The Process Pathway Model of bacterial growth

The growth profile of microorganisms in an enclosed environment, such as a bioreactor or flask, is a well studied and characterized system. Despite a long history of examination, there are still many competing mathematical models used to describe an output of the microorganisms, namely the number of bacteria as a function of time. However, these descriptions are either purely phenomenological and give no intuition as to the biological mechanisms underlying the growth curves, or extremely complex and become computationally unfeasible at the population level. In this paper, we develop the Process Pathway Model by modifying a model of sequential processes, which was first used to model robustness in metabolic pathways, and demonstrate that the Process Pathway Model encapsulates many features and temperature dependence of bacterial growth. We verify the predictions of the model against growth data for multiple species of microorganisms, and confirm that the model generates accurate predictions on temperature dependence of bacterial growth. The model has five free parameters, and the simplifying assumptions used to build the model are built upon biologically realistic notions. The Process Pathway Model accurately models a microorganism’s growth profile at an intermediate level of complexity that is computationally feasible. This model can be used as both an conceptual model for thinking about systems of bacterial growth, as well as a computational model that operates at level of complexity that is amenable to large scale simulation. This balance in accuracy and intuitiveness was accomplished by using realistic biological assumptions to simplify the underlying biology, which may point the way forward for future models of this type.


Introduction
The growth profile of microorganisms in an enclosed environment, such as a bioreactor or flask, known 1 as batch culture, is a commonly used and well studied and characterized system [8,20]. It has ap-2 plications to many fields, including food science, microbiology, experimental evolution, and bioreactor 3 engineering [4, 5, 15, 24]. However, despite a long history of examination, there are still many compet-4 ing mathematical models used to describe the output of the system, namely the number of bacteria as 5 a function of time. Furthermore, these descriptions are either purely phenomenological, which give no 6 biological intuition into the mechanisms underlying the growth curves, or extremely complex, becoming 7 computationally unfeasible at the population level. The most common empirical models are the Logistic, 8 Gompertz, van Impe, and Baranyi-Roberts, which describe growth profiles as concentrations over time integration. By adding temperature dependence to the simple model of Kacser and Burns [11], our model, 15 termed the Process Pathway Model, encapsulates many features and temperature dependence of bacterial 16 growth. Our results demonstrate that a relatively simple mechanistic model can be used to accurately 17 describe and predict the dynamics of a complex biological system while maintaining biological relevance, 18 computational tractability, and broad applicability.

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Previous models of bacterial growth have been developed to predict bacterial growth rates as functions 20 of time and temperature [2,3,9,21,22,25]. Additional features that models are designed to predict are lag 21 time, which is the time the bacteria spend in a stationary state before growing; carrying capacity, which 22 is the maximal concentration that the bacteria grow to; and maximal growth rate, all as functions of 23 variables such as temperature and pH. In addition to closed-form equations, differential equation models 24 have also been applied to the problem of bacterial growth rates. The most common of these models is the  Baranyi-Roberts and van Impe models have had some success is predicting population level phenomena 30 by approximating underlying processes, but these models remain highly phenomenological with little 31 ability to extract biological insight into key parameters, such as the lag time [1,14,[17][18][19]25]. Other   Each process in the chain of N processes is governed independently by Michaelis-Menten kinetics, 59 3 resulting in a flux, φ i between processes given by: for i = 1, 2, 3, ..., N , where S N is the concentration of the output of process N . For this model we 61 assume no flux into S 0 and the flux out of S N is given by a linear death rate. The dynamics for the 62 variables S i is then given by: for i = N ( Fig. 1 Panel A). S 0 represents the concentration of the limiting resource, and is subsequently 66 reduced as it is consumed by the process generating S 1 . The initial value of S 0 is a free parameter, while 67 the initial value of all other S i are set to 0 to model an initial state before growth has begun. The  Fig. S1).

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Here, the final process is taken to represent the progress of the final metabolic pathway in the chain, in 71 this case that of reproduction. The parameter D represents the natural death rate of the population.  The primary features of the temperature dependence that was salient for the model were the peak and 82 the minima of the temperature dependence.

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It is important to note here that while the equation for temperature dependence contain multiple  In order to determine the optimal value for N , the number of processes in the chain, we compared 91 experimental data from growth of Escherichia coli to the predictions of the model, yielding a optimal 92 prediction of N = 8 (Supplemental Figures: Fig. S2). This is in line with predictions of the diameter of

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Applying the Pathway Process Model to data obtained from E. coli at a single temperature resulted 98 in an accurate prediction of growth (Fig. 2 Panel A). The prediction was comparable to existing models of 99 bacterial growth (Supplemental Figures: Fig. S4). Furthermore, the application of the Pathway Process 100 model to data obtained of growth under a fluctuating temperature profile (Fig. 2 Panel B) fit the data 101 as well as existing phenomenological models (Supplemental Figures: Fig. S5) 102 In order to test the ability of the model to predict the effects of temperature on the properties of 103 bacterial growth, we determined a single set of parameters that best matched the data for growth rate,  Table S1 is shown in (Supplemental Figures: Fig. S5). It should be for all values are given in Table S1.