Long-term, repeated measurements of individual synaptic properties have revealed that synapses can undergo significant directed and spontaneous changes over time scales of minutes to weeks. These changes are presumably driven by a large number of activity-dependent and independent molecular processes, yet how these processes integrate to determine the totality of synaptic size remains unknown. Here we propose, as an alternative to detailed, mechanistic descriptions, a statistical approach to synaptic size dynamics. The basic premise of this approach is that the integrated outcome of the myriad of processes that drive synaptic size dynamics are effectively described as a combination of multiplicative and additive processes, both of which are stochastic and taken from distributions parametrically affected by physiological signals. We show that this seemingly simple model, known in probability theory as the Kesten process, can generate rich dynamics which are qualitatively similar to the dynamics of individual glutamatergic synapses recorded in long-term time-lapse experiments in ex-vivo cortical networks. Moreover, we show that this stochastic model, which is insensitive to many of its underlying details, quantitatively captures the distributions of synaptic sizes measured in these experiments, the long-term stability of such distributions and their scaling in response to pharmacological manipulations. Finally, we show that the average kinetics of new postsynaptic density formation measured in such experiments is also faithfully captured by the same model. The model thus provides a useful framework for characterizing synapse size dynamics at steady state, during initial formation of such steady states, and during their convergence to new steady states following perturbations. These findings show the strength of a simple low dimensional statistical model to quantitatively describe synapse size dynamics as the integrated result of many underlying complex processes.
Cooperative interactions, their stability and evolution, provide an interesting context in which to study the interface between cellular and population levels of organization. Here we study a public goods model relevant to microorganism populations actively extracting a growth resource from their environment. Cells can display one of two phenotypes - a productive phenotype that extracts the resources at a cost, and a non-productive phenotype that only consumes the same resource. Both proliferate and are free to move by diffusion; growth rate and diffusion coefficient depend only weakly phenotype. We analyze the continuous differential equation model as well as simulate stochastically the full dynamics. We find that the two sub-populations, which cannot coexist in a well-mixed environment, develop spatio-temporal patterns that enable long-term coexistence in the shared environment. These patterns are purely fluctuation-driven, as the corresponding continuous spatial system does not display Turing instability. The average stability of coexistence patterns derives from a dynamic mechanism in which the producing sub-population equilibrates with the environmental resource and holds it close to an extinction transition of the other sub-population, causing it to constantly hover around this transition. Thus the ecological interactions support a mechanism reminiscent of self-organized criticality; power-law distributions and long-range correlations are found. The results are discussed in the context of general pattern formation and critical behavior in ecology as well as in an experimental context.
Interactions between microorganisms can have a crucial effect on their population dynamics. Typically, interactions are mediated through the environment by molecules and proteins that are products of cell metabolism and physiology; they therefore reflect the internal dynamics of the single cell. In this work we aim to integrate single-cell properties of gene expression that affect indirect interactions between microorganisms under challenging conditions, into a quantitative model of population dynamics. Specifically we address the problem of a microbial population secreting a protein that can actively extract a growth-limiting resource, such as a simple sugar or iron, from the environment. The genes coding for the protein can undergo random epigenetic transitions between active and silenced states, and can be repressed by the product of their reaction. We model cooperative and competitive interactions between protein producing and non-producing phenotypes by nonlinear dynamical systems and analyze them both in terms of asymptotic states and of transient dynamics. Our model shows that phenotypic transitions allow a stable coexistence of the two phenotypes, and enables us to make predictions regarding the conditions required for such coexistence and the typical timescales of transient dynamics. It also shows how repression by the reaction product induces a feedback at the population-environment level that can result in limit cycle dynamics. The relation of these results to experiments are discussed.
The engineering of microorganisms to produce a variety of extracellular enzymes (exoenzymes), for example for producing renewable fuels and in biodegradation of xenobiotics, has recently attracted increasing interest. Productivity is often reduced by "cheater" mutants, which are deficient in exoenzyme production and benefit from the product provided by the "cooperating" cells. We present a game-theoretical model to analyze population structure and exoenzyme productivity in terms of biotechnologically relevant parameters. For any given population density, three distinct regimes are predicted: when the metabolic effort for exoenzyme production and secretion is low, all cells cooperate; at intermediate metabolic costs, cooperators and cheaters coexist; while at high costs, all cells use the cheating strategy. These regimes correspond to the harmony game, snowdrift game, and Prisoners Dilemma, respectively. Thus, our results indicate that microbial strains engineered for exoenzyme production will not, under appropriate conditions, be outcompeted by cheater mutants. We also analyze the dependence of the population structure on cell density. At low costs, the fraction of cooperating cells increases with decreasing cell density and reaches unity at a critical threshold. Our model provides an estimate of the cell density maximizing exoenzyme production.
Many membrane channels and receptors exhibit adaptive, or desensitized, response to a strong sustained input stimulus. A key mechanism that underlies this response is the slow, activity-dependent removal of responding molecules to a pool which is unavailable to respond immediately to the input. This mechanism is implemented in different ways in various biological systems and has traditionally been studied separately for each. Here we highlight the common aspects of this principle, shared by many biological systems, and suggest a unifying theoretical framework. We study theoretically a class of models which describes the general mechanism and allows us to distinguish its universal from system-specific features. We show that under general conditions, regardless of the details of kinetics, molecule availability encodes an averaging over past activity and feeds back multiplicatively on the system output. The kinetics of recovery from unavailability determines the effective memory kernel inside the feedback branch, giving rise to a variety of system-specific forms of adaptive response-precise or input-dependent, exponential or power-law-as special cases of the same model.
Biological cells in a population are variable in practically every property. Much is known about how variability of single cells is reflected in the statistical properties of infinitely large populations; however, many biologically relevant situations entail finite times and intermediate-sized populations. The statistical properties of an ensemble of finite populations then come into focus, raising questions concerning inter-population variability and dependence on initial conditions. Recent technologies of microfluidic and microdroplet-based population growth realize these situations and make them immediately relevant for experiments and biotechnological application. We here study the statistical properties, arising from metabolic variability of single cells, in an ensemble of micro-populations grown to saturation in a finite environment such as a micro-droplet. We develop a discrete stochastic model for this growth process, describing the possible histories as a random walk in a phenotypic space with an absorbing boundary. Using a mapping to Polyas Urn, a classic problem of probability theory, we find that distributions approach a limiting inoculum-dependent form after a large number of divisions. Thus, population size and structure are random variables whose mean, variance and in general their distribution can reflect initial conditions after many generations of growth. Implications of our results to experiments and to biotechnology are discussed.
The copy number of any protein fluctuates among cells in a population; characterizing and understanding these fluctuations is a fundamental problem in biophysics. We show here that protein distributions measured under a broad range of biological realizations collapse to a single non-gaussian curve under scaling by the first two moments. Moreover, in all experiments the variance is found to depend quadratically on the mean, showing that a single degree of freedom determines the entire distribution. Our results imply that protein fluctuations do not reflect any specific molecular or cellular mechanism, and suggest that some buffering process masks these details and induces universality.
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