Journal of Applied Bacteriology Symposium Supplement 1992,73, 09s-95s

ModelIing microbial ecosystems M.J. Bazin and J.I. Prosser’



Microbiology Group King’s College London and Department of Molecular and Cell Biology, Marischal College, University of Aberdeen. UK I

Mechanistic vs predictive models, 92s Multicomponent models, 92s Conclusions, 94s

1. Introduction, 89s 2. An example, 89s 3. The specific growth rate, 90s 4. Growth as a function of temperature, 91s

5. 6. 7. 8.

1. INTRODUCTION

itself which also serves as a mechanism for evaluating the efficiency of the model. The other type of model we have termed mechanistic. The function of such models is to aid in our understanding about how the system works. The prototype in this case is an experimental system. Mechanistic models serve as vehicles for generating predictions from hypotheses. Experiments can be undertaken to test these predictions. Experiments are performed with the prototype which represents some aspect of the ecosystem under stud! but differs from the natural ecosystem in that one or more parameters can be controlled by the person undertaking the experiment. In this sense then we can define an experimenl as an operation to test the predictions of a hypothesis Occasionally, hypotheses can be rejected because such experiments have already been performed or because the predictions are not physically possible. In the modelling work that we have undertaken, we have mostly been concerned with mechanistic models. Therefore much of this article will be about such models.

The term ‘mathematical model’ is usually taken to mean a set of equations which are supposed to represent what has been termed a prototype (Salmon & Bazin 1988). The prototype is not actually the system under study but a representation of it. For the purposes of this article, we would like to discuss mathematical models of microbial ecosystems using the simplifications shown in Fig. 1 in which two sorts of model are indicated. The one that has the most direct association with the system under study we have called a predictive model. The function of such a model is to forecast future events. In the context of food microbiology, for example, a model of this type may be used to predict the shelf-life of a product under a given set of environmental conditions. T h e prototype for the model is the ecosystem

References, 94s

Experi rnent o I microcosm

2. AN EXAMPLE

Environmenta t e s t system

Fig. 1 Diagrammatic representation of mechanistic and predictive models and the systems used to test them

Correspondence to :ProJ M . J . Bazin, Microbtology Group, King’s College London, Campden Hill Road, London W8 7AH, U K .

First, let us consider a simple example of the developmeni: of a mechanistic model concerning the growth of a singlespecies microbial population in a well-mixed ecosystem. We have observed that the cells divide into two and this result!; in an increase in the population density. Based on this observation we propose the hypothesis that the time between cell divisions is constant. We call this time the generation time and denote it by the symbol, t. It is difficult to test the hypothesis directly, for instance by observation under a microscope, because that would involve removing the cells from their well-mixed environment. Wc therefore develop an experimentally testable prediction a!; follows. Denoting the number of cells at time zero as N o , then after each time period T, the population will increase in the following progression :

No 2N0 4N0 8No 1 6 N o . . .

90s M.J. B A Z I N AND J . I . PROSSER

which can also be written as

2'N, 2 l N , 22N0 2 3 N , 24N0... It follows that after n generations, the population density becomes

population density. That is to say as the population increases in size, so the specific growth rate gets smaller. This forms the basis of what is called the Logistic equation :

1 dN r -r--N N dt k

p=--

N , = 2"N, As n is the time elapsed, t, divided by the generation time, we have

Thus dN r -= r N - - N 2 dt k

t

ln(N,/N-,) = - In 2 t

(3)

If we define p = IR 2/t, then In(NJNo) = p t

(4)

N , = No el''

(5)

and

N , _ 1-dN, d In -_ dt

N , dt

-P

Thus, p is the per capita rate of increase of the population or the specific growth rate and can be calculated by dividing the growth rate, dN,/dt, by the number of cells present at time t, N , . From eqn. ( 5 ) we obtain In N , = In

N, + p t

(9)

Here r and k are constants called, in population ecology, the intrinsic rate of increase and the carrying capacity of the environment. In microbiology r is the maximum specific growth rate and k is related to the yield. Solution of the equation gives the growth curve shown in Fig. 2. Here we get a much more realistic picture of what actually happens in a well-mixed microbial system. T h e sigmoid curve which tends to a constant equilibrium value, k, is a fairly good representation of what happens in a batch culture. There are a great many other equations that have been proposed to represent the specific growth rate of a microbial population. Probably the most widely employed is that developed by Monod (1942). This expression relates the specific growth rate of the population to the concentration, S, of a single, growth-limiting nutrient and takes the form

17)

This represents the prediction generated from the hypothesis. I t implies that the population will grow exponentially without limit and if the logarithm of the numbers are plotted against time, a linear relationship will result and the slope of the line produced will be p . So, starting with one cell of an organism with a fixed generation time of 30 min, the population density is predicted to be in about 130 h. This is the estimated number of protons in the observable universe so on the basis of the prediction the hypothesis that the generation time remains constant can justifiably be rejected.

where pmaxis the maximum specific growth rate, comparable to r in the Logistic equation, and K, is known as the saturation or Monod constant. Simulation studies have shown that it gives a good approximation to nutrient forming an enzyme-nutrient complex which breaks down to release the enzyme and form biomass (Bazin et al. 1990). Thus there is at least a quasi-mechanistic basis for the

3. THE SPECIFIC G R O W T H R A T E In this rather trivial exercise we have identified what is probably the key element in dynamic models of microbial growth and interactions. This is the specific growth rate. The hypothesis we have just rejected stated that the generation time uas constant. If this was the case, the specific growth rate would have been constant too. However, the hypothesis was rejected. If the specific growth rate is not constant it must change, presumably by getting smaller. The simplest new hypothesis we can propose makes the specific growth rate a linear, decreasing function of the

0

t Fig. 2 Growth of a microbial population at concentration, x, as

predicted by the Logistic equation

MODELLING MICROBIAL ECOSYSTEMS 91s

model and it has been used widely especially in association with chemostat studies. T h e disadvantage of the Monod equation is that it contains the variable, S, so that for a complete description of a microbial culture, in addition to an equation for microbial population density, another equation for change in S must be provided. This results in a set of two differential equations the solution of which is usually obtained by numerical integration. Another expression for specific growth rate, and one that seems to have found favour as a predictive model in food microbiology, is that developed in the last century by Gompertz to describe the growth of human populations (Zweitering et a/. 1990). In this case, the relationship takes the form p = ac . exp[ -exp(b - ct)]exp(b - ct)

( 1 1)

T h e parameters a, b and c have been interpreted in terms of the carrying capacity, the maximum specific growth rate and the lag phase in a batch culture by Zweitering et al. (1990). We have not been able to find a mechanistic basis for this equation and, despite its popularity, reservation should be expressed about its use in modelling microbial growth. Equation ( 1 1) contains three exponentials each built on the other. This means that the solution is very sensitive to the parameters. Indeed, a very small change in one of them can have a very large effect on the result. One reason for the popularity of the Gompertz equation is that it usually gives a better fit to batch culture data than the Logistic equation and does not require numerical integration as is usually the case with the Monod function. The reason for the former is that the Gompertz equation contains more parameters than the Logistic equation.

4. GROWTH AS A FUNCTION TEMPERATURE

OF

Nevertheless, the Gompertz equation has been used with apparent success in modelling the effect of temperature, pH and NaCl concentration on microbial growth (Gibson et af. 1988). Other predictive models for the effect of temperature have been developed by McMeekin et al. (1988). Several predictive empirical models were developed by these workers and they can be compared with the mechanistic model, developed much earlier by Topiwala & Sinclair

(1971). One of the equations developed by 'McMeekin et al. (1988) to describe the effect of temperature on growth is: r d~ = b(T - Tmin>{l- ex~Cc(T- TrnaJl) (12) where T is temperature, Tminand T,,, are the minimum and maximum temperatures at which growth occurs and b and c are constants. This equation is a refinement of a pre-

vious model called the square root model and, probably, has now been superseded by other models. Within our definition, this is a predictive model and, within the temperature ranges tested, appears to work well. Topiwala & Sinclair (1971) took a different approach. Rather than embarking on what some people would describe as a curve-fitting exercise, these authors used the Monod expression to describe microbial growth and proposed that the coefficients in this model, i.e. the maximum specific growth rate and the saturation constant, were Arrhenius-type functions of temperature. For example, K, was replaced by :

--1 - K exp( K,( r )

-A)

where T is absolute temperature, R is the gas constant, E is the activation energy and K is described as a 'collision' factor. Topiwala & Sinclair (1971) used equations similar to this one to replace the coefficients in the Monod equation. These were then used in two equations of balance to represent the experimental system they used. This was a chemostat and the equations described the change in biomass and limiting nutrient concentration expected on the basis a,f the hypothesis he proposed, i.e. that the coefficients were Arrhenius functions of temperature. The resulting model consisted of a set of two differenti;il equations. As is common in many such biological modek the equations were of a type called non-linear. This usually implies that no direct solution of the equations exists and numerical methods must be used in association with a cornlputer to approximate a solution. In order to do this-to get a prediction that can be tested experimentally-values fcbr all of the constants in the model must be estimated. Topiwala & Sinclair (1971) undertook this parameter estimation by growing Klebsiella pneumoniae in a chemostat and emmating steady state biomass and nutrient concentrations at different temperatures. By equating the differential equations of the model to zero, they were able to obtain algebraic expressions relating predicted steady-state values to the coefficients of their model. With such relationships and their steady-state data they applied statistical methods to estimate the values of the coefficients. They then had a set of two differential equations with estimates of the coefficients they contained. In order to test the model, they simulated the effect of changing the tern!perature of their system by numerical integration. Thns provided predictions of how the biomass and nutrient concentrations would behave during the transient period after a temperature change. The predictions were then compared with the results of experiments. As was expected, steady-state data fitted the experimental results well. However, when the temperature was

92s M.J. B A Z l N AND J.I. PROSSER

increased from 25‘ to 35”C, the change in the experimental system from one steady state to the new one was slower than that predicted by the model. Topiwala & Sinclair (1971) modelled this lag by employing a time-dependent function to describe what the! termed the ‘effective’ temperature (7“).In practice, this meant replacing temperature in their equations with the function:

Here T is a retardation constant and when it was assigned a value of 1.43, the transient data after the increase in temperature were represented reasonably well. However, the final steady state was over-estimated. A complication arose when temperature shifts in the opposite direction were tested. When the temperature was reduced from 35‘ to 2 5 T , the best value of z was found to be 0.83. Thus the behaviour of the system was not symmetrical N-ith respect to shift-up and shift-down in temperature.

5. MECHANISTIC VS PREDICTIVE MODELS

The mechanistic approach of Topiwala & Sinclair (1951) seems to leave more questions unanswered at the conclusion of their work than at the beginning and they did not obtain a particularly good fit to their data. On the other hand, the more arbitrary approach of McMeekin et a / . (1988) yielded much tighter fitting curves. For predictive purposes, within the rangc of temperatures tested, the latter workers seem to have the better model. This leads us on to a basic question about modelling for which we have no answer. I t is commonly assumed that the closer a model is to describing the underlying mechanisms of a process, the better will be its predictive capacity. Is this assumption valid ? It does not seem to be supported by the work we have just described. We might also ask what is the role of mechanistic modelling? If we know quantitatively how factors such as temperature and osmotic pressure affect lag periods, spore germination and specific growth rate through an understanding of microbial physiology, we may be able to explain differences in effects on different groups of organisms. This in turn may lead to improved efficiency in sterilization and storage strategies. hlechanistic modelling provides the basis for such an approach but mechanistic models of cell phy-siology are often too complex for routine predictive use. It may be possible, however, to simplify such models or, alternatively, they may be of value in determining the degree of confidence we can place in simple predictive models, and the conditions under which they may be used.

6. MULTICOMPONENT MODELS

So far in this article we have been very restrictive in our treatment of ecosystem models. We have considered only single species growing in a well-mixed system and then only the dynamics of the situation. Most real ecosystems are not well mixed and contain more than one species. Interactions between the species must be considered and also the effect of gradients which occur in heterogeneous systems. Before saying something about these factors, let us refer back to the chemostat experiments of Topiwala & Sinclair (1971). Firstly, by using the Monod function which relates the concentration of a growth-limiting nutrient to specific growth rate, they were considering one part of the abiotic environment, i.e. the concentration of the nutrient. Secondly, by using a chemostat they were reflecting a very important physical aspect of most, but not all ecosystems; that is to say their thermodynamically open nature. isolated systems exchange neither energy nor matter with their surroundings. Artificially constructed microcosms have this property. Thermodynamically open ecosystems exchange both matter and energy. Most microbial ecosystems are of this type and in modelling them this property seems sometimes to be forgotten. However, it is an important property because the rate of exchange between a microbial ecosystem and its surroundings can considerably modify genetic, physiological and morphological properties of the microorganisms. For example, in chemostat culture the size of cells, their metabolic activity and their ability to undergo genetic transformation are all functions of the rate of flow of nutrient through the reaction vessel (Williams 1967; Sevinc et al. 1990). Closed systems exchange only energy with their surroundings. Batch cultures are usually regarded as closed and, in many cases, ecosystems consisting of micro-organisms growing on food may be regarded in a similar way. Food ecosystems, like the soil and many other environments, are not homogeneous. This has a significant effect with respect to the type of mathematics used in modelling the dynamics of such systems. Kinetic functions such as those describing specific growth rate need not alter but the dependent variables, such as biomass and nutrient concentrations, vary with respect to position as well as time. This can be illustrated with the diagrammatic representation of the rhizosphere shown in Fig. 3. The rhizosphere is the region of soil close to the plant root and is microbially highly active. In order to consider microbial population dynamics in this region we need to consider movement of nutrients from the root by diffusion and down the soil column by leaching. Position is specified by two coordinates, distance down and radially. These are independent variables. The third independent variable is time.

MOD ELL1NG M I C R O B I A L ECOSYSTEMS 93s

I

Movement of substrate In by leoching

--

-'l;t.l

3ut by diffusion

O u t by leaching /J

Fig. 3 Simplified representation of a rhizosphere ecosystem. The inner cylinder represents the plant root from and to which nutrients diffuse. The outer cylinder represents the region of the rhizosphere. The inset is a representation of a theoretical compartment illustrating flow and diffusion of microbial substrates

Dependent variables such as nutrient concentration and biomass change with respect to all three. The resulting equations are partial differential equations rather than ordinary differential equations such as those used by Topiwala & Sinclair (1971) for a well-mixed system. Non-linear partial differential equations are considerably more difficult to handle than ordinary differential equations although suitable computer routines are becoming more readily available. For many food ecosystems, sets of partial differential equations are likely to be the most appropriate. However, for a regular colony growing on a surface only one spatial co-ordinate may be necessary. The other peculiar feature of such a system, as with some biofilms, is that the appropriate measure of biomass may not be concentration but amount per unit area. With the introduction of more than one independent variable, theoretical consideration of ecosystem dynamics

becomes more complex. The equations for changes in concentration of a solute moving down a column of soil and their numerical solution are surprisingly complicatedl. However, because of the importance of such movemenl, numerous attempts have been made to construct predictive models for this and associated processes. An example o'f such a model is LEACHM, developed by Wagenet fir Hutson (1989) in the USA. The basic model contains no terms specifically relating to microbial activity. However, because it is so relatively easy to understand, we have been able to modify it to include such activity. A representative simulation of microbial predator-prey activity at three different depths of soil is shown in Fig. 4. The significant aspect of this result is the irregularity of the predicted behaviour of the prey and predator populations. The environment of most organisms includes other organisms and interactions between species are important characteristics of ecosystems. We have investigated theoretical aspects of microbial predation partly, at leasr, because the resulting equations are ideal for testing the robustness of models. Let us examine an example of just how complicated a simple ecosystem with microbial interactions becomes. Consider first of all a predator populatioin such as amoebae or ciliate protozoa feeding on a bacterial prey species in a well-mixed thermodynamically open ecosystem. As mentioned earlier, the behaviour of such a system is dependent on the flux through it. If we idealize the situation to that of a chemostat, the control parameter is the dilution rate, i.e. the rate of addition of medium divided by the volume of the vessel. At relatively high dilution rates, the predator and prey populations change smoothly and reach steady-state concentrations. At intermediate dilution rates, both populations show damped oscillations and, at low dilution rates, sustained oscillations called limit cycles occur. Thus both the quantitative and qualitative behaviour of the interaction depend on the dilution rate olf the system. In many natural situations, prey and predator populations will interact in a volume in which the dilution rate will vary. For instance, consider a soil pore after it has rained and it has filled with water. The leaching process will supply and deplete the ecosystem and gradually the pore will empty or fill when it rains again. An idealization of the latter case is a fed-batch culture in which the rate of supply of nutrient is constant but the volume increases from a minimum value to a maximum value. Thus the dilution rate starts out relatively large and then decreases in magnitude. A predator-prey system in a fed-batch culture may therefore start out changing smoothly with time, then start to show damped oscillations and finally oscillate continuously. The emptying and filling process that takes place in nature is mimicked by a repeated fed-batch culture in which the vessel empties to the minimum volume after the

94s M.J .

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maximum volume is reached and the cycle is repeated. At the end of the first cycle in such a system when the populations are oscillating continuously, the volume decreases and the dilution rate becomes relatively large. Thus the effect of the flux is to get the populations to change smoothly but they are unable to react quickly enough. Effectively, after the vessel empties each time, the initial conditions are different. This 'confuses' the organisms and the behaviour becomes erratic and non-repeatable. T h e sort of dynamics that result appear to be chaotic. Under such circumstances predictability is lost. Modelling in terms of mechanistic models with such dynamics cannot be effectively performed in the manner undertaken by Topiwala & Sinclair (1971) as results cannot be compared with predictions in the same way. In terms of predictive models, the situation is worse. If chaos occurs, the system becomes, by definition, unpredictable. From a practical point of view, the occurrence of chaos could be of great significance.

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We have attempted to present an overview of some aspects of ecosystem modelling that we think are of importance. We have not mentioned stochastic models in which the probabilities of events are examined or the ways in which mathematical analysis of some sorts of models sometimes helps to gain an understanding of the system under study such as Liapounov type analysis which has been used to investigate the relationship between biodiversity and ecosystem stability (Gardner & Ashby 1970). Application of mathematics to microbiology has increased over the past two or three years, especially in the formulation of mathematical models. We are sure this trend will continue and be of value both for learning about basic microbiological phenomena and for practical forecasting of future events. We are sure of this not just because we think that modelling is a sensible way of investigating biological phenomena but also because modelling research is cheap !

Fig. 4 Simulation of microbial predator-prey activity in the soil using LEACHM. T h e program of Wagenet & Hutson (1989) was modified by Elizabeth Scott, Department of Mathematics, King's College London, and the results are printed with her kind Substrate; , prey; ' . ' . ., predator. permission. (a) 0-200 mm; (b) 2 0 0 0 mm; (c) 400-600 mm. Profile details : Water flux density (mm/d) 0.1368E 03 Water content 0.2500E + 00 8. REFERENCES Substrate in flow (mg/l) 0.8330E + 02 N o crops B A Z I N ,M.J., G R A Y ,S . & R A S H I T ,E. (1990) Stability Indigenous substrate present properties of microbial populations. In Microbial Growth Dynamics ed. Poole, R.K., Bazin, M.J. & Keevil, C.W. pp. Initial profile data: 127-143. Oxford : IRL Press. Soil layer Substrate Prey Predator G A R D E N EM.R. R. & A S H B Y , W.R. (1970) Connectance of (mg/kg dry soil) 1 04000E + 00 0.1670E + 01 0.1670E + 01 large dynamic (cybernetic) systems: Critical values for stability. 2-16 0.0000E + 00 0.1670E + 01 0.1670E + 01 N u t u r e 228, 784-785. @ma* (l/d) 1440 9.60 G I B S O NA , . M . , BRATCHELL, N . & ROBERTS, T.A. (1988) K, (mg/dm3) 4.00 12.00 Predicting microbial growth : growth responses of salmonellae c 0.48 0.48 in a laboratory medium as affected by pH, sodium chloride and Y 0.40 0.50 storage temperature. International Journal of Food Microbiology Time step (d) 0.lOOOE- 01 6, 155-178. ?- -

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MODELLING M I C R O B I A L ECOSYSTEMS 95s

M C M E E K I N ,T . A . , O L L E Y , J . & R A T K O W S K YD.A. , (1988) Temperature effects on bacterial growth rates. In Physiological Models in Microbiology, Vol. 1. ed. Bazin, M.J. & Prosser, J.I. pp. 75-89. Boca Raton, Florida: CRC Press. MONOD,J . (1942) Recherches sur la Croissance des Cultures Bacteriennes. Paris : Hermann et Cie. S A L M O NI,. & B A Z I N ,M . J. (1988) The role of mathematical models and experimental ecosystems in the study of microbial ecology. In Handbook of Laboratory Model Systems f o r Microbial Ecosystems, Vol. 2. ed. Wimpenny, W.T. pp. 235-252. Boca Raton, Florida : CRC Press. S E V I N CM , . S . , B A I N B R I D G B.W. E, & B A Z I N ,M . J . (1990) Genetic transformation and cell morphology of Bacillus subtilis in Mg' +-limited chemostat culture. Microbios 62, 29-35.

T O P I W A LH. A, & SINCLAIR C.G. , (1971) Temperature relationships in continuous culture. Biotechnology and Bioengineering 13, 795-813. W A G E N E TR , . J . & H U T S O N ,J . L . (1989) L E A C H M : A process-based model of water and solute movement, transfirmations, plant uptake and chemical reactions in the unsaturated zone. Version 2.0. Continuum 2. New York State Water Resources Institute, Cornell University, Ithaca, New York. W I L L I A M S F, . M . (1967) A model of cell growth dynamics. Journal of Theoretical Biology 15, 19C207. Z W I E T E R I N GM, . H . , J O N G E N B U R G E R , I . , R O M B O U T S , F . M . & V A N ' T R I E T , K . (1990) Modeling the bacterial growth curve. Applied and Environmental itftcrobiologp 56, 1875-1881.

Modelling microbial ecosystems.

Journal of Applied Bacteriology Symposium Supplement 1992,73, 09s-95s ModelIing microbial ecosystems M.J. Bazin and J.I. Prosser’ ’ Microbiology Gr...
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