A mathematical study of the efficacy of possible negative feedback pathways involved in neuronal polarization.

Neuronal polarization, a process wherein nascent neurons develop a single long axon and multiple short dendrites, can occur within in vitro cell cultures without environmental cues. This is an apparently random process in which one of several short processes, called neurites, grows to become long, while the others remain short. In this study, we propose a minimum model for neurite growth, which involves bistability and random excitations reflecting actin waves. Positive feedback is needed to produce the bistability, while negative feedback is required to ensure that no more than one neurite wins the winner-takes-all contest. By applying the negative feedback to different aspects of the neurite growth process, we demonstrate that targeting the negative feedback to the excitation amplitude results in the most persistent polarization. Also, we demonstrate that there are optimal ranges of values for the neurite count, and for the excitation rate and amplitude that best maintain the polarization. Finally, we show that a previously published model for neuronal polarization based on competition for limited resources shares key features with our best-performing minimal model: bistability and negative feedback targeted to the size of random excitations.

Emergent spatiotemporal population dynamics with cell-length control of synthetic microbial consortia.

The increased complexity of synthetic microbial biocircuits highlights the need for distributed cell functionality due to concomitant increases in metabolic and regulatory burdens imposed on single-strain topologies. Distributed systems, however, introduce additional challenges since consortium composition and spatiotemporal dynamics of constituent strains must be robustly controlled to achieve desired circuit behaviors. Here, we address these challenges with a modeling-based investigation of emergent spatiotemporal population dynamics using cell-length control in monolayer, two-strain bacterial consortia. We demonstrate that with dynamic control of a strain's division length, nematic cell alignment in close-packed monolayers can be destabilized. We find that this destabilization confers an emergent, competitive advantage to smaller-length strains-but by mechanisms that differ depending on the spatial patterns of the population. We used complementary modeling approaches to elucidate underlying mechanisms: an agent-based model to simulate detailed mechanical and signaling interactions between the competing strains, and a reductive, stochastic lattice model to represent cell-cell interactions with a single rotational parameter. Our modeling suggests that spatial strain-fraction oscillations can be generated when cell-length control is coupled to quorum-sensing signaling in negative feedback topologies. Our research employs novel methods of population control and points the way to programming strain fraction dynamics in consortial synthetic biology.

Delay-induced uncertainty for a paradigmatic glucose-insulin model.

Medical practice in the intensive care unit is based on the assumption that physiological systems such as the human glucose-insulin system are predictable. We demonstrate that delay within the glucose-insulin system can induce sustained temporal chaos, rendering the system unpredictable. Specifically, we exhibit such chaos for the ultradian glucose-insulin model. This well-validated, finite-dimensional model represents feedback delay as a three-stage filter. Using the theory of rank one maps from smooth dynamical systems, we precisely explain the nature of the resulting delay-induced uncertainty (DIU). We develop a framework one may use to diagnose DIU in a general oscillatory dynamical system. For infinite-dimensional delay systems, no analog of the theory of rank one maps exists. Nevertheless, we show that the geometric principles encoded in our DIU framework apply to such systems by exhibiting sustained temporal chaos for a linear shear flow. Our results are potentially broadly applicable because delay is ubiquitous throughout mathematical physiology.

Heterogeneity Improves Speed and Accuracy in Social Networks.

How does temporally structured private and social information shape collective decisions? To address this question we consider a network of rational agents who independently accumulate private evidence that triggers a decision upon reaching a threshold. When seen by the whole network, the first agent's choice initiates a wave of new decisions; later decisions have less impact. In heterogeneous networks, first decisions are made quickly by impulsive individuals who need little evidence to make a choice but, even when wrong, can reveal the correct options to nearly everyone else. We conclude that groups comprised of diverse individuals can make more efficient decisions than homogenous ones.

Bayesian Evidence Accumulation on Social Networks.

To make decisions we are guided by the evidence we collect and the opinions of friends and neighbors. How do we combine our private beliefs with information we obtain from our social network? To understand the strategies humans use to do so, it is useful to compare them to observers that optimally integrate all evidence. Here we derive network models of rational (Bayes optimal) agents who accumulate private measurements and observe the decisions of their neighbors to make an irreversible choice between two options. The resulting information exchange dynamics has interesting properties: When decision thresholds are asymmetric, the absence of a decision can be increasingly informative over time. In a recurrent network of two agents, the absence of a decision can lead to a sequence of belief updates akin to those in the literature on common knowledge. On the other hand, in larger networks a single decision can trigger a cascade of agreements and disagreements that depend on the private information agents have gathered. Our approach provides a bridge between social decision making models in the economics literature, which largely ignore the temporal dynamics of decisions, and the single-observer evidence accumulator models used widely in neuroscience and psychology.

Delayed feedback model of axonal length sensing.

A fundamental question in cell biology is how the sizes of cells and organelles are regulated at various stages of development. Size homeostasis is particularly challenging for neurons, whose axons can extend from hundreds of microns to meters (in humans). Recently, a molecular-motor-based mechanism for axonal length sensing has been proposed, in which axonal length is encoded by the frequency of an oscillating retrograde signal. In this article, we develop a mathematical model of this length-sensing mechanism in which advection-diffusion equations for bidirectional motor transport are coupled to a chemical signaling network. We show that chemical oscillations emerge due to delayed negative feedback via a Hopf bifurcation, resulting in a frequency that is a monotonically decreasing function of axonal length. Knockdown of either kinesin or dynein causes an increase in the oscillation frequency, suggesting that the length-sensing mechanism would produce longer axons, which is consistent with experimental findings. One major prediction of the model is that fluctuations in the transport of molecular motors lead to a reduction in the reliability of the frequency-encoding mechanism for long axons.

Fast decisions reflect biases, slow decisions do not.

Decisions are often made by heterogeneous groups of individuals, each with distinct initial biases and access to information of different quality. We show that in large groups of independent agents who accumulate evidence the first to decide are those with the strongest initial biases. Their decisions align with their initial bias, regardless of the underlying truth. In contrast, agents who decide last make decisions as if they were initially unbiased, and hence make better choices. We obtain asymptotic expressions in the large population limit that quantify how agents' initial inclinations shape early decisions. Our analysis shows how bias, information quality, and decision order interact in non-trivial ways to determine the reliability of decisions in a group.

Stochastic switching of delayed feedback suppresses oscillations in genetic regulatory systems.

Delays and stochasticity have both served as crucially valuable ingredients in mathematical descriptions of control, physical and biological systems. In this work, we investigate how explicitly dynamical stochasticity in delays modulates the effect of delayed feedback. To do so, we consider a hybrid model where stochastic delays evolve by a continuous-time Markov chain, and between switching events, the system of interest evolves via a deterministic delay equation. Our main contribution is the calculation of an effective delay equation in the fast switching limit. This effective equation maintains the influence of all subsystem delays and cannot be replaced with a single effective delay. To illustrate the relevance of this calculation, we investigate a simple model of stochastically switching delayed feedback motivated by gene regulation. We show that sufficiently fast switching between two oscillatory subsystems can yield stable dynamics.

The space between us: Modeling spatial heterogeneity in synthetic microbial consortia dynamics.

A central endeavor in bioengineering concerns the construction of multistrain microbial consortia with desired properties. Typically, a gene network is partitioned between strains, and strains communicate via quorum sensing, allowing for complex behaviors. Yet a fundamental question of how emergent spatiotemporal patterning in multistrain microbial consortia affects consortial dynamics is not understood well. Here, we propose a computationally tractable and straightforward modeling framework that explicitly allows linking spatiotemporal patterning to consortial dynamics. We validate our model against previously published results and make predictions of how spatial heterogeneity impacts interstrain communication. By enabling the investigation of spatial patterns effects on microbial dynamics, our modeling framework informs experimentalists, helps advance the understanding of complex microbial systems, and supports the development of applications involving them.

A closed-loop multi-scale model for intrinsic frequency-dependent regulation of axonal growth.

This article develops a closed-loop multi-scale model for axon length regulation based on a frequency-dependent negative feedback mechanism. It builds on earlier models by linking molecular motor dynamics to signaling delays that then determine signal oscillation period. The signal oscillation is treated as a front end for a signaling pathway that modulates axonal length. This model is used to demonstrate the feasibility of such a mechanism and is tested against two previously published reports in which experimental manipulations were performed that resulted in axon growth. The model captures these observations and yields an expression for equilibrium axonal length. One major prediction of the model is that increasing motor density in the body of an axon results in axonal growth-this idea has not yet been explored experimentally.

Delay-induced uncertainty in the glucose-insulin system: Pathogenicity for obesity and type-2 diabetes mellitus.

We have recently shown that physiological delay can induce a novel form of sustained temporal chaos we call delay-induced uncertainty (DIU) (Karamched et al. (Chaos, 2021, 31, 023142)). This paper assesses the impact of DIU on the ability of the glucose-insulin system to maintain homeostasis when responding to the ingestion of meals. We address two questions. First, what is the nature of the DIU phenotype? That is, what physiological macrostates (as encoded by physiological parameters) allow for DIU onset? Second, how does DIU impact health? We find that the DIU phenotype is abundant in the space of intrinsic parameters for the Ultradian glucose-insulin model-a model that has been successfully used to predict glucose-insulin dynamics in humans. Configurations of intrinsic parameters that correspond to high characteristic glucose levels facilitate DIU onset. We argue that DIU is pathogenic for obesity and type-2 diabetes mellitus by linking the statistical profile of DIU to the glucostatic theory of hunger.

Quantitative imaging and dynamics of tumor therapy with viruses.

Cancer therapy remains challenging due to the myriad presentations of the disease and the vast genetic diversity of tumors that continuously evolve and often become resistant to therapy. Viruses can be engineered to specifically infect, replicate, and kill tumor cells (tumor virotherapy). Moreover, the viruses can be "armed" with therapeutic genes to enhance their oncolytic effect. Using viruses to treat cancer is exciting and novel and in principle can be used for a broad variety of tumors. However, the approach is distinctly different from other cancer therapies since success depends on establishment of an infection within the tumor and ongoing propagation of the oncolytic virus within the tumor itself. Therefore, the target itself amplifies the therapy. This introduces complex dynamics especially when the immune system is taken into consideration as well as the physical and other biological barriers to virus growth. Understanding these dynamics not only requires mathematical and computational models but also approaches for the noninvasive monitoring of the virus and tumor populations. In this perspective, we discuss strategies and current results to achieve this important goal of understanding these dynamics in pursuit of optimization of oncolytic virotherapy.

Spatiotemporal Dynamics of Synthetic Microbial Consortia in Microfluidic Devices.

Synthetic microbial consortia consist of two or more engineered strains that grow together and share the same resources. When intercellular signaling pathways are included in the engineered strains, close proximity of the microbes can generate complex dynamic behaviors that are difficult to obtain using a single strain. However, when a consortium is not cultured in a well-mixed environment the constituent strains passively compete for space as they grow and divide, complicating cell-cell signaling. Here, we explore the temporal dynamics of the spatial distribution of consortia cocultured in microfluidic devices. To do this, we grew two different strains of Escherichia coli in microfluidic devices with cell-trapping regions (traps) of several different designs. We found that the size of the traps is a critical determinant of spatiotemporal dynamics. In small traps, cells can easily signal one another, but the relative proportion of each strain within the trap can fluctuate wildly. In large traps, the relative ratio of strains is stabilized, but intercellular signaling can be hindered by distances between cells. This presents a trade-off between the trap size and the effectiveness of intercellular signaling, which can be mitigated by increasing the initial seeding of cells in larger traps. We also built a mathematical model, which suggests that increasing the number of seed cells can also increase the strain ratio variability due to an increased number of strain interfaces in the trap. These results help elucidate the complex behaviors of synthetic microbial consortia in microfluidic traps and provide a means of analysis to help remedy the spatial heterogeneity inherent to different trap types.

Diffusive transport in the presence of stochastically gated absorption.

We analyze a population of Brownian particles moving in a spatially uniform environment with stochastically gated absorption. The state of the environment at time t is represented by a discrete stochastic variable k(t)∈{0,1} such that the rate of absorption is γ[1-k(t)], with γ a positive constant. The variable k(t) evolves according to a two-state Markov chain. We focus on how stochastic gating affects the attenuation of particle absorption with distance from a localized source in a one-dimensional domain. In the static case (no gating), the steady-state attenuation is given by an exponential with length constant sqrt[D/γ], where D is the diffusivity. We show that gating leads to slower, nonexponential attenuation. We also explore statistical correlations between particles due to the fact that they all diffuse in the same switching environment. Such correlations can be determined in terms of moments of the solution to a corresponding stochastic Fokker-Planck equation.

A frequency-dependent decoding mechanism for axonal length sensing.

We have recently developed a mathematical model of axonal length sensing in which a system of delay differential equations describe a chemical signaling network. We showed that chemical oscillations emerge due to delayed negative feedback via a Hopf bifurcation, resulting in a frequency that is a monotonically decreasing function of axonal length. In this paper, we explore how frequency-encoding of axonal length can be decoded by a frequency-modulated gene network. If the protein output were thresholded, then this could provide a mechanism for axonal length control. We analyze the robustness of such a mechanism in the presence of intrinsic noise due to finite copy numbers within the gene network.