Chemostat Theory

8/7/2019 Chemostat Theory

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J. Theoret. Biol . (1967) 14,74-101

Theory of the Chemostat-Study of the Conditions underwhich a Population of Micro-organisms and its Mutantwill Coexist in the ChemostatC. RENNEBOOG-SQUILBIN~

Laboratoire de Physiologie Animale, Universite Libre de Bruxelles, Belgium(Received 30 July 1966)

When a population of micro-organisms giving rise to non-reverting mutantsis allowed to grow in a chemostat and when the mutation rate is assumedto be proportional either to the growth rate or to absolute time, threedifferent fina l situations will occur. The first one is that both populationsare washed out because the flow rate was badly chosen (too fast). The secondis that the mutant having a selective advantage replaces the normal popula-tion in the culture vessel. The third possibility is that both populationsgrow simultaneously and that between their respective concentrations, aconstant ratio gets established. We have studied this last case in detai land have shown the importance of the flow rate in the establishment of astable ratio.

1. IntroductionWhen a population of micro-organisms (which will be called normal) givesrise to mutant organisms, two eventualities are to be considered. One is thatthe proportion of mutants produced per unit time is proportional to thespecific growth rate of the cells generating them as it is usually admitted(Braun, 1953), the other is that the proportion of mutant cells appearing perunit time is constant as it has been established by Novick & Szilard (195Oa.195 1) in the case of mutation to phage T5 resistance.The specific growth rate ~1 s defined as

1 drrI-l = nht

where n is the concentration of the cells and t the time (Herbert, Elsworth &Telling, 1956). As shown by Monod (1942), y is an increasing function ofthe concentration of an essential growth rate limiting substrate S, which isexpressed by M.Sl = k+S

t Aspirant du Fonds National de la Recherche Scientifique.74


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THEORY OF THE CHEMOSTAT 75M being the growth rate constant (i.e. the maximum value of p at saturationlevels of substrate) and k a saturation constant which depends on the affinityof the cells for the substrate. When k is small, the affinity is great. Moser(1957) has given another form for the dependence of ,u upon the concentrationof a limiting growth factor:

M.SX

where x is a numerical constant (exponent of S). The general conclusions weshall find in this paper do not depend on the expression we choose asthe unknown S (or S) will not appear in the equation expressing the ratio--m/n as function of the dilution rate D (section 2(B)), provided that thevalues of x are identical for both organisms (normal and mutant). Moser(1957) showed that this assumption was essentially true for Escherichiacoli strain S/l t and its prevalent mutant B/ 1 fwhen grown under tryptophancontrol.We shall study the behavior of a population of microorganisms growingin a chemostat and giving rise to nonreverting mutants. The chemostat is aculture vessel whose volume remains constant by means of an overflow andwhose nutrient medium is constantly renewed. This nutrient medium containsan excess of all growth factors except one, which is limiting. The basicprinciples of the chemostat have been given for the first time by Monod (1950)and Novick & Szilard (1950aJ).We can ask under what conditions a normal population and a derivedmutant form may coexist in equilibrium in the culture vessel keeping in mindthat we have only the possibility of imposing the flow rate of the nutrientmedium and the initial concentration of the limiting substrate.The choice of the appropriate flow rate is of the highest importance becauseit determines the specific growth rates of both populations and, consequently,controls the proportion of mutants present at the steady state. We call thesteady state, the state where the concentration of normal cells and theconcentration of mutated cells no longer vary. The fact that a steady state ispossible does not, however, imply that the system will ever reach this state(Volterra, 1931), but it is possible to demonstrate in this particular case (cf.Appendix) that when the proportion of mutants approaches the steady statevalue, this value will finally be reached.We were prompted to this study by experiments with Saccharomycescerevisiae and its petite cytoplasmique mutant. As this mutation is veryfrequent (about 0402 mutant per organism and per generation time, Ephrussi,1951), the accumulation of mutants in the chemostat may occur after a shorttime even when the latter have no great selective advantage. We shall first


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76 C. RENNEBOOG-SQUILBINconsider the case which is admitted to apply to this situation, i.e. the mutationrate is proportional to the growth rate and then more briefly, the case wherethe mutation rate is constant per unit time.

2. The Mutation Rate is Proportional to the Growth Rate

We may write(A) EQUATIONS OF THE PHENOMENON

2 = YP,where I is the proportion of mutant cells which appears per unit time (time- )and y a positive constant always inferior to the unity. Let n be the concentra-tion of normal cells in the culture vessel (cells x ml.-I); m, the concentrationof mutant cells in the culture vessel (cells x ml.-); ,uI, the specific growthrate of the normal population, which is the mean growth rate of the cells ofthe population (time-); pZ, the specific growth rate of the mutant popula-tion (time-); D, the flow rate divided by the culture volume, which is calledthe dilution rate (time-r) (Herbert et al., 1956); S,, the initial concentrationof the limiting substrate in the nutrient medium (moles xml.-); S, theconcentration of the limiting substrate in the nutrient medium (moles x ml. - );l/Y,, the yield constant of the normal population for the limiting substrate(cells x mole-), (Monod (1942) showed that there is a simple relationshipbetween growth and utilization of substrate dn/dt = - I/ Y ds/dt, where l/ Yis known as the yield constant); l/ Y,, the yield constant of the mutant popu-lation for the limiting substrate (cells x mole-).

Mathematically, the problem may be stated as follows. During a timeinterval At which is supposed very small, the normal population gives rise to(1 -y)pl n.At

new normal cells per unit volume. During that same time, an amount normalcells proportional to II, D and At is removed from the culture vessel. Thevariation of n is thenAFI = (l-y)pl~~.At--nD.At, (11

The mutant population is increased by division of mutant cells and bymutation of normal cells. We get(&n~+y~l~~).At

new mutant cells per unit volume. During that same time, a fraction of themutant cells is removed. Finally, the variation of T F Z isAFIT = (~c,m+~~L1~~).At-~)~D.At. (2)

For the variation of the concentration of the limiting substrate, we know


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THEORY OF THE CHEMOSTAT 77that during the time interval At and per unit volume, the normal cells areresponsible for A S=-In Ar

Y l .and the mutant cells are responsible for

A 2 S= -h, AtY, . *A concentration S, of limiting substrate enters the culture vessel, while aconcentration 5 leaves it, hence

A,S = D(S,-S).At.The total variation of the limiting substrate concentration is thusAS= D(S,-S)++n .At.

1 2 1 (3)From equations (l), (2) and (3), we deduce the following differential equations

f dndt= Rl-~h-Dl~~dm

= +D(S,-S) _ !!;.! _ !!?.1 y2The general solution of this system is

(4)

(6)

We see that the general solution of this system of differential equations isextremely complicated. Consequently, we shall not study the evolution ofthe phenomenon toward the steady state but the conditions which must befulfilled by the different factors and especially by the dilution rate D if the


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78 C. RENNEBOOG-SQUILBINsteady state with the two populations present is to be possible. The steadystate will be reached when at the same timer drz-=dt 0

Equations (4), (5) and (6) become

I[jY1(l-y)-D]fi = 0

(j2 - D)iG+ yj& ii: = 0

I-- __D(S, S) - F;! _ 5; = 0.

1 2The letter with a bar represents the value of the variable at the steady state. Theinteresting caseswhere we satisfy conditions (13), (14) and (15)are the following :--p2fnii=& j2=D, s=S,- -.-.

Y2DThe normal population has been washed out of the culture vessel. Only themutant population remains in the chemostat. --(ii) (l-y),&=D, D-p,=j&;, &s,-S-E. 1 2We shall develop this last case in which the two populations remain in thechemostat. In the Appendix we shall show that, when n, m and S approachthe steady state values, continuing oscillations about the steady state areexcluded and the steady state will be attained.(B) STUDY OF THE INFLUENCE OF THE DILUTION RATE D ON THE STEADY STA TE

WHENCE # 0~~~16 # 0Case (ii) gives us the equations

riD-J2 =yjilz. (17)From equation (16) we may define a superior limit for D. We have a maximumvalue S, for S. Consequently, the maximum value that pr may reach is

MI.%plr = 6%


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THEORY OF THE CHEMOSTATwith M1 as growth rate constant of the normal cells andconstant of the normal cells for the limiting substrate. Itdilution rate we choose must be inferior to the value

D -M,(~-Y)S,r- k,+S, *

79kr as saturationfollows that the

(18)--Equation (17) leads to another limit for D. The term r&(n/m) beingpositive, D must be greater than &. (It is recalled that & is positive bydefinition.) Let us develop the condition D > &.Relation (16) becomes M,.S 1-_ zz -DD.k,i-S l-y

We know that_ M,.S =k,+S (20)

with M2 as growth rate constant of the mutant cells and k, as saturationconstant of the mutant cells for the substrate. Putting the value of 3 derivedfrom equation (19) in equation (20), we find

It is easy to see that the curve of jiz as function of D is a hyperbola. Theasymptotes

and the slope of the tangent at the origin

define, as shown by Figs 1,2,3 and 4, the conditions which the dilution rate Dmust fulfil to be superior to j&.


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80 C. RENNEBOOG-SQUILBIN

r-FIG. 1. The specific growth rates at the steady state of the normal and the mutant popula-

tions, fi l (-) and pa (-), k,as functions of the dilution rate D when - < 1 andkaM1 F < M1(l - y) . The intercept of p, and of the dashed line (----) pz = D determines2the limiting value D, of the dilution rate,

M+MI(I - :,)D, _ ..m+-m~- -,g-1

FIG. 2. The specific growth rates at the steady state of the normal and the mutant popula-tions, p1 and pz, as functions of the dilution rate Dwhen 2 > 1 and Ma : < Ml(l --I+.a aFor the signifkation of the curves, see Fig. 1.


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THEORY OF THE CHEMOSTAT 81

0 DFIG. 3. The specific growth rates at the steady state of the normal and the mutant popula-

tions, ,c, and pz, as functions of the dilution rate D when z < I and Ma : > M, (1 - r).a aFor the signification of the curves, see Fig. I.

/v //// ////Ib D1andM.$-M+~).For the signification of the curves, see Fig. 1.

T. B. 6


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82 C. RENNEBOOG-SQUILBINWe notice that equation (16) has as a consequence that ,Er is always greater

than ji2 when D > ,ii2.In the particular case where k,/k, = 1, equation (21) becomesM2 D~ = M,(l-y)

The inequality D > ,i& is satisfied only when M, < M,(l - y).We may summarize the data of Figs 1, 2, 3 and 4 as follows : a steady statewith E/E different from infinity is possible only in the three following cases :

(i) when k,.-~ 1kzM2$ < M,(l-y).

2provided that D < D,;(iii) when k,

li;>1

M+ >M,(l -:,J,2provided that D, < D -c D,.Putting the value of p1 from equation (16) and of p2 from equation (21) in--equation (17) we obtain the relation between the ratio m/n and the dilution rate :

Figures 5,6 and 7 represent the variations of Ejfi as function of D in the onlythree possible cases.


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THEORY OF THE CHEMOSTAT 831IIIIIII

IIIII0 4 LFIG. 5. The proportion of mutants at the steady state, fi jf i as function of the dilution rate

D when $1 < 1 and M, L: < M1(l - y) . (--) ?%/A; (----) asymptotes whichlimit the ra:ge of dilution rates for which the steady state is possible. It is recalled that Dmust also be inferior to D, = &Cl - Y)Sp--, The existence of this second limiting valuekz+S,depends on the intrinsic constants o f the.normal population.

I------

FI G. 6. The proportion of mutants at the steady state, Sz/rias function of the dilution rateDwhen~>IandM2~iM1(l-;~).D~= MI (1 - YW ,k,+Sr For the signification of the2curves, see Fig. 5.


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84 C. RENNEBOOG-SQUILBINi-~----.-. ._-I/

1 1!

\ET ,

FIG. 7. The proportion of mutants at the steady state, ti/ti as function of the dilutionrate D when 2 > 1 and M, F > M 1 1 - ;A). D, :- M1$-&!@. For the signification ofthe curves, seeFig. 5. 2 I T

--We notice that, when k,/k, = 1, the fraction m/n is independent ofD. Any value of dilution rate is thus allowed provided that it is smallerthan

A suitable dilution rate being chosen, let us briefly see the influence of theother constants (which we cannot change because they represent inherentproperties of the considered micro-organism) on the value reached by theratio E/E at the steady state. It is easy to demonstrate from equation (22)- -.that in the three cases cited above, m/n is an increasing function of y betweenthe y values zero and 1. Taking the derivatives of iii/@ as function of k,/kz, wefind that it is always positive and thus, the greater is k,/k, (i.e. the lower isthe affinity of the normal population for the limiting substrate in respect tothe one of the mutant population), the greater is the ratio FE/R at the steadystate, all the other constants remaining the same. In the same manner,the derivatives of G/E as function of M, and as function of Mz show usthat the proportion of mutants increases with the growth rate constant ofthe normal population and decreases when the growth rate constant ofthe mutant population increases. These results were of course logicallyexpected.


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THEORY OF THE CHEMOSTAT 85(C) NUMERICAL EXAMPLES

We shall illustrate those theoretical data by some numerical examples. Inall those examples, we have deliberately chosen an abnormally high mutationrate. Notwithstanding this, conditions of culture can still be found compatible--with a ratio m/n different from infinity.In Fig. 8, we have chosen very close values for the growth rate constants ofboth populations (M, = 0+3/hr, M, = 0+294/hr) and we have taken different

3

1

FIG. 8. The proportion of mutants at the steady state, rTi/iias function of the dilution rateD for different kl/k2 ratios. S, = 40.10-Bm oles/l., y = lOma, kl = 10-Bm oles/l.,MI = 0.3/hr, Ma = 0.294/hr, D, = 0.290/h r. For curves (l), (2), (3), (4), (5), (6), (7), (8)and (9), the ratios kI/kz are respectively 0.1, 0.5, 0.91, 0.99, I, lW1 , 1.0102, 1.1 and 10.k,/k,. When the normal population has a very low affinity for the limitingsubstrate compared with those of the mutant population (kl/k2 S l), thedilution rate must be very large if both populations are to coexist, but thiscondition is often incompatible with its superior limit (0, = 0*29/hr). Whenon the contrary, the normal population has a very high affinity (kl/k2 G l),--it is logical that the fraction m/n remains very small. For values of kl/kz closeto 1, appreciable quantities of both populations will be present at the steadystate. When k,/kd is slightly greater than 1, very slow dilution rates give high


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X6 C. RENNEBOOG-SQUILBIKlevels of mutants while when k,jk, is slightly smaller than I. they induce aratio Z/E smaller than 1, but in both casesdilution rates near the superiorlimit D, lead to a proportion of mutants of about 1.If the growth rate constant Mz of the mutant population is much smaller--than that of the normal population, the ratio of mutant cellsm/n at the steadystate will be very small except for large values of k,/k, and within a limitedrange of dilution rates (Fig. 9).

FIG. 9. The proportion of mutants at the steady state, N2jfi as function of the dilutionrate D for different kl/ka ratios. Same S,, y, k,, MI and D, as in Fig. 8. M, = 0,03/hr. Forcurves (1) and (2), the ratios kI/kz are respectively 10 and 100.

If the growth rate constant M, of the mutant population is much higherthan that of the normal population, washing out of the normal populationwill result except for very low values of k,/k, (Fig. 10).The influence of M, on the physionomy of the curves expressing iir/ii asfunction of D appears clearly by comparing Figs 8, 9 and 10.

We shall now see n more detail in Fig. 11, the effect of the constants in theonly interesting case, .e. when Mz approachesM, and when k,/k, hasa valueclose to 1. The comparisons betweencurves (1) and (2) between curves (3) and(4) or between curves (6) and (7) shows the influence of slight differences


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! II

FIG. 10. The proportion of mutants at the steady state, fi/ ti as function of the dilutionrate D for different kl/kz ratios. Same S,, y, k,, MI and D, as in Fig. 8. Mz = 3/hr. Forcurves (l), (2), (3) and (4) the ratios kI/kz are respectively 10e6, 10-a, 0.09 and 0.098.

FIG. 11. The proportion of mutants at the steady state, rii/ri as function of the dilutionrate D. Same S,, y and kI as in Fig. 8. The other characteristics of the curves are given inTable 1.


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C. RENNEBOOG-SQUILBINTABLE 1

Characterisks of the curves of Fig. I1Curve ML!(hr-I) k,/k, LA(hr- )

I 0.3 0.2967 0.9; 0.2902 0.3 0.294 0.99 0.2903 0.3 0.2967 I.0102 0.2904 0.3 0.294 I.0102 0.2905 1 .5 I.47 1.0102 I.456 0.3 0.3003 0.91 0.2907 0.3 0.2967 0.91 0.2908 0.3 0.2967 I 0.290

between the M,. The way of the shift of the curve is that we expect logically.For curve (3) we see that dilution rates smaller than 0.2667 result in the wash-ing out of all normal cells. The effect of k,/k, is made evident by curves (1).(3) and (7), or (2) and (4). In curve (5), M, and M, are respectively fivefoldgreater than in curve (4). We see that as long as the difference betweenM,(l-y) and M2(kl/k2) is small compared with D(k,/k, - l), the ratioE/E is fivefold greater in curve (5) than in curve (4).

(D) CONCLUSIONThe conclusion of this study is that the choice of the dilution rate followingthe particularities (growth rate constant, affinity for the limiting substrate,

mutation rate) of the two populations present is of the greatest importancewhen their affinity for the limiting substrate is not the same. On the contrary,the dilution rate (provided it is inferior to the maximum rate) has no effectwhen the affinities of both populations for the substrate are identical.3. The Mutation Rate I is Constant per Unit T ime

(A) EQUATIONS OF THE PHENOMENONThe equations of the phenomenon are established in the same manner as insection 2. They become

r dil/ 5 = (p1-a-D,J7dm- = ( /A2 - D)m + indt


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THEORY OF THE CHEMOSTAT 89At the steady state,

I(Jil -A-D)ii = 0 (26)

(,E,-D)Ei+Aii = 0 (27)

As in section 2, the interesting cases where we satisfy conditions (26), (27)and (28) are the following: --P2 J'lJl=o, ji2= D, s=s,---- Y,D'Only the mutant population remains in the chemostat.(ii)Here, the number of normal and mutant cells differs from zero. We shalldevelop this case.

(B) STUDY OF THE INFLUENCE OF THE DILUTION RATE ON THE STEADYSTATE when fi # 0 AND 15i # 0

The equations we shall study are thusD=&-A (29)

Equation (29) gives an immediate condition for the dilution rate. Thesubstrate concentration being inferior to S,., ,iI1 will never exceedMI .S,k,

so that D has a maximum valueD, = MI .Srk,+s,-l-

For reasons of convenience, we shall not express the value of Ci/ti as func-tion of D, as in section 2, but the value of ii/Z as function of D/LIn the particular case of kl/kz = 1, Z/Kz as function of D/A is a straight line

rij??i is positive provided that M2 < M, and D//z > M,/(M, -MJ.T.B. 7


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90If we say

C. RENNEBOOG-SQUILBIN

we find from equation (29) and (30)ii L2--=X-y---.Ei 1+ 3.x+1- -.

(31)We immediately see that n/m IS the difference between a straight line whoseequation is a = x and a hyperbola which corresponds to fi,/A and whoseequation is

y = -LL.1 +;$!j

The graphic representation of such functions is very easy.The asymptotes y = L,and x = - L, - 1 define the conditions which the dilution rate must satisfy inorder that the ratio la/k is positive at the steady state.There exist four cases, illustrated in Figs 12, 13, 14 and 1.5.

0 :j AFIG. 12. The specific growth rates at the steady state of the normal and the mutantpopulations expressed as &/A and izz/J. as functions of the dilution rate expressed asx = D/Awhen k,/kz P 1, The proportion of normal cells at the steady state ii/t% (curve (3))is directly obtained by subtracting curve (1): y = &/A from the dashed line )I = D/ j..The solid straight line (2) represents PI/l.


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THEORY OF THE CHEMOSTAT 91

FIG. 13. The speciiic growth rates at the steady state of the normal and the mutant popula-tions expressed as p,/A and Q,I as functions of the dilution rate expressed as x = D/l whenkJkz < 1 and IL,I < 1. For the signification of the curves, see Fig. 12.

FIG. 14. The specific growth rates at the steady state of the normal and the mutant popula-tions expressed as pJ,I and ii,/1 as functions of the dilution rate expressed as x = D/Awhenkl/kl < 1, jLll > 1 and (1 +L1 --L,)-I-~LL~ > 0. For the signification of the curves,see Fig. 12.


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92 C. RENNEBOOG-SQUILBIK

0 :FIG. 15. The specific growth rates at the steady state of the normal and the mutant popula-tions expressed as &/,I and bz/A as functions of the dilution rate expressed as .Y= D/A whenkl/kz < 1, ]Lll > 1 and (l+L,-LL,)2+4Lz < 0. For the signification of the curves,see Fig. 12.

--Since n/m must be positive, the only dilution rates allowed are those atwhich y is smaller than X. For that reason, the case of Fig. 15 is excludedwhile the case of Fig. 13 must be rejected because y (i.e. &/A) is negative.In Fig. 12, x must be superior to x1 and in Fig. 14, x must be comprisedbetween x1 and x2. x1 and x2 are defined as the solutions to the equation

-(1+L1-L,)-J(1+L,-LJ2+4L2x2 = -- ____-2It must be noticed that at the steady state the specific growth rate ,Ei of thenormal population necessarily exceeds the specific growth rate fiZ of themutant population when D > iI2 (cf. equation (29)).The situation is summarized in the following manner. When the affinity of

the normal population is smaller than that of the mutant population (Fig. 12),--the fraction n/m will increase with the dilution rate provided that this latterdoes not exceed the maximum value allowed by the conditions of the experi-ment. This is comprehensible because the difference between x and )? (i.e.


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THEORY OF THE CHEMOSTAT 93also between jil and j&) increases with the dilution rate. The differencebetween Figs 14 and 15 is less immediate. It lies only in the sign of the func-tion F = (1 +Li -L2)2 +4L,, which is responsible for the existence or non-existence of intercepts between the hyperbola y and the straight line 01.(i) Function F may be expressed as function of k1/k2

The denominator being always positive except when k,/k, = 1, we are onlyinterested in the numerator N which is the equation of a parabola. Itsderivative N is first negative and then positive. That means that the concavityof the parabola is directed towards the top and that function F(kl/k2) iseither always positive or negative for k1/k2 values comprised between(k,/k,), and (kl/k2)2 and positive for k,/k, values outside those limits,(k,/k,), and (k,/k2)2 being the roots of N.

The influence of M,, M2 or y on F, all the other terms remaining respec-tively constant, may be studied in the same manner as the k,/k, influence.We find that M1 values must be outside M1, and M12, withM,,=M2$($1)2+2/~-l)i~M2,M,,=M,$($+,/-~:)~


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94 C. RENNEBOOG-SQUILBINM, values must be outside M,, and MZ2, with

kzand y values must be outside yi and y2, with .---

42M,M22__.----

k1- 1kz4;M,M,1

in order that the steady state with n/m different from zero may exist.In other terms, certain values only of the intrinsic constants of the popu-lations are compatible with the proposed state.

Example 1(C) NUMERICAL EXAMPLES

M, = l+/hr, M, = 1*47/hr, A = O*Ol/hr. The roots (k,/k,), and (kl/k2)2are imaginary numbers, hence F(k,/k,) is positive for any k,/k, (cf. Fig. 14).In Table 2 we find the abscissae of the intercepts of y and CI for differentk,/k,. It is recalled that the dilution rate must be inferior to 1*45341/hr (orx < 145.341) if we choose S, = 40.10m6 moles/l. and k, = low6 moles/l.and that the permitted values of x are comprised between x1 and x2 whenk,/k2 is smaller than 1, or are greater than xi when k,/k, is greater than 1.We see that when k,/k, values reach the neighborhood of 10, x1 becomesgreater than 145.341. Hence we can say if the affinity of the mutant popula-tion is tenfold higher than that of the normal population, no dilution ratesare compatible with an equilibrium with both populations in presence.--Figure 16 shows the ratio n/m as function of D for some k,/k, values ofTable 2.


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96 C. RENNEBOOG-SQUILBIN

FIG. 16. The proportion of normal cells at the steady state, ti/tiias function of the dilutionrate expressed as x = D/1.. S, = 4O.1O-6 moles/l., 1. =: 10e2/hr. k, =: lo- moles/l..M1 = 1.5/hr, M, = 1.47/hr. For curves (l), (2), (3) and (4), the ratios k,/kz are respec-tively 0.1, 0.9, 1.01 and 1.1.To illustrate which conditions of M,, M, and y must be satisfied for thegiven value k,/k, = O-1, we shall search the limits outside of which onemust stay in order to have the studied steady state. These limits areM1, = 90*8/hr and M,, = 89*4/hr, M,, = 91*6/hr and M,, = 91*4/hr,

Yi = 0.78 and yZ = 2.88. The given values M, = 1..5/hr, M, = 1*47/hr andy = 0.01 are outside those limits as expected.Example 2

M1 = O-3/hr, M, = 0*294/hr, I = O-Ol/hr. Those values lead to a(k,/k,), = 0.99546 and a (kl/kJ2 = 0.91416. Inside this interval, F(kl/kZ)being negative (cf. Fig. 15), the steady state with both populations in theculture vesseldoes not exist. Outside this interval, as the maximum dilutionis 0*283/hr (or x,,, = 28.3), the steady state will only be attained for k,/k,values smaller than (k,/k,) = O-91416. In Table 3 llustrated by Fig. 17 forsome k,/k,, we find the abscissae f the intercepts of y and CI or differentk,/kz when these ntercepts exist.

4. General ConclusionTo summarize the influence of the dilution rate D on the proportion ofmutants at the steady state, it must first be recalled that the specific growthrates of the normal and of the mutant populations, ,Er and ,C2are increasing


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THEORY OF THE CHEMOSTAT

20/---

97

xFIG. 17. The proportion of normal cells at the steady state, A/E as function of the dilutionrate expressed as x = D/l. SameS,, 1, k, as in Fig. 16. M, = 0.3, M, = 0.294. For curves(I), (2) and (3), the ratios kl/kz are respectively 0.1, 0.5 and 0.9.

functions of D and that obviously ji I must always be superior to /Yi2 f thenormal population is to coexist with the mutant population.When the mutation rate per unit time L is constant (section 3), the propor---.tion of normal organisms at the steady state (n/m) mcreases or decreases withthe difference A,u = ,i& -j& (cf. Figs 12 and 14).When 1 is proportional to the growth rate (section 2), two factors interactto influence the proportion of mutants at the steady state when the dilutionrate D increases. These are Ap, as above, but also /I, which is proportional top1 and hence to D. This latter fact explains why, in Fig. 5, for low dilutionrates E/E decreases (the inverse Z/E increases) though Ap increases (cf.Fig. 1). The most striking effect of the interaction between 2 and AP is thatZ/C is independent of the dilution rate when the affinities of both populationsfor the limiting substrate are the same (kl/k2 = I), ~1and /1 being in this caseboth proportional to D.A last thing to observe is that only the ratio k,/k, of the saturation con-stants occurs in the equations. It follows that the absolute value of k, doesnot matter while the absolute value of k, sets a limit to the maximum dilutionrate allowed.We may say in all generality that some pairs of microorganisms formed bya given population and a mutant derived from it will never coexist in thechemostat because of the values of the intrinsic constants which characterizethem, while other pairs will coexist within a limited range of dilution rates.

We thank Dr R. Hamers for his helpful advice and we are very grateful toDr J Renneboog for his mathematical help and his encouragement.


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98 C. RENNEBOOG-SQUILBINREFERENCES

BRAUN, W. (1953). Bacterial Genetics. Philadelphia, London: W. B. Saunders Company.EPHRUSSI. B. (1951). In Genetics in the Twentieth Centurv. Dunn. New York: TheMacmillan Company.HERB ERT, D., ELSW ORTH, R. & TELLING, R. C. (1956). J. grn. hficrobiol. 14, 601.MONOD, J. (1942). La croissance des cultures bactkriennes. Paris: Hermann et Cie.MONOD, J. (1950). Ann. Inst. Pasteur, 79, 390.MOSER, H. (1957). Cold Spring Harb. Sytnp. quant. Biol. 22, 121.NO~KK, A. & SZILARD, L. (195Oa). hoc. natn. Acad . Sri. U.S.A. 36, 708.NOVICK. A. & SZILARD. L. (1950b). Science. N.Y. 112. 715.NOWZK; A. & SZKARD; L. ,

since [(l - y),ul - D] n is a function of S, and n, (p2 - D)m+ ypl n andD(S,- S) - $ n - F Ii1

1 2are functions of S, n and tn.We know that r dn dAi1dt dt

! dtn dAt7t4 ~..~~_ __~ dt dt


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THEORY OF THE CHEMOSTAT 99Considering that An, Am and AS are very small respectively compared withfi, Z and S, the Taylors series corresponding to fi(S, n), fi(S, n, m) and,&(S, n, m) may be limited to the terms of the first degree. Hence

(32)dAndt ~j-~(S,ii) + @An + @ASdm

i

dt =jl(S, E, E) +dSdt Nfl(S,ti,%) + e)An + ($)A, + g)A& (34)

where

represent respectively the values at the steady state of the partial differentialsof the functionsfr, f2 and& with respect to n, m and S.If we say

c=ji2-Dd=.,(g)+YR(!$

g-;2

h D-where

($$) and ($$)represent respectively the values at the steady state of the derivatives of the


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100 C. RENNEBOOG-SQUILBINfunctions pI and ,uZwith respect to S, the equations (32), (33) and (34) become

f dAnI IryaAS

(35)dAm1 dr 2: bbrr + cAm + dAS (36)

(37)The resolution of this system of three homogeneous linear differentialequations with constant coefficients can classically be reduced to the resolution

of one differential equation of the third order with constant coefficients. Byeliminating Am and AS, we obtain the following equation for An:d3An-- _dt3 (c + h) ::T + (hc - dg - aj) f$i + a(fc - bg)Arl N 0. (38)

In order that An = eXt s a solution of the equation (38), x must be a root ofthe equationlx3 - (c + h)%2 $ (hc -dg-CrJ)cc+a(fc-hg)=O.

The Routh-Hurwitz criterion defines the following conditions of stability:- (c + h) > 0 (39)

a(fc - by) > 0 (40)a(bg +Jk) - (c + h)(ch - dg) > 0. (41)

(B) ANAL YSIS OF THE THRE E CONDITIONS OF STA BILIT Y- -(9 - (42)Seeing that -($!) and (i%)are respectively-equal to.

which are always positive (because x > 0, S > 0, jI, or ji2 > 0, PI orj& < M, or M,), the expression (42) is positive if D is superior to ,ii2, acondition that we have established in section 2(B).(ii) G-la(fc - bg) = (1 - y)E as( >[ (D - &)!$ -t ypl F .-12 (43


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THEORY OF THE CHEMOSTAT 101For the same reasons, the expression (43) is always positive.

(iii) That the inequality (41) is satisfied when l/Y, > r/Y, results fromthe fact that then (bg+fh) is positive and the second term (c+h)(ch-dg) isnegative.When l/Y, < y/Y,, the verification of the inequality (41) implies ratherlaborious calculations. After some transformations, the inequality (41) isequivalent to

-ti /*( >[s 1uJ - P2) -t- (& - :-> P2P - P2)] (44)2 1which is always verified for (D > ji2).

(C) CONCLUSIONThe three conditions of stability being satisfied, we may say that theconsidered system (i.e. two populations of microorganisms, one of whichderives from the other), when near the steady state, will tend to approach itmore and more. Thus oscillations about the steady state without reaching itare excluded.

Chemostat Theory

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