I. rheor. Biol. (1977) 68, 415-435

Predictions of Cellular Growth Patterns by a Feedback Model S. V. Ho AND M. L. SHULER Sclraol of Chemical Engitzeerirtg, CorneN University, Ithaca, New York 14853, U.S..d. (Rcccived

17 March

1916, urld ill rccisedform

16 March

1977)

A mathematical model for the growth of an individual bacterium incorporating feedback control of nutrient uptake is proposed. An alternative model based on a constant number of transport sites with constant activity has previously been explored (Kubitschek, 1968ab 197Ou, 1971; Kubitschek. Freedman & Silver, 1971). A feedback model is capable of predicting the same growth pattern as the constant number of transport sites model.

1. Introduction Over the last 50 years numerous attempts have been made to determine the growth pattern of individual cells. The very diverse patterns observed have been reviewed by Collins & Richmond ( 1962), Kubitschek (1970a), Mitchison (1071). The simplest forms, linear and exponential, are very difficult to directly distinguish without very precise data (Kubitschek, 1970~; Mitchison, 1971). Consequently, Kubitschek (1970a) has argued that linear growth for the first two-thirds of the cell cycle may be general for all cell types when grown under conditions of balanced growth (Campbell, 1957) and that other observed patterns could be due to the use of experimental techniques that were too insensitive to distinguish among the possible growth patterns. Mitchison (1971) suggests that linear growth exists for a number of species but is not the predominant form. Anderson &Bell (1971) have argued strongly that for some mammalian cells linear growth is not consistent with the data. Kubitschek (1968a,b, 1971), using synchronous culture and the measurement of nutrient uptake rate, has concluded that for Escherichia coli the cell growth was linear for approximately the first two-thirds of the cell cycle followed by a period of rapidly increasing growth until division. A similar pattern has been observed for E. cofi by Adler, Fisher & Hardigree (1969). More recently measurements of the length of E. coli (Zaritsky & Pritchard, 1973) and Bacillus suhtilis (Sargent, 1975) have shown that the extension of the cell envelope occurs at a constant rate which increases abruptly when new 415

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“growth zones” are formed. The formation of new zones is apparently related to the completion of DNA replication. Consequently, cell extension would under many nutritional conditions appear to be linear with an abrupt increase in rate in the last half to quarter of the cell cycle. Of course, the rate of length extension is the same as the rate of volume growth only if cell width is constant. Although a constant cell width during the cell cycle is likely (Marl, Harvey & Trentini, 1966) only a 3 “/; variation in width during a cell’s lifetime could change the volume growth from linear to exponential (Collins & Richmond, 1962). Ward & Glaser (1974) have measured cell voiumc for E. coli during synchronized growth and found volume increase to be linear with a doubling in rate when new rounds of DNA synthesis were initiated. Thus, linear volume growth for one-half to three-fourths of the growth cycle followed by a more rapid increase in volume appears to be a plausible growth pattern for a number of bacteria. Kubitschek (196&z, 1970a, 1971) and Kubitschek, Freedman & Silver (1971) have proposed that for such bacterial cells in a balanced growth situation the number of sites for the transport of material into the cell remains constant until just prior to cell division when the number of sites must double. This model assumes the rate of transport per site is constant and that a constant rate of nutrient uptake implies linear growth. An alternate explanation, also recognized by Kubitschek (1971) is that the number of sites varies as the surface area does but the activity of the sites varies under feedback control from the level of intracellular nutrient pools. Hennaut, Hilger & Grenson (1970) have data suggesting that for several transport systems in yeast the amount of transport system per unit surface area is constant. For some organisms the concentrations of nutrients in the pools are known to rise during the first part of the cell cycle and to decrease before the cell divides (Kubitschek, 1970b; Mitchison, 1971 ; Huzyk & Clark, 1971). Intuitively one would then expect that the individual sites under feedback control would decrease in activity during the first part of the cell cycle and increase to the original level as the cell approached division. Since the number of sites would be continuously increasing, the overall rate of uptake could be essentially constant for the first part of the cell cycle and increase rapidly as the cell approaches division. Kubitschek (1971), however, has favored the constant number of sites theory because of the greatel inherent biological complexity of feedback models and the wide range of conditions over which feedback systems would have to work. However, certain conceptual problems are inherent in the constant number of sites model. Material is being continuously excreted from the cell. Only if the difference between uptake and excretion is constant will growth be linear. Since the macromolecular fraction, i.e. as protein and RNA, is

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formed exponentially (Ecker & Kokaisl, 1969; Kubitschek, 19706; Mitchison, 1971) and waste material is made primarily from synthesis reactions, it is not clear that the rate of waste excretion will be constant. Consequently, a constant rate of uptake need not imply linear growth. The constant number of sites model also requires that the activity of a transport site be constant. Since feedback inhibition of some permeases has been established (Mitchison, 1971: Kaback & Hong, 1973 ; Wan, Floyd $2 Hatch, 1975), the assumption of constant activity of a transport site i5 questionable. Since glucose uptake is constitutive of particular interest is the feedback inhibition of glucose transport by high-energy compounds (Kepes. 1964) or by glucose-l-P and glucose-6-P (Kaback & Hong, 1973). Such observations are more consistent with a feedback model than with a constant number of sites model. The purpose of this paper is to test quantitatively whether a feedback model can give a growth pattern similar to that predicted by the constant number of sites model. Also, the potential effect of ccl1 size and morpholog> on thcsc growth patterns is explored.

2. The Model The general conception of the model is sketched in Fig. I. A: and A: are the two main nutrients in the medium: for example, AT could be a nitrogen source and AT the carbon-energy source. Once inside the cell those nutrient5 are denoted as A, and A?. A, and A2 then react enzymically to form precursors, P, which consist of amino acids, nucleotides, etc. P is further reacted to form macromolecules M (proteins, RNA, DNA, polysaccharides. and lipids). Since all of these above reactions require energy, part of A2 transported into the cell is decomposed to E (wastes such as COz, H,O) to suppI) the energy needed. E formed then diffuses back into the medium and. once

A:-

A;-

1.“.

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outside, is denoted as E*. These observations can be written on a mash baiis as a series of pseudochemical reactions enzymically, i.e. M, catalyzed.

crA, -t/M2 + P, P+

M:

(1) (2)

M

where a and /I are stoichiometric coefficients. It is also assumed that ‘/ gr-awns A, arc consumed for energy in the formation of one gram of P and S grams A, for the formation of one gram of M. The model further makes the following assumptions. (1) Cellular components such as sulfur, phosphorous, etc. arc available in great excess compared to cell’s needs and need not be explicitly accounted for; (2) The internal reactions of the ccl1 arc not limited by diffusion within the cell; (3) A: is taken into the cell by facilitated transport. R,,, the rate of transport of A, per unit of membrane surface, can bc cxpresscd in the form (Cirillo, 1962) :

where ,4, = mass of substance A, in the cell at time t, KI = transport coefficient for A,, C,*, = external concentration of A,, Kz, = saturation constant for AT transport, and V = cellular volume of a single cell at time t. The actual mechanism of uptake of cations (particularly NH:) has not been well established although a mechanism similar to ion exchange is possible (Damadian, 1973). Facilitated transport is based 011 an absorptiondesorption mechanism as is ion exchange. Transport based on equation (3) will show feedback effects. A rise in intracellular A, concentration will decrease the rate of A, uptake. (4) At is taken up through the mechanism of active transport which is the transport of nutrients into the cell against a concentration gradient with the consumption of energy provided by the cell. Glucose is known to be transported across bacterial membranes via an cnzyrnic system and to be under apparent feedback inhibition by an end product of glucose metabolism (Kepes, 1964; Kaback & Hong, 1973). Consequently, we poslulatc that the rate of transport of AT into the cell per unit membrane area, R,,?, can bc written as

where A, = the mass of substance A, in the cell. 1’ =: maximum rat2 oi transport, C,:, = external concentration of A,, K,:.. Kb,, L saturation constants, Ki = inhibition constant due to internal concen;ration of A,,

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bf = the total mass of macromolecules in the cell at time t. The term (&Z/V)/&, + M/V) is added to satisfy the expectation that the rate of transport would be negligible when there is no catalyst (M) present. For the cases considered in this paper this term is saturated with respect to M/V and equals approximately one. Parnas & Cohen (1976) have also used an uptake e.upression that includes a similar inhibition term. Since active transport requires energy, we require that 5 g of A, be consumed to provide the energy to transport 1 g of A, into the cell. 15) The transport of E to the ouside is inherently very rapid; consequently, no E will accumulate in the cell and there will be no inhibition of any intracellular reactions by waste products. (6) The cell density, p or gram cell dry mass/cm3 of cell volume, is constant throughout the cell cycle. Constant density has been observed for 1:. coli, Proteus vulgaris, and Salmonella typhimurium throughout the division cycle (Schaechter, Williamson, Hood & Koch, 1962). [This, however, is not general as Mitchison (1971) reports density variations in yeast and Streptococwrfaecalis.] The following relationship for cell density will be used : T = pV or V = T/p, (5) where T is the total cell dry mass at time t. In addition to the above assumptions the model will be formulated with the following conventions : (I ) a dot above a quantity indicates the time derivative of that quantity. e.g. A, = dA,/dt; (2) .4 prime indicates a dimensionless mass quantity achieved by dividing the component mass by the total cellular dry weight at the beginning of the cell cycle. (3) Let b be the surface to volume ratio for an individual cell. (For a cell with geometries other than that of an infinitely long filament b will vary with time.) The component mass balances then can be written as:

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where k, and g(m are rate constants for precursor and macromolecule formation and Ki is the saturation constant for component j. Equations (6) through (10) are hypothesized to be the result of many enzymic reactions. The factor [M’/T’/(K,/+M’/T’)] is added as in equation (4) to satisfy the expectation of zero reaction rate with no enzyme present and will be approximately constant for the cases discussed in this paper. The use of multisubstrate rate form is not new; a double-substrate rate form for the growth of micro-organisms has been successfully used by Megee, Drake, Fredrickson & Tsuchiya (1972). The boundary conditions for the above set of differential equations can bc obtained as a consequence of the assumption that the culture is in balanced growth (Campbell, 19.57). A typical cell in balanced growth culture can bc postulated to have all its components doubled at the same time r, when cell division is expected to occur.

3. Solution Strategy for Filamentous Cells-Linear

Model

The above non-linear equations reduce to a linear form for filamentous cells growing under A, limiting conditions with P also in reaction limiting concentrations. For a long filamentous cell where the end area is a negligibly small portion of the cell surface area and where the cell grows in length only. b will-be approximately constant. Under these conditions equations (6) through (10) simplify to : A’, = -(K;‘+ak;)A; A; = -[k;p~+j?)]A;

+K;T’.

-+@‘+[(I M’ = p,P’,

(II)

-r)R,,h/p]T’.

(12) (13)

P’ = k;A; -p;P’, i-’ = -(K’,‘+yk;)A;

-+,!,,P’+[K;

(14) +(l-r)R,,bjp]T’,

(1.5)

where k,lp

1 + P’/T’

(16)

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-- ---- ---



(17)

If we assume that &, k’,, K;, KY, and RA> are approximately constant during the cell cycle, equations (ll), (l4), and (15) constitute a set of lineal first-order differential equation s with constant coefficients. The boundary condition5 are : PI,=, = 2P’j,,, = 2lJ;, A’,J,=, = 2A’;[,,, T’j,=, = 2T’],=”

= 3A;“. = 2.

(20)

It is desired to solve for the initial mass concentration A;,, Pb, and the doubling time z given the coefficients in the differential equations. This is not simple because the coefficients also involve the initial concentrations. As it turns out, however, for this set of linear first-order differential equations imposed by the boundary conditions stated, the solution reduces to one exponential term (Appendix B), that is: A;(f)

wlhere 1) is the specific growth

= it;, e”‘,

(21)

P’(t) = Pb e”‘.

(22)

T’(f)

(23)

=;I e@,

rate (hr-‘).

This implies that:

M’(f)

= Mb e”,

(24)

A;(t)

= A;, e”‘.

(25)

For an infinitely long filamentous celi all the ceil’s components and the cell mass itself would grow exponentially. Consequently, the mass concentration of each component in the cell would be constant during the cell cycle and equal to its initial concentration as shown above. With this in mind, we can come back to the set of equations (1 l)-(16) and solve th;ern by one simple integration for each equation. The task of computing the initial mass concentrations of cell components given the characteristic constants of the system is greatly simplified by this approach. Solving these equations and applying the boundary conditions, we obtain the following relations : t-26) A;, = @A,. blp-~k’dl~~ 4,

= CU-5)R+p-(y+W,

-h.&l/r~,

(27)

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Mb = A,/P, Pb = e; -&J/II

SFIULER

(28) 1

i29)

where :

P = CRA,+(~-~)RA~I~/P-Y~;-~~~

(30)

and k;, ~6, RA2, and R,, are evaluated in terms of initial concentrations. The values of the initial concentrations are determined by a trial-and-error method using equations (26) through (30) and a simple digital computer program with the constraints :

0 < A;,, A;,, Pb, Mb < 1.0 and A;, < WP. 4. Solution Strategy for the Non-linear Model The non-linear model [equations (6)-(lo)] applies to rod-shaped and spherical cells as well as for filamentous cells in environments when the assumptions used for the linear model will not hold. For bacilliary cells it was observed that growth would occur only by elongation (Marr et al., 1966). Spherical cells are assumed to maintain that geometry throughout the growth cycle. Thus, b will be a function of time for both rods and spheres. Numerical techniques were required for the non-linear model. The method is to use the initial concentrations obtained from the linear model as a first guess for solving the set of equations (6)-(10). The system of differential equations is then solved numerically on a digital computer using a Rutta-Kunge and Predictor-Corrector combined method (McCracken & Dorn, 1964). Cell growth as a function of time is thus simulated. As the amount of M’ is doubled, all components are divided by T’ to obtain a new set of initial conditions. This is similar to cell division in unbalanced growth. The process then starts over again until the requirement that every component is doubled at the same time is met. This method converges rather rapidly (from two to six iterations for a relative accuracy of about 0.5%). Although some modification can be used to speed up the convergence process (McCracken & Dorn, 1964), the approach discussed above has a physical significance. This method may be used to simulate the comparative changes in cell size and composition the cell might experience when exposed to a new abiotic environment. Values of the constants used in the calculation are tabulated in Table 1. These values are expected to be typical of a small fast growing bacterium. Most values are surmised from existing data on growth and composition. The value of K,, however, was varied until the cell had typical values for the

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1

Values of constants usedfor calculation Parameters

Values 0.3 g/cm3 2~10-~ cm/h 3.6~: 1O-5 cm/h 4.5 hr3.0 hr-

1 1

0904 (dimensionless) 0402 0.01 0.1

0.05

003001 0405 0.1 0.9 0.4 0.1 0.4

percent of the macromolecular fraction. Independent variables were Cz, and C$ and were considered constant throughout the division cycle. A constant external environment is consistent with the assumption of balanced growth and can be obtained in a steady-state chemostat. The justification for the choice of constants used is given in Appendix A.

5. Results and Discussion

This paper will be restricted to results obtained with A, as the limiting substrate and A, in great excess. The solution of this problem is simpler and less expensive. The same type of results have been obtained for the A, lirniting cases checked. The predictions of the model concerning the effect of limiting substrate variations on specific growth rate was first examined. Such prediction can be easily checked with well established observations. If the model and these data were not consistent, the model would have little creditability when extended to predictions on growth patterns where the data are not so well established. Simple computer experiments were performed to examine predictions on specific growth rate.

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The relationship of specific growth rate to the csternal concentration of‘ A, is given in Fig. 2 for various cell geometries. All cells show saturation type kinetics but the maximum growth rate, pmaxr achieved is a function of cell type. Filamentous cells have the highest I)(,,,~~since the surface to vol~~mc ratio, b, is constant throughout. Since b decreases during the cell cycle, rod shaped cells show a lower pmax. A spherical cell of the same initial vol~~mc as the bacillus cell above (length to diameter ratio = 3: 1) shows an even lower r(lmaxdue to the lower average 6. The effect of size is shown in Fig. 3 for spherical cells using a LineweaverBurk type plot. Similar results were found with filamentous and rod-shaped cells. The deviation from linearity at high C,*, indicates a more sudden change from first order to zero order kinetics than would be predicted from Monod, (1949) kinetics. This rapid transition has been observed for many organisms and a “three constant” model (Dabes, Finn & Wilke, 1973) has been proposed that accounts for this behavior. Their model can be derived by considering diffusion control of growth or as a consequence of having two slow reactions in a sequence of intracellular enzymatic reactions. As might be anticipated, larger cells grow more slowly because of the decreased surface to volume ratio, b. Adams & Hansche (1974) have found that for yeast growing under conditions of nutrile transport limitation the surface to volume ratio apparently controls the rate of cell reproduction. Larger cells grew more slowly as in Fig. 3. If the rate of utilization of a nutrient is limiting, then cell size did not control the growth rate. Extension of the linear portion of the curves results in a plot analogous to non-competitive inhibition for enzyme kinetics. Since the model could make reasonable predictions of gross cultural behavior, it was extended to predictions on the mode of individual cellular growth. For long filamentous cells whose b (surface-volume ratio) is approximate]) constant during the cell cycle, cell mass and all other components grow exponentially at the same specific growth rate. This behavior was shown earlier in this paper to be exactly true for the case of very low intracellular concentrations of A, and P with a high concentration of AZ. Computer experiments show that exponential behavior (to at least three significant figures) is generally observed irrespective of the assumption concerning intracellular concentrations. For a filamentous cell in balanced growth the concentration of the cell’s components would remain constant throughout the cell cycle. If it is postulated that the mechanism for cell division results from the two contradictory forces between the tendency for cell to grow in size and the tendency for cell to keep its intracellular components at the same proportion (Dean & Hinshelwood.

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I;IG. 2. The specific growth rate is plotted against the dimensionless corccrttration of the limiting substrate. Growth of cells with three different geometries arc shown for comparison. Filamentous and bacillus cells were assumed to have the same initial surface to volume ratio h = 6 x 10” cm-‘. Rod shaped and spherical cells were assumed to have the same initial volume, V, = 0.785~,3. To insure that the cell is A, limited CE,,i/j was chosen as 200 C,*,/p throughout. I unit = 3.2:: 10m4.

12IO-;18 r T

4‘,$/‘,,

/’ ,’ I ,.I / ,’ I ,,/’ ‘;

FIG. 3. Lineweaver-Burk plot for spherical cells with different initial volumes: one. two, and four times 0.785 ~2. The straight line portions of the curves were extended. They coincide at a point on the horizontal axis, which is an analogous lo non-competiti,+,c inhibition enzyme kinetics. 1 unit = 3115.

426

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1966), this would mean that a filamentous cell will grow indefinitely without dividing. Perret (1960) also stressedthis point. The phenomenon has been observed for many bacterial cells, as Aerobactcr aeropvzes (Dean & Hinshelwood, 1966, p. 383). Also, observations on the growth pattern of fiiamentous cells (unbalanced conditions) have shown exponential growth (Kubitschek, 1970b). For bacillary and spherical cells whose b’s are decreasingas the cell grows, growth is neither exactly linear nor exponential. Increase in cell mass,and therefore volume if cell density is constant, is found to be close to linear growth (< 1% deviation) for about two-thirds of the cycle; synthesis of macromolecules is on the other hand closer to exponential growth (with a relative difference at less than I %) (Fig. 4). These growth patterns are in accord with experimental data for nonfilamentous cells e.g., Kubitschek, 1970a. Thus, a feedback model appears to be as capable of explaining the observed growth patterns as the constant number of transport sites model is. The feedback model also predicts, for cells with geometries other than long filament, a fluctuation in the size of precursor pools and of macromolecules between divisions. Pool size of precursors first increasesand then decreases during the cell cycle; the reverse is true for the macromolecular portion of the cell (Fig. 5). This ty’pe of fluctuation in cell composition during the cell

FIG. 4. Increase in cell mass (T) and macromolecular mass (M) during the cell cycle. The solid lines are predicted by the model. The dotted lines are for exactly exponential growth. The dashed line is for linear growth of a spherical cell mass during the first twothirds of the cell cycle determined by a least squares analysis.

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FIG. 5. Adjustment of intracellular levels of cell components as the cell was transferred to a new medium twice as rich in the limiting substrate. Note that essentially balanced growth was obtained after about three generation times.

division cycle has been observed for a number of organisms (Kubitschek, 1970b; Mitchison, 1971) but is probably not universal (Mitchison, 1971). Since much of the support for a linear growth pattern and a constantnumber-of-uptake-sites model has been deduced from the measurement of nutrient uptake using radioactively-tagged compounds (Kubitschek, 1968~; 1970~; Kubitschek et al., 1971), we have calculated the rate of uptake of A, in the computer experiments. The relative rate of A, uptake decreases during the first 10% of the cell cycle to a value approximately 85 to 90% of the original rate. Then the A, uptake rate recovered from this minimum and increased linearly. The initial rate of uptake was again achieved approximately one-third of the way through the division cycle. Kubitschek (1968a, 1970b) has correlated most of his uptake data with a straight line of zero slope for about 75 y0 of the cell cycle. The data generally scatter about 20% from this line. While the precision of the data may justify only a straight line correlation. other forms of uptake could exist that would not be distinguished by this experiment. For example, a rate of uptake which decreased slightly ( p,!,,.

This reduces to

Although K,,,/p was chosen to be smaller than K&p and normally A;, is lower than Pb, in some extreme region, the above ratio might be comparable to one. It was, therefore, thought to be prudent to assign a higher value fol k,lp. (4) Value for the maximum transport of AZ, v, was approximated from the data for glucose uptake measured by Brown & Barton (1974). Since the data were given as rate of uptake per cell, an E. coii with a size of0.5 ftm in diameter and 1*O pm in length was assumed to convert data into rate of uptake per unit of cell surface area. The value of (K,/p) (transport of AT) was manipulated until it yielded realistic values for the level of intracellular components. The saturation constant for A, uptake, K&, was set equal to 3 mg/l which is approximately the value Ezzell & Dobrogosz (1975) have measured for cc-methylglucoside (a glucose analog) uptake in a CAMP deficient E. coli mutant when cultured in high external levels of CAMP. Ki, the inhibition constant, was arbitrarily set equal to Kz,. The saturation constant, K,*,/p, for A, transport was chosen to be twice the saturation constant for AL reactions. This insured that feedback control of A, transport would occur only if the A, pool was sufficient to insure near maximal reaction rates.

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(5) The values for the saturation constants were assigned based on the expected relative concentrations of intracellular components. Since A, is required at 10% of the level, A,, the saturation constant for A,, was chosen as one-tenth of that for AZ. KAI, KA2, and K, were picked so that the pool level in the cell would be about 5 to 10% of the cell’s dry weight. KM values were chosen such that the macromolecular terms would be saturated and approximately constant during the cell cycle for all normal growth conditions. (6) It was assumed that 0.1 gm A, reacts with 0.9 g A, to form I .O g of‘ precursor (i.e. r = 0.1, /I = 0.9). This is close to the relative amounts of carbohydrates and nitrogen found in macromolcculcs of bacterial ccl]\ (Aiba, Humphrey & Millis, 1973). ‘~3.6. 4 are the amount of A, converted to wastes to obtain energy for cell metabolism. Values of y, 0) need be considered. There may bc zero, enc. or more realistic t’s, For a giccn 7j we require that FZ, = 0. Since Fii = 0 and alI other F,, # 0 (i # ,j). the only non-Lero component of Z, is ZUj. Example: z,,,i = I, F,, = 0 00 0 0 Fz2 0 22. F 33. Zo, 0 0 F,, Only when Z,, = Z,, = 0 then: ,_1

0.Z”, = F Zo, I[ 1

Therefore for ,j = 1, %,j =

1 I

1 0 0

where 1 is arbitraril>, set. Knowing 2, for a given T,~,we can calculate the desired initial condition

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PATTERNS

vector by : so that j(zj) = cry, at the time t = zj. Note an important point that since the a-ling time was found to be: ?j = In a/Sj, this implies that for a possible set of initial conditions to satisfy the boundary conditions, the solution is always reduced to one exponential term. For example, the general solution for one component is y(t) = i

Ui es!‘.

i=l

One of the cc-ling times is zli = In g/S,. We then have: Y(Q) = NY(O)

(BlOa)

2”” = a jl

(BlObI

or igl

ui

(ail.

Let’s take the kth term out of the summation

on both sides:

uk eSkrk+ i ai eSiTk= aa,+cr i (a,), i=i’ i=i’ where i’ = 1, 2 . . . n, except k. Since T, = In cx/S, * uk eSkrr = cia,. Equation (B12) becomes i$, 4 eT

E2 $, ui.

(B12)

(B13’)

Since zk is a fixed number, Si varies with each coefficient and eSirk # c1 The only way equation (B13) is satisfied is that all coefficients ai have to be identically zero: ai E 0. The solution from equation (B9) becomes y(t) = uk es”‘. 0314) APPENDIX

C

Nomenclature

Al A,: b

mass of A, (nitrogen source) in the cell (g) mass of A, (carbon source) in the cell (g) ratio of surface area to volume for the cell at time, r (cm-‘) 29

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concentrations of A, and A, in the abiotic environment k/cm31 mass of E (wastes) in the cell (g) transport coefficient for the rate of A, uptake (g/cm’ h) rate constant for the formation of P from A, and A, (g/h cm31 KA,, KAL, KM, K, saturation constant for A,, AZ, M, and P, respectively

KG

K M3 Ki A4 P R Al R A2 s T V

Wm3)

saturation constant for the transport of A, into the cell (g/cm31 saturation constant for the active transport of A, into the cell (g/cm3) saturation constant for M in the active transport of A, into the cell (g/cm”) inhibition constant due to internal concentration of A2 (g/cm31 mass of M (macromolecules) in the cell (g) mass of P (precursors) in the cell (g) rate of A, uptake per unit surface area of the cell (g/h cm”) rate of A, uptake per unit surface area of the cell (g/h cm*) surface area of the cell (cm2) cell dry mass (g) cell volume (cm3)

Greek letters a B Y

fraction of 1 g of A, that is required to form 1 g of P fraction of 1 g of A, that is required to form 1 g of P fraction of 1 g of A, consumed to supply energy for the synthesis of 1 g of P fraction of 1 g of A, consumed to supply energy for the active transport of 1 g of A, fraction of 1 g of A2 consumed to supply energy for the synthesis of 1 g of M cell density (g cell dry mass/cm3 cell volume) specific growth rate (h-l) rate constant for the formation of M (g/h cm3) doubling time (h) maximum rate of transport of A, into the cell (g/h cm’)

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Subscripts 0 Superscripts I

implies concentration

of a component

at cell age 0

indicates a component mass divided by the initial cell mass, (T,). (A prime when used with a rate constant indicates an abbreviated form defined appropriately in the text, i.e. equations (16)-(19).) indicates a material in the abiotic environment indicates time derivative

29*

Predictions of cellular growth patterns by a feedback model.

I. rheor. Biol. (1977) 68, 415-435 Predictions of Cellular Growth Patterns by a Feedback Model S. V. Ho AND M. L. SHULER Sclraol of Chemical Engitzee...
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