applies when the nucleic acid drivingthe reaction remains unpaired as in RNA driven reactions. The third function

(WHATOR) has a form which can be varied by changing the value of the exponent,and is useful when the apparent order of the reaction is not known or there is

a need to test the heterogeneity of the reacting comnonents. When the

exponent n=1, it reduces to second order form. When high quality single

component hydroxyapatite measurements are analyzed, the least squares solution

yields n=1.0 (11). As n becomes large this equation approaches first orderand values of n as high as 10 yield a form which is indistinguishable from

first order. The next functional form (WHTCMP) is applicable to DNA

reassociation when the reaction is assayed by S-1 nuclease (11,12) and

expresses the fraction of the length of the DNA sequence present which remains

single stranded. The last equation (NUFORM) applies when the rate of

reassociation of tracer with the driver and the rate of driver renaturation

itself differ. The amount of driver available is assumed to follow the

equation (1+kCot)-n. A value of n=.44 is usually used and gives a good

approximation to the actual capacity of the remaining single stranded regions

to reassociate with tracer molecules (13).

The non-linear least squares program described in this paper is in use on

a PDP-10 timesharing computer. The program provides for interactive input but

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has been designed for ease of conversion to a batch processing system. A

typical run of the program is presented in Appendix I, with user responses

underlined. In a Batch processing environment the underlined inputs would be

submitted on cards.

The algorithm used in the program was developed by Marquardt (10) fornon-linear least squares problems. The implementation of the algorithm

provides two additional facilities: 1) the ability to hold any number or

combination of parameters constant in order to find the best solution with the

remaining variable parameters; and 2) the ability to substitute any function

for the NUFORM function by replacing a subroutine. Parameter fixing can

provide important insight into the uniqueness of the solution parameter set.

The function substitution option allows more complex functions to be fit to

data as more comolex phenomena - e.g. rate retardation for single strands on

duplexed molecules (10) - are studied.

ANALYSIS OF HYBRIDIZATION DATAThe program strategy

This non-linear least squares program is designed to converge on a

solution yielding parameters which minimize the least squares deviation of the

function from a set of data. There are two main concerns: whether the final

solution is biased by the input parameter estimates, and whether further

iterations will improve the solution. A solution is indeoendent of inout

parameter estimates and insensitive to further iterations if it has converged.

Convergence is indicated by the amount of change in the RMS from one iteration

to the next and by the change in the DELMX parameter. Because of the Taylor'sseries approximation technique used by this program, the parameter values

converge very rapidly in the neighborhood of a solution. Although this

neighborhood may be difficult to find for the first few iterations (thealgorithm's strategy uses a "gradient" technique to find the neighborhood),once found, successive iterations will improve the oarameter values by one to

two significant figures with each iteration. This imorovement is reflected in

a 10- to 100- fold decrease in DELMX with successive iterations. This rapiddecrease in DELMX may occur while the RMS changes very little. Attemots to

improve the solution beyond convergence may cause the message CORRECTIVE

ITERATIONS EXCEEDED to be printed.

Possible problemsThis orogram rapidly converges to a unique solution from a wide range of

parameter estimates when the data provide adequate constraints. New parameterestimates are always better (in the least squares sense) than the previous

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values but the parameters may become negative during the process. Inadequate

termination data may allow the FINAL parameter to become negative while the

FRACTION parameter for the slowest component increases without bound. Fixing

the FINAL fraction unreassociated (presumably near zero) or close scrutiny of

the DELMX values to find regions of local convergence may solve the problem.Often when the series of iterations do not immediately converge the

program hesitates in a region of local convergence. When pressed to improve

the RMS the program may go off in a direction (such as negative parameters)where termination is impossible. Usually DELMX starts at 1.0 to 5.0, then

decreases to 0.02 to 0.1 as the fit converges (the largest parameter change is

less than 2-10%). DELMX may then jump by a factor of 50 to 200 and start on

another (possibly unterminated) path. The small DELMX value indicates a local

low RMS region which might provide good values for the parameters but theymust be carefully examined. To determine the quality of the parameter values,plot the solution.

Occasionally the program returns negative rate constants for components

of the reaction. This is usually due to an attempt by the program to remove

that component from the solution. Plotting the solution usually shows why

this was done. Single erroneous points that do not show in the plot should be

searched for and perhaps a new solution should be attempted with fewer

components.

Parameter fixing

Data from other measurements may supply fixed values for parameters in a

solution. For example, chemical measurements of genome size may be used to

establish the rate constant of the single copy component. Slave "mini-cot"experiments (3) often provide the most reliable determinations of repetitive

and single copy rate constants which may be used as fixed values to calculatethe fraction of the genome associated with the different rate components in a

total reassociation curve.

Fixing the single copy rate constant at a value determined from the

genome size or minicot analysis may be particularly important for DNA which

contains a small fraction repeated 2-20 fold. This low repetition class isvirtually indistinguishable from single copy DNA but can increase the apparent

single copy rate constant by a factor of two.Parameter fixing also provides information about the variety of similar

least squares solutions which describe the data equally well. A graph of RMS

vs. a set of fixed values of a parameter can be informative, particularly if

it turns out to be a very shallow curve near the minimum.

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TABLE: EFFECT OF GAUSSIAN NOISE ON PARAMETER VALUES AND ERROR ESTIMATESSuccessive trials with random noise

Trial Number ofpoints

F:

Value Parameters used 0Error to generate data 0

SolutionError

SolutionError

SolutionError

SolutionError

SolutionError

SolutionError

Parameter valuesat solution

inal f1 ki f2 k2

.200~ 0.300 0.1000 0.300 10.00I U LU`

1 40 o:. 1b6 o:25 0.0536 0:82832bg0.017 0.037 o0.0289 0. 38 .382 40 o.19 0o.309 0.106 0.297

0.019 0.038 0.048 0.038 6.3 40 0.212 0.801 0.149 0. 1 lt.

0.015 0. 32 0.058 0.8? 7.44 40 o.208 o 0o061 0 82

0.017 0.040 0.039 0:841 2.475 40 o.180 0.340 0.110 0.302 10.2

0.015 0.035 0.039 0.035 4.2sum of 200 0.200 0.293 0.103 0-.300 9.28

trials above 0.007 0.017 0.020 0.017 1 .81

RMS

0.0350

0.0466

0.0392

0.0382

0.0379

0.0403

a Noise factor equal to EFS described in the text. For this example, F is0.2

Parameter statistics

Parameter standard deviations and correlation coefficients calculated by

the program provide information about the range of parameter values which may

adequately describe the data. These numbers, along with the RMS and DELMX,

indicate the significance of the parameter values obtained at the least

squares solution.

The parameter standard deviation and correlation coefficient calculation

is similar to standard deviation and correlation coefficient calculations in

linear regression analysis. To obtain the standard deviations and correlation

coefficients the data covariance matrix is inverted. The calculation assumes

that the Taylor's series linear approximation is accurate in the neighborhood

of the solution (indicated by a low DELMX at convergence). To examine theusefulness of the parameter standard deviations and correlation coefficients,

artificial test data were generated. Some examples of solutions using the

second order (FINGER) function are shown in Table 2.

The data set for each of these analyses was calculated using a Gaussian

random number generator for the final unreacted quantity as follows:

C/Co = GAUS(F,FSD) + ifi(1+kiCot)1(GAUS is a computer subroutine 'which generates a series of values following a

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Gaussian distribution with average E and standard deviation ES2.)This method of generating variation in the "data" is a good analogue of

actual fluctuations from measurement to measurement, such as the binding of

DNA to hydroxyapatite. The fluctuations observed in Table 2 for the various

solutions and the standard deviations shown are probably representative ofwhat would happen in repetitions of actual measurements which showed a

comparable RMS. It is clear that the standard deviations of the rate

constants are much larger than those of the component quantities. Whenanalyses of this sort are carried out with components only a factor of ten

apart in rate constant (instead of the factor of 100 for the exampleillustrated in Table 2) the rate constant fluctations are very large. It can

be seen from Table 2 that there is a reasonable quantitative relationshipbetween the fluctuations from set to set and the calculated standard

deviations.

This table exhibits one of the characteristic oroblems of fittingreassociation kinetic data and indicates the insight into the accuracy ofindividual parameter estimates provided by the standard deviation

calculations. It is clear that where two kinetic components are present, even

a factor of one hundred apart, very accurate data are needed to obtain

estimates of rate constants with moderate accuracy.

DISCUSSIONWe have described a powerful and flexible program for the least squares

analysis of the kinetics of reassociation. This program has been used for the

analysis of an extensive series of measurements (3,4,5,10,11,13). Least

squares analysis is necessary for reproducible and clear interpretation ofsuch measurements. We refer to these paoers for examples of use and

interpretation, while in this discussion we focus on oroblems of

over-interpretation or misinterpretation of the solutions. As powerful

analytical tools of this sort are developed it becomes very easy to simplyaccept the "output" and lose touch with its meaning.

It is important to remember that any successful least squares strategy

provides parameter values which "fit the data better" in the "least squares

sense". The solution does not guarantee either physical or biological

reality. ComDonents may be used to fit small peculiarities in the data and

have little physical meaning. This is usually evident from a plot of the fit.Parameter values may also be the fluke of a particular set of data and have

little absolute importance. The parameter standard deviations calculated bythe program provide confidence limits for the Darameter values given the data.

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Where it is known from other measurements that a DNA fraction is

homogeneous or where a single copy fraction has been purified the least

squares analysis provides an excellent means for evaluating its rate constant

and permits a more accurate calculation of its complexity. In general it is

not known that components are homogeneous and often the values of parameters

derived from the least squares solutions may be averages of a set of

unresolved components. In such a case the solution makes an excellent model

of the set of repetitive sequences which may be used in a variety of

calculations even though the true individual components are not known. Where

an important issue rests on the potential heterogeneity of a component other

tools must be used. For example fractionation of double from single standed

DNA could be carried out at the midpoint of its reassociation and the rate of

reassociation of the two fractions carefully compared.

Another consideration is as important as questions of parameter meaning:

solution uniqueness. If a parameter can change over a wide range without

affecting the quality of the fit measured by the RMS or GOODNESS OF FIT,conclusions based on a specific parameter value are suspect. In many cases,

this problem will be indicated by a high parameter standard deviation. More

quantitative insight into the significance of a particular parameter value can

be gained by plotting the RMS or GOODNESS OF FIT criterion as a function of

the best fit using different fixed values for the parameter in question. For

example, if the best fit of the data gives a rate constant of 0.05 for a slow

component with a GOODNESS OF FIT of 0.02, other solutions should be found with

the rate constant fixed at 0.15 and 0.015. If the variation in the GOODNESS

OF FIT is less than 10% over this range of rate constants, no conclusions can

be based on the 0.05 value which would be different for the other values. If

the GOODNESS OF FIT changes by 50- to 100% with variations in the parameter

value the value is probably uniquely determined by the data. While the method

of fixing one parameter value and varying the others to find the best least

squares solution can be used for any parameter, it is particularly importantwhen measuring rate constant parameter values. Data with two rate constants

differing by a factor of 10-30 can usually be described by a very wide range

of rate constants.

In summary, once a fully convergent least squares solution has been

established for a set of data we must consider four issues of interpretation:1. There may be systematic errors in the measurement such as DNA degradation

or unknown fragment size. 2. The individual set of data may be atypical,particularly if only a few measurements are available and the standard

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deviations are then a ooor measure of the possible error. Deviations inreassociation kinetic measurements are typically not due to random statistical

variables such as sampling from a population or radioactive decay, but aremore likely due to variations in assay procedures such as hydroxyapatite batch

or temperature or volume of samples. This weakens the significance of thestandard deviations. 3. The components may not represent true individualrate components but be averages of unresolved sets of components. 4. The

least squares minimum in the region of the solution may be shallow. In that

case, the set of solution parameter values may not uniquely fit the data;other parameter sets with substantially different values might provide equallyvalid interpretations of the data.

In many cases such problems can be shown to be of minor quantitativesignificance. Least squares solutions such as those generated by this programrepresent the best presently known approach to the interpretation of

reassociation and hybridization kinetics.

The program described in this paper is available as a punched card deckfrom the authors. Two versions are available, one for a DECsystem-10timesharing computer and a second for an IBM 370 Batch processor. Both

programs are written in FORTRAN and can be easily modified for other systems;the two programs differ in minor system dependent features. Inquiries shouldbe directed to William Pearson, California Institute of Technology, 101

Dahlia, Corona del Mar, CA 92625.

Also Staff Member, Carnegie Institute of Washington

REFERENCES1 Davidson, E. H. and Britten, R. J., (1973) Quart. Rev. Biol.

46,565-6132 Davidson, E. H., Galau, G. A., Angerer, R. C. and Britten, R. J.,

(1975) Chromosoma 51,253-2593 Britten, R. J., Graham, D. E. and Neufeld, B. R. (1974) in Methods in

Enzymology (L. Grossman and K. Moidave, eds.) Vol. 29 Part E pp.363-416

4 Galau, G. A., Britten, R. J. and Davidson, E. H., (1974) Cell 2,9-205 Galau, G. A., Klein, W. H., Davis, M. M., Wold, B. J. Britten, R. J.

and Davidson, E. H. (1976) Cell 7,487-5056 Bishop, J. O., Morton, .J. G., Roshbash, M. and Richardson, M. (1974)

Nature 250,199-2047 Ryffel, G. U. and McCarthy, B. J. (1975) Biochemistrv 14,1379-13848 Britten, R. J. and Kohne, D. E. (1966) Carnegie Inst. Wash. Yearbook

65,73-1069 Wetmir, J. G. and Davidson, N. (1965) J. Mol. Biol. 31,349-370

10 Britten, R. J. and Davidson, E. H., (1976) Proc. Nat. Acad. Sci. US73,415-419

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11 Smith, M. J., Britten, R. J. and Davidson, E. H. (1975) Proc. Nat.Acad. Sci. US 72,4805-4809

12 Morrow, J. (1974) Ph. D. Thesis, Stanford Universitv13 Davidson, E. H., Hough, B. R., Klein, W. H. and Britten, R. J.

(1975) Cell 4,217-23814 Marquardt, D. W. (1963) J. Soc. Indust. Appl. Math. 11,431-441

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