ON RELATIONSHIPS BETWEEN VARIOUS DISTRIBUTION FUNCTIONS IN BALANCED UNICELLULAR GROWTH

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1 BULLETIN O:F MATHEMATICAL BIOPHYSICS VOLU~E 30, 1968 ON RELATIONSHIPS BETWEEN VARIOUS DISTRIBUTION FUNCTIONS IN BALANCED UNICELLULAR GROWTH 9 D. RAMKRISHNA, A. G. ~REDRICKSOI~ AND H. M. TSUCHIYA Department of Chemical Engineering, University of Minnesota, Minneapolis, Minnesota The relationships between various size distributions in balanced exponential growth of a batch culture of microorganisms are presented. Starting from the partial differential integral equations (Eakman et al., 1966; Fredrickson et al., 1967) derived for the growth of a microbial culture expressions are obtained for the growth rate of organisms of specific size and size range. These expressions were first obtained by Collins and Richmond (1962) by an entirely different method. Also derived are equations which link probability functions, which are basic to the growth of a microbial culture, with other size distributions that can be estimated experimentally. Collins and Richmond (1962) have analyzed the exponential growth of Bacillus cereus by examining the length distributions of cells of the entire population, of dividing cells, and of organisms newly formed by fission of parent cells. The purpose of their analysis is to obtain the rate of elongation of organisms of specific lengths (or ranges of length) under submerged culture conditions by experimental estimation of the densities of the distributions enumerated above. Their analysis does not begin with the partial differential (integral) equations that have been derived for describing a growing population in terms of a characteristic size variable (Eakman et al., 1966; Fredrickson et al., 1967). It is of interest to see whether the results obtained by Collins and Richmond can also be derived from these differential equations. Let WL(I, t) dl be the number of organisms per unit volume of culture of 319

2 320 D. RAMKRISHNA, A. G. FREDRICKSOI~ AND H. M. TSUCHIYA length between 1 and 1 + dl at time t. Then in balanced exponential growth, WL can be described by the following partial differential integral equation. where ~WL ~(w~lq)) f~ ~--V + ~l = 2 ~(l')p(1, l')w~(r, t) dl' - ~(1)W~. (1) L(l) = Growth rate of cells of length 1. a(1) dt = Transition probability of division during time dt for a cell of length 1 at time t. p(/, l') dl = Probability that a daughter cell formed from a mother cell of length l' has length between l and l + dl. If length is conserved at fission, then the partitioning function p(l, l') must satisfy the conservation relation p(1, l') = p(l' - l, l'). It is emphasized that the functions J~(1), and p(1, 1') are neither explicitly nor implicitly time dependent in balanced growth, nor will they depend on other indices of physiological state, such as cell age, in balanced growth. The functions may depend on environmental factors, provided that these are not changing with time. These points are discussed fully by Fredrickson et al. (1967). Thus WL(I, t) can be written as: WL(1, t) = n(t)2(1) where n(t) is the number of organisms per unit volume and 2(l), retaining the nomenclature of Collins and Richmond, represents the density of the length distribution of the entire population during balanced growth. The total number of organisms is determined by dn d--t -- kn, (2) where k dt can be interpreted as the transition probability that an arbitrary cell of unspecified length will divide during the time interval between t and t +dt. Clearly from the total probability theorem it is true that k = f: a(1)2(l) dl. (3) Relation (3) can also be obtained by integrating (1) with respect to 1 from 0 to co and comparing the result with (2). Consider the various terms in equation (1). The right side of the equation has two terms. The first term represents the rate at which cells of length 1 are formed from dividing cells of length greater than l, and the second term repre-

3 UNICELLULAR GROWTH 321 sents the rate of division of cells of length I. We define a probability density #(1) so that ~b(1) dl represents the probability that a newly formed cell has length between l and 1 + dl. (This density is independent of time since growth is balanced.) Thus the first term on the right-hand side of equation (1) can also be written as 2kn(t)r that is, 2 ~(r)p(~, r)w~.(r, t) dr = 2kn(0r (4) f; Equation (4) may be regarded as the defining equation for ~(l). If one defines r dl to be the probability that a dividing cell has length between l and l + dl then the second term on the right-hand side of equation (1) can be replaced by kn(t)r that is, w~(~, t)~(~) = ~n(t)r (5) Equation (5) may be regarded as the defining equation for r Rewriting equation (1) using (2), (4), and (5) yields a(~)-n + n(t) [a(1)l] -- 2~(t)~(l) - kn(t)r (6) Dividing equation (6) by n(t) and using (2) one gets d d~ Ea(~)L,] = k[2~(l) - r - a(l)]. (7) Equation (7) was also obtained by Harvey, Marr, and Painter (1967) by differentiating the result of Collins and Richmond.* Integrating equation (7) one gets the result g = k fl [2~(r) - r - ;~(r)] dr/~(l) (s) which was derived by Collins and Richmond by a quite different method. Moreover the differential equation (7) can be integrated subject to the condition that 2(0) = 0 to obtain ;~(l) ~ ~ exp [2~(r) - r at. (9) Equations (8) and (9) may be used to derive the expressions obtained by Collins and Richmond for an "average growth rate" for organisms of length over a noninfinitesimal range (l - a) to (1 + a). This is done as follows: * Harvey, Marl" and Painter have also provided a clear derivation of Collins and Richmond by systematic argument.

4 (10) $22 D. RAMKRISHNA, A. G. FREDRICKSON AND H. M. TSUCHIYA Let the average growth rate for 1 in the desired range (l - a) to (1 + a) be denoted by v; i.e./~(l) = v for (l - a) < 1 < (l + a). Expression (9) can be written for (1 + a) and (1 - a) and subtracting the two one gets on a slight rearrangement of the exponential terms: exp (2ka/v) ~(1 + a)l(l + a) - 2(1 - a)l(1 - a) -- fexp - Substituting from equation (8) for the terms in the left-hand side of equation (10) and replacing the integral on the right side of (10) by a trapezoidal approximarion the result of Collins and Richmond can be obtained. It is interesting to observe that equations (5) and (9) provide a means for the experimental determination of the transition probability a(1). Just as/~ is a basic quantity representing the rate of growth, a(l) is basic to the phenomenon of reproduction. Thus ~(1) = ~, Cq)s (11) f: exp (k fz 7)[2r r dl' The right-hand side of (11) contains quantities that can be determined experimentally so that the transition probability a(l) can be evaluated. Eakman,~ Fredrickson and Tsuchiya relate a(l) to another probability density h(1) where h(1) dl represents the a priori probability that a cell selected at random will have length between I and l +dl when it divides. They show that aq) = r~h(l)z (12) 1 - Jo h(~') dl' It is possible to compute h(l) from the integral equation (12) knowing a(1) from (11). The importance of h(1) lies in the fact that models for cell division obtain h(1) rather than r or a(l) (for example, Rahn's model in terms of cell age, 1932). Thus models for h(1) could be compared with experimental estimation of h(1) from equations (I1) and (12). We are grateful to Dr. Ernesto Trucco for suggesting that this note be published. We also thank Professor H. D. Landahl for his helpful comments. ~f Eakman eg al. used mass instead of length in their work.

UNICELLULAR GROWTH 323 LITERATURE Collins, J. F. and M. H. Richmond. 1962. "Rate of Growth of Bacillus cereus Between Divisions." J. Gen. Mierobiol., 28, 15-33. Eakman, J. M., A. G. Fredrickson and H.

5 UNICELLULAR GROWTH 323 LITERATURE Collins, J. F. and M. H. Richmond "Rate of Growth of Bacillus cereus Between Divisions." J. Gen. Mierobiol., 28, Eakman, J. M., A. G. Fredrickson and H. M. Tsuchiya "Statistics and Dynamics of Microbial Cell Populations." Chem. Eng. Prog. Syrup. Series No. 69, 62, Fredriekson, A. G., D. Ramkrishna and H. M. Tsuchiya "Statistics and Dynamics of Procaryotie Cell Populations." Math. Biosciences, 1, Harvey, R. J., A. G. Marr and P. R. Painter "Kinetics of Growth of Individual Cells of Escherichia coli and Azotobacter agilis." J. Bacteriol., 93, Rahn, O "A Chemical Explanation of the Variability of the Growth Rate." J. Gen. Physiol., 15, RECEIVED

Kinetics of Growth of Individual Cells of

JOURNAL OF BACTERIOLOGY, Feb., 1967, p. 605-617 Vol. 93, No. 2 Copyright @ 1967 American Society for Microbiology Printed in U.S.A. Kinetics of Growth of Individual Cells of Escherichia coli and Azotobacter