Optimal protein production by a synthetic microbial consortium: coexistence, distribution of labor, and syntrophy

The bacterium E. coli is widely used to produce recombinant proteins such as growth hormone and insulin. One inconvenience with E. coli cultures is the secretion of acetate through overflow metabolism. Acetate inhibits cell growth and represents a carbon diversion, which results in several negative effects on protein production. One way to overcome this problem is the use of a synthetic consortium of two different E. coli strains, one producing recombinant proteins and one reducing the acetate concentration. In this paper, we study a mathematical model of such a synthetic community in a chemostat where both strains are allowed to produce recombinant proteins. We give necessary and sufficient conditions for the existence of a coexistence equilibrium and show that it is unique. Based on this equilibrium, we define a multi-objective optimization problem for the maximization of two important bioprocess performance metrics, process yield and productivity. Solving numerically this problem, we find the best available trade-offs between the metrics. Under optimal operation of the mixed community, both strains must produce the protein of interest, and not only one (distribution instead of division of labor). Moreover, in this regime acetate secretion by one strain is necessary for the survival of the other (syntrophy). The results thus illustrate how complex multi-level dynamics shape the optimal production of recombinant proteins by synthetic microbial consortia.

Resource allocation accounts for the large variability of rate-yield phenotypes across bacterial strains

Different strains of a microorganism growing in the same environment display a wide variety of growth rates and growth yields. We developed a coarse-grained model to test the hypothesis that different resource allocation strategies, corresponding to different compositions of the proteome, can account for the observed rate-yield variability. The model predictions were verified by means of a database of hundreds of published rate-yield and uptake-secretion phenotypes of Escherichia coli strains grown in standard laboratory conditions. We found a very good quantitative agreement between the range of predicted and observed growth rates, growth yields, and glucose uptake and acetate secretion rates. These results support the hypothesis that resource allocation is a major explanatory factor of the observed variability of growth rates and growth yields across different bacterial strains. An interesting prediction of our model, supported by the experimental data, is that high growth rates are not necessarily accompanied by low growth yields. The resource allocation strategies enabling high-rate, high-yield growth of E. coli lead to a higher saturation of enzymes and ribosomes, and thus to a more efficient utilization of proteomic resources. Our model thus contributes to a fundamental understanding of the quantitative relationship between rate and yield in E. coli and other microorganisms. It may also be useful for the rapid screening of strains in metabolic engineering and synthetic biology.

Optimal proteome allocation and the temperature dependence of microbial growth laws

Although the effect of temperature on microbial growth has been widely studied, the role of proteome allocation in bringing about temperature-induced changes remains elusive. To tackle this problem, we propose a coarse-grained model of microbial growth, including the processes of temperature-sensitive protein unfolding and chaperone-assisted (re)folding. We determine the proteome sector allocation that maximizes balanced growth rate as a function of nutrient limitation and temperature. Calibrated with quantitative proteomic data for Escherichia coli, the model allows us to clarify general principles of temperature-dependent proteome allocation and formulate generalized growth laws. The same activation energy for metabolic enzymes and ribosomes leads to an Arrhenius increase in growth rate at constant proteome composition over a large range of temperatures, whereas at extreme temperatures resources are diverted away from growth to chaperone-mediated stress responses. Our approach points at risks and possible remedies for the use of ribosome content to characterize complex ecosystems with temperature variation.

Global dynamics of the chemostat with overflow metabolism

Fast growing E. coli cells, in glucose-aerobic conditions, excrete fermentation by-products such as acetate. This phenomenon is known as overflow metabolism and has been observed in a diverse range of microorganisms. In this paper, we study a chemostat model subject to overflow metabolism: if the substrate uptake rate (or the specific growth rate) is above a threshold rate (different from zero), then secretion of a by-product happens. We assume that the presence of the by-product has an inhibitory effect on the growth of the microorganism. The model is described by a non-smooth differential system of dimension three. We prove the existence of at most one equilibrium (or steady-state) with presence of microorganism, which is globally stable. We use these results to discuss the performance of chemostat-type systems to produce biomass or recombinant proteins.

Optimal bacterial resource allocation: metabolite production in continuous bioreactors

We show novel results addressing the problem of synthesizing a metabolite of interest in continuous bioreactors through resource allocation control. Our approach is based on a coarse-grained self-replicator dynamical model that accounts for microbial culture growth inside the bioreactor, and incorporates a synthetic growth switch that allows to externally modify the RNA polymerase concentration of the bacterial population, thus disrupting the natural process of allocation of available resources in bacteria. Further on, we study its asymptotic behavior using dynamical systems theory, and we provide conditions for the persistence of the bacterial population. We aim to maximize the synthesis of the metabolite of interest during a fixed interval of time in terms of the resource allocation decision. The latter is formulated as an Optimal Control Problem which is then explored using Pontryagin's Maximum Principle. We analyze the solution of the problem and propose a sub-optimal control strategy given by a constant allocation decision, which eventually takes the system to the optimal steady-state production regime. On this basis, we study and compare the two most significant steady-state production objectives in continuous bioreactors: biomass production and metabolite production. For this last purpose, and in addition to the allocation parameter, we control the dilution rate of the bioreactor, and we analyze the results through a numerical approach. The resulting two-dimensional optimization problem is defined in terms of Michaelis-Menten kinetics, and takes into account the constraints for the existence of the equilibrium of interest.

Enhanced production of heterologous proteins by a synthetic microbial community: Conditions and trade-offs

Synthetic microbial consortia have been increasingly utilized in biotechnology and experimental evidence shows that suitably engineered consortia can outperform individual species in the synthesis of valuable products. Despite significant achievements, though, a quantitative understanding of the conditions that make this possible, and of the trade-offs due to the concurrent growth of multiple species, is still limited. In this work, we contribute to filling this gap by the investigation of a known prototypical synthetic consortium. A first E. coli strain, producing a heterologous protein, is sided by a second E. coli strain engineered to scavenge toxic byproducts, thus favoring the growth of the producer at the expense of diverting part of the resources to the growth of the cleaner. The simplicity of the consortium is ideal to perform an in depth-analysis and draw conclusions of more general interest. We develop a coarse-grained mathematical model that quantitatively accounts for literature data from different key growth phenotypes. Based on this, assuming growth in chemostat, we first investigate the conditions enabling stable coexistence of both strains and the effect of the metabolic load due to heterologous protein production. In these conditions, we establish when and to what extent the consortium outperforms the producer alone in terms of productivity. Finally, we show in chemostat as well as in a fed-batch scenario that gain in productivity comes at the price of a reduced yield, reflecting at the level of the consortium resource allocation trade-offs that are well-known for individual species.

In nature, microorganisms occur in communities comprising a variety of mutually interacting species. Established through evolution, these interactions allow for the survival and growth of microorganisms in their natural environment, and give rise to complex dynamics that could not be exhibited by any of the species in isolation. The richness of microbial community dynamics has been leveraged to outperform individual species in biotechnological production processes and other processes of high societal value. Yet, in view of their complexity, natural communities are difficult to study and control. In order to overcome these issues, a rapidly growing research field concerns the rational design and engineering of synthetic microbial consortia. Despite the great potential of synthetic microbial consortia, and significant efforts devoted to their mathematical modelling and analysis, a detailed understanding of how enhanced production can be achieved, and at what cost, is still unavailable. In this work, based on a quantitative model of a prototypical synthetic microbial consortium, we determine precise conditions under which a consortium outperforms individual species in the production of a recombinant protein. Moreover, we identify the inherent trade-offs between productivity and efficiency of substrate utilization.

Reducing a model of sugar metabolism in peach to catch different patterns among genotypes

Several studies have been conducted to understand the dynamic of primary metabolisms in fruit by translating them into mathematics models. An ODE kinetic model of sugar metabolism has been developed by Desnoues et al. (2018) to simulate the accumulation of different sugars during peach fruit development. Two major drawbacks of this model are (a) the number of parameters to calibrate and (b) its integration time that can be long due to non-linearity and time-dependent input functions. Together, these issues hamper the use of the model for a large panel of genotypes, for which few data are available. In this paper, we present a model reduction scheme that explicitly addresses the specificity of genetic studies in that: (i) it yields a reduced model that is adapted to the whole expected genetic diversity (ii) it maintains network structure and variable identity, in order to facilitate biological interpretation. The proposed approach is based on the combination and the systematic evaluation of different reduction methods. Thus, we combined multivariate sensitivity analysis, structural simplification and timescale-based approaches to simplify the number and the structure of ordinary differential equations of the model. The original and reduced models were compared based on three criteria, namely the corrected Aikake Information Criterion (AIC_C), the calibration time and the expected error of the reduced model over a progeny of virtual genotypes. The resulting reduced model not only reproduces the predictions of the original one but presents many advantages including a reduced number of parameters to be estimated and shorter calibration time, opening new promising perspectives for genetic studies and virtual breeding. The validity of the reduced model was further evaluated by calibration on 30 additional genotypes of an inter-specific peach progeny for which few data were available.

Modeling the bioconversion of polysaccharides in a continuous reactor: A case study of the production of oligogalacturonates by Dickeya dadantii

In the quest for a sustainable economy of the Earth's resources and for renewable sources of energy, a promising avenue is to exploit the vast quantity of polysaccharide molecules contained in green wastes. To that end, the decomposition of pectin appears to be an interesting target because this polymeric carbohydrate is abundant in many fruit pulps and soft vegetables. To quantitatively study this degradation process, here we designed a bioreactor that is continuously fed with de-esterified pectin (PGA). Thanks to the pectate lyases produced by bacteria cultivated in the vessel, the PGA is depolymerized into oligogalacturonates (UGA), which are continuously extracted from the tank. A mathematical model of our system predicted that the conversion efficiency of PGA into UGA increases in a range of coefficients of dilution until reaching an upper limit where the fraction of UGA that is extracted from the bioreactor is maximized. Results from experiments with a continuous reactor hosting a strain of the plant pathogenic bacterium Dickeya dadantii and in which the dilution coefficients were varied quantitatively validated the predictions of our model. A further theoretical analysis of the system enabled an a priori comparison of the efficiency of eight other pectate lyase-producing microorganisms with that of D. dadantii Our findings suggest that D. dadantii is the most efficient microorganism and therefore the best candidate for a practical implementation of our scheme for the bioproduction of UGA from PGA.

Principal process analysis of biological models

BACKGROUND

Understanding the dynamical behaviour of biological systems is challenged by their large number of components and interactions. While efforts have been made in this direction to reduce model complexity, they often prove insufficient to grasp which and when model processes play a crucial role. Answering these questions is fundamental to unravel the functioning of living organisms.

RESULTS

We design a method for dealing with model complexity, based on the analysis of dynamical models by means of Principal Process Analysis. We apply the method to a well-known model of circadian rhythms in mammals. The knowledge of the system trajectories allows us to decompose the system dynamics into processes that are active or inactive with respect to a certain threshold value. Process activities are graphically represented by Boolean and Dynamical Process Maps. We detect model processes that are always inactive, or inactive on some time interval. Eliminating these processes reduces the complex dynamics of the original model to the much simpler dynamics of the core processes, in a succession of sub-models that are easier to analyse. We quantify by means of global relative errors the extent to which the simplified models reproduce the main features of the original system dynamics and apply global sensitivity analysis to test the influence of model parameters on the errors.

CONCLUSION

The results obtained prove the robustness of the method. The analysis of the sub-model dynamics allows us to identify the source of circadian oscillations. We find that the negative feedback loop involving proteins PER, CRY, CLOCK-BMAL1 is the main oscillator, in agreement with previous modelling and experimental studies. In conclusion, Principal Process Analysis is a simple-to-use method, which constitutes an additional and useful tool for analysing the complex dynamical behaviour of biological systems.

Mathematical modelling of microbes: metabolism, gene expression and growth

The growth of microorganisms involves the conversion of nutrients in the environment into biomass, mostly proteins and other macromolecules. This conversion is accomplished by networks of biochemical reactions cutting across cellular functions, such as metabolism, gene expression, transport and signalling. Mathematical modelling is a powerful tool for gaining an understanding of the functioning of this large and complex system and the role played by individual constituents and mechanisms. This requires models of microbial growth that provide an integrated view of the reaction networks and bridge the scale from individual reactions to the growth of a population. In this review, we derive a general framework for the kinetic modelling of microbial growth from basic hypotheses about the underlying reaction systems. Moreover, we show that several families of approximate models presented in the literature, notably flux balance models and coarse-grained whole-cell models, can be derived with the help of additional simplifying hypotheses. This perspective clearly brings out how apparently quite different modelling approaches are related on a deeper level, and suggests directions for further research.

Dynamical Allocation of Cellular Resources as an Optimal Control Problem: Novel Insights into Microbial Growth Strategies

Microbial physiology exhibits growth laws that relate the macromolecular composition of the cell to the growth rate. Recent work has shown that these empirical regularities can be derived from coarse-grained models of resource allocation. While these studies focus on steady-state growth, such conditions are rarely found in natural habitats, where microorganisms are continually challenged by environmental fluctuations. The aim of this paper is to extend the study of microbial growth strategies to dynamical environments, using a self-replicator model. We formulate dynamical growth maximization as an optimal control problem that can be solved using Pontryagin’s Maximum Principle. We compare this theoretical gold standard with different possible implementations of growth control in bacterial cells. We find that simple control strategies enabling growth-rate maximization at steady state are suboptimal for transitions from one growth regime to another, for example when shifting bacterial cells to a medium supporting a higher growth rate. A near-optimal control strategy in dynamical conditions is shown to require information on several, rather than a single physiological variable. Interestingly, this strategy has structural analogies with the regulation of ribosomal protein synthesis by ppGpp in the enterobacterium Escherichia coli. It involves sensing a mismatch between precursor and ribosome concentrations, as well as the adjustment of ribosome synthesis in a switch-like manner. Our results show how the capability of regulatory systems to integrate information about several physiological variables is critical for optimizing growth in a changing environment.

Microbial growth is the process by which cells sustain and reproduce themselves from available matter and energy. Strategies enabling microorganisms to optimize their growth rate have been extensively studied, but mostly in stable environments. Here, we build a coarse-grained model of microbial growth and use methods from optimal control theory to determine a resource allocation scheme that would lead to maximal biomass accumulation when the cells are dynamically shifted from one growth medium to another. We compare this optimal solution with several cellular implementations of growth control, based on the capacity of the cell to sense different physiological variables. We find that strategies maximizing growth in steady-state conditions perform quite differently in dynamical conditions. Moreover, the control strategy with performance close to the theoretical maximum exploits information of more than one physiological variable, suggesting that optimization of microbial growth in dynamical rather than steady environments requires broader sensory capacities. Interestingly, the ppGpp alarmone system in the enterobacterium Escherichia coli, known to play an important role in growth control, has structural similarities with the control strategy approaching the theoretical maximum. It senses a discrepancy between the concentrations of precursors and ribosomes, and adjusts ribosome synthesis in an on-off fashion. This suggests that E. coli is adapted for environments with intermittent, rapid changes in nutrient availability.

Links between topology of the transition graph and limit cycles in a two-dimensional piecewise affine biological model

A class of piecewise affine differential (PWA) models, initially proposed by Glass and Kauffman (in J Theor Biol 39:103-129, 1973), has been widely used for the modelling and the analysis of biological switch-like systems, such as genetic or neural networks. Its mathematical tractability facilitates the qualitative analysis of dynamical behaviors, in particular periodic phenomena which are of prime importance in biology. Notably, a discrete qualitative description of the dynamics, called the transition graph, can be directly associated to this class of PWA systems. Here we present a study of periodic behaviours (i.e. limit cycles) in a class of two-dimensional piecewise affine biological models. Using concavity and continuity properties of Poincaré maps, we derive structural principles linking the topology of the transition graph to the existence, number and stability of limit cycles. These results notably extend previous works on the investigation of structural principles to the case of unequal and regulated decay rates for the 2-dimensional case. Some numerical examples corresponding to minimal models of biological oscillators are treated to illustrate the use of these structural principles.

Global stability of enzymatic chains of full reversible Michaelis-Menten reactions

We consider a chain of metabolic reactions catalyzed by enzymes, of reversible Michaelis-Menten type with full dynamics, i.e. not reduced with any quasi-steady state approximations. We study the corresponding dynamical system and show its global stability if the equilibrium exists. If the system is open, the equilibrium may not exist. The main tool is monotone systems theory. Finally we study the implications of these results for the study of coupled genetic-metabolic systems.

Global stability of reversible enzymatic metabolic chains

We consider metabolic networks with reversible enzymatic reactions. The model is written as a system of ordinary differential equations, possibly with inputs and outputs. We prove the global stability of the equilibrium (if it exists), using techniques of monotone systems and compartmental matrices. We show that the equilibrium does not always exist. Finally, we consider a metabolic system coupled with a genetic network, and we study the dependence of the metabolic equilibrium (if it exists) with respect to concentrations of enzymes. We give some conclusions concerning the dynamical behavior of coupled genetic/metabolic systems.

Stabilizing effect of cannibalism in a two stages population model

In this paper we build a prey-predator model with discrete weight structure for the predator. This model will conserve the number of individuals and the biomass and both growth and reproduction of the predator will depend on the food ingested. Moreover the model allows cannibalism which means that the predator can eat the prey but also other predators. We will focus on a simple version with two weight classes or stage (larvae and adults) and present some general mathematical results. In the last part, we will assume that the dynamics of the prey is fast compared to the predator's one to go further in the results and eventually conclude that under some conditions, cannibalism can stabilize the system: more precisely, an unstable equilibrium without cannibalism will become almost globally stable with some cannibalism. Some numerical simulations are done to illustrate this result.

Positive and Negative Circuits in Dynamical Systems

We state precisely and demonstrate two conjectures of R. Thomas following which (a) the existence of a positive circuit in the oriented interaction graph of a differential system is a necessary condition for the existence of several steady states, and (b) the existence of a negative non-oriented circuit of length at least two is a necessary condition for the existence of a stable periodic orbit.

A theoretical exploration of birhythmicity in the p53-Mdm2 network

Experimental observations performed in the p53-Mdm2 network, one of the key protein modules involved in the control of proliferation of abnormal cells in mammals, revealed the existence of two frequencies of oscillations of p53 and Mdm2 in irradiated cells depending on the irradiation dose. These observations raised the question of the existence of birhythmicity, i.e. the coexistence of two oscillatory regimes for the same external conditions, in the p53-Mdm2 network which would be at the origin of these two distinct frequencies. A theoretical answer has been recently suggested by Ouattara, Abou-Jaoudé and Kaufman who proposed a 3-dimensional differential model showing birhythmicity to reproduce the two frequencies experimentally observed. The aim of this work is to analyze the mechanisms at the origin of the birhythmic behavior through a theoretical analysis of this differential model. To do so, we reduced this model, in a first step, into a 3-dimensional piecewise linear differential model where the Hill functions have been approximated by step functions, and, in a second step, into a 2-dimensional piecewise linear differential model by setting one autonomous variable as a constant in each domain of the phase space. We find that two features related to the phase space structure of the system are at the origin of the birhythmic behavior: the existence of two embedded cycles in the transition graph of the reduced models; the presence of a bypass in the orbit of the large amplitude oscillatory regime of low frequency. Based on this analysis, an experimental strategy is proposed to test the existence of birhythmicity in the p53-Mdm2 network. From a methodological point of view, this approach greatly facilitates the computational analysis of complex oscillatory behavior and could represent a valuable tool to explore mathematical models of biological rhythms showing sufficiently steep nonlinearities.

A Simple Unforced Oscillatory Growth Model in the Chemostat

In a chemostat, transient oscillations in cell number density are often experimentally observed during cell growth. The aim of this paper is to propose a simple autonomous model which is able to generate these oscillations, and to investigate it analytically. Our point of view is based on a simplification of the cell cycle in which there are two states (mature and immature) with the transfer between the two dependent on the available resources. We use the mathematical global properties of competitive differential systems to prove the existence of a limit cycle. A comparison between our model and a more complex model consisting of partial differential equations is made with the help of numerical simulations, giving qualitatively similar results.

Piecewise-linear Models of Genetic Regulatory Networks: Equilibria and their Stability

A formalism based on piecewise-linear (PL) differential equations, originally due to Glass and Kauffman, has been shown to be well-suited to modelling genetic regulatory networks. However, the discontinuous vector field inherent in the PL models raises some mathematical problems in defining solutions on the surfaces of discontinuity. To overcome these difficulties we use the approach of Filippov, which extends the vector field to a differential inclusion. We study the stability of equilibria (called singular equilibrium sets) that lie on the surfaces of discontinuity. We prove several theorems that characterize the stability of these singular equilibria directly from the state transition graph, which is a qualitative representation of the dynamics of the system. We also formulate a stronger conjecture on the stability of these singular equilibrium sets.

Hybrid bounded error observers for uncertain bioreactor models

In this paper, we build bounded error observers for a common class of partially known bioreactor models. The main idea is to construct hybrid bounded observers "between" high gain observer, which has an adjustable convergence rate but requires perfect knowledge of the model, and asymptotic observer which is very robust towards uncertainty but has a fixed convergence rate. An hybrid bounded error observer which reconstructs the two state variables is constructed considering two steps: first step is similar to a high gain observer meaning that fast convergence rate but error depending on the knowledge of the model are obtained; second step is a switch to an observer similar to the asymptotic one meaning that fixed convergence rate towards an error as small as desired is obtained. Thus, a better convergence rate of estimated variables than the classical asymptotic observer is obtained.

A tunable multivariable nonlinear robust observer for biological systems

This paper presents a robust nonlinear asymptotic observer with adjustable convergence rate with a great potential of applicability for biological systems in which the main state variables are difficult and expensive to measure or such measurements do not exist. This observer scheme is based on the classical asymptotic observer, which is modified to allow the tuning of the convergence rate. It is shown that the proposed observer provides fast and satisfactory estimates when facing load disturbances, system failures and parameter uncertainty while maintaining the excellent robustness and stability properties of the classical asymptotic observer. The implementation of the tunable observer is carried out by numerical simulations of a mathematical model of an anaerobic digestion process used for wastewater treatment. The key results are examined and further developed.

Qualitative simulation of genetic regulatory networks using piecewise-linear models

In order to cope with the large amounts of data that have become available in genomics, mathematical tools for the analysis of networks of interactions between genes, proteins, and other molecules are indispensable. We present a method for the qualitative simulation of genetic regulatory networks, based on a class of piecewise-linear (PL) differential equations that has been well-studied in mathematical biology. The simulation method is well-adapted to state-of-the-art measurement techniques in genomics, which often provide qualitative and coarse-grained descriptions of genetic regulatory networks. Given a qualitative model of a genetic regulatory network, consisting of a system of PL differential equations and inequality constraints on the parameter values, the method produces a graph of qualitative states and transitions between qualitative states, summarizing the qualitative dynamics of the system. The qualitative simulation method has been implemented in Java in the computer tool Genetic Network Analyzer.

A discrete, size-structured model of phytoplankton growth in the chemostat: introduction of inhomogeneous cell division size

We introduce inhomogeneous, substrate dependent cell division in a time discrete, nonlinear matrix model of size-structured population growth in the chemostat, first introduced by Gage et al. [8] and later analysed by Smith [13]. We show that mass conservation is verified, and conclude that our system admits one non zero globally stable equilibrium, which we express explicitly. Then we run numerical simulations of the system, and compare the predictions of the model to data related to phytoplankton growth, whose obtention we discuss. We end with the identification of several parameters of the system.

A size-structured, non-conservative ODE model of the chemostat

We study a class of size-structured, ODE models of growth in the chemostat, that take into account cell maintenance and substrate dependent cell mortality. Unlike most classical chemostat models, they are supposed to be non-conservative, in the sense that they do not verify the mass conservation principle. However, using a change of time scale, we are able to obtain qualitative results. Then, using a Lyapunov functional, we prove the global stability of the non-trivial equilibrium. Some examples of the possible structure of the models are given to finish with.

Hybrid neural modelling of an anaerobic digester with respect to biological constraints

A hybrid model for an anaerobic digestion process is proposed. The fermentation is assumed to be performed in two steps, acidogenesis and methanogenesis, by two bacterial populations. The model is based on mass balance equations, and the bacterial growth rates are represented by neural networks. In order to guarantee the biological meaning of the hybrid model (positivity of the concentrations, boundedness, saturation or inhibition of the growth rates) outside the training data set, a method that imposes constraints in the neural network is proposed. The method is applied to experimental data from a fixed bed reactor.

Non-linear qualitative signal processing for biological systems: application to the algal growth in bioreactors

We present in this paper a qualitative method to validate and monitor the structure of a non-linear model with respect to experimental data, under some hypotheses. This method is broadly independent of the analytical formulation of the model, and depends only on the qualitative structure (the signs of the Jacobian matrix). The temporal sequences of the extrema of a filtered experimental signal are compared with the transitions allowed by a graph. In particular, we show that the usual moving average of the outputs follows this transition graph. We apply this method to compare models of algal growth in a bioreactor with experimental data.

Transient behavior of biological loop models with application to the Droop model

In this paper we study the transient behavior of a class of nonlinear differential systems verifying sign conditions through the succession of extrema of the state variables. This analysis does not depend, for the main part, on the analytical formulation of the model. The possible scenarios of sequences for the extrema, are represented on a graph and can be compared with the experimental data to validate the model. An application to the Droop model illustrates this method; we obtain as a result the global stability of the equilibrium and the possible successions of the extrema.

Global behavior of n-dimensional Lotka-Volterra systems

The behavior of Lotka-Volterra systems is studied using as tools the results from positivity and auxiliary functions that decrease along the trajectories. One typical result is that if a decomposition of the interaction matrix into a product of a symmetric and an off-diagonal nonnegative matrix is possible, then all the trajectories either go to equilibria or cannot remain in any compact set of the interior of the positive orthant.

Effect of activity on the selective stabilization of the motor innervation of fast muscle posterior latissimus dorsi from chick embryo

The role of neuromuscular activity in the maturation of the motor innervation was investigated in the fast focally innervated posterior latissimus dorsi (PLD) muscle of the chick embryo. The axonal supply in the PLD motor nerve, and the focal multiple innervation of the endplates were described on days 15 and 16 of embryonic life in normal and experimental embryos. In the first series of experiments, chick embryos were paralyzed by repeated injections between days 4 and 10 in ovo of the curare-like agent, flaxedil. Twice more axons in the PLD motor nerve and about twice more nerve terminal profiles at the endplates in the PLD muscles were found in paralyzed than in control embryos. In a second series of experiments, electrodes were implanted around the spinal cord of 7-day-old embryos and electric pulses delivered at 0.5 Hz frequency from day 10 to days 15-16 of incubation. At day 15.5, no change was observed in the axonal supply in the PLD motor nerve of stimulated embryos, while a two-fold decrease was observed in the number of motor nerve terminal profiles per endplate in the corresponding PLD muscle. The statistical distribution of the number of motor nerve terminal profiles per endplate was described from complete semi-serial sections in the PLD muscle from normal, paralyzed and stimulated chick embryos. In these three cases, the distribution of supernumerary nerve terminal profiles followed a Poisson law after one nerve ending had been subtracted from the number of nerve endings counted per endplate.

Selective stabilization of muscle innervation during development: a mathematical model

The biochemical model presented concerns a critical step of the development of skeletal muscle innervation. After invasion of the muscle by exploratory motor axons, several nerve terminals converge from different motoneurons onto each muscle fibre at a single endplate. During the following weeks the redundant innervation disappears: a single nerve ending per muscle fibre becomes stabilized. The model is based on the assumption that the numbers of motoneurons and of muscle fibres remain constant during this evolution and that the selective stabilization of the adult connectivity results from the competition of the active nerve terminals for a postsynaptic retrograde factor mu. At the peak of the multiple innervation, the synthesis of mu by the muscle fiber stops, possibly as a consequence of muscle electrical and/or mechanical activity. The stock of mu becomes limited; a retrograde trans-synaptic diffusion of mu from the muscle to the nerve endings takes place. Within each nerve ending, mu enters into a chemical autocatalytic reaction which results in the production of a presynaptic stabilization factor s. The nerve impulses reaching the nerve terminal initiate this reaction. Any given nerve terminal become stabilized when the concentration of s reaches a threshold value. The mathematical analysis of the model shows that there exists a unique solution which is physically acceptable. Its application and computer simulation predict that only one nerve terminal becomes stabilized per muscle fibre. The model accounts for the experimental observations that the reduction in size of the motor units is not necessarily accompanied by a reduction in the variability of their size. The model also accounts for the acceleration or delay in regression which follows modifications of the chronic activity of the nerve endings and for the variability of the pattern of innervation observed in isogenic organisms. Plausible biochemical hypotheses concerning the factors engaged in the "selective stabilization" of the nerve-endings are discussed.