We use mathematical modeling to study the proliferation dynamics of CD4+ T cells within an immune response. This proliferation is driven by the autocrine reaction of helper T cells and interleukin-2 (IL-2), and regulated by natural regulatory T cells (nTregs). Previous studies suggested that a fratricidal mechanism is necessary to eliminate helper T cells post-infection. Contrary to this, our mathematical analysis establishes that the depletion of these cells is due to two pivotal factors: the saturation in the proliferation rate of helper CD4+ T cells at high IL-2 concentrations, and the activation rate of nTregs outpacing their death rate. This yields an excitable process, such that the proliferation starts once the helper T cell population passes a threshold. Additionally, we find that when the proliferation of nTregs lags behind their mortality, induced regulatory T cells (iTregs) are crucial to curbing the proliferation of helper CD4+ T cells.

We propose that chaotic Glass networks (a class of piecewise-linear Ordinary Differential Equations) are good candidates for the design of true random number generators. A Glass network design has the advantage of involving only standard Boolean logic gates. Furthermore, an already chaotic (deterministic) system combined with random “jitter” due to thermal noise can be used to generate random bit sequences in a more robust way than noisy limit-cycle oscillators. Since the goal is to generate bit sequences with as large a positive entropy as possible, it is desirable to have a theoretical method to assess the irregularity of a large class of networks. We develop a procedure here to calculate good upper bounds on the entropy of a Glass network, by means of symbolic representations of the continuous dynamics. Our method improves on a result by Farcot (2006), and allows in principle for an arbitrary level of precision by refinements of the estimate, and we show that in the limiting case, these estimates converge to the true entropy of the symbolic system corresponding to the continuous dynamics. As a check on the method, we demonstrate for an example network that our upper bound after only a few refinement steps is very close to the entropy estimated from a long numerical simulation.

A broad array of experimental techniques have been used to determine the interactions between genes that regulate key cellular processes such as differentiation, metabolism and the cell cycle. The experimental studies are often complemented by development of models of varying degrees of complexity. We consider the ‘inverse problem’: to determine the underlying interactions based solely on the observed dynamics. In earlier work, we considered a specific class of ordinary differential equations that are continuous analogues of a Boolean switching network. We developed techniques to analyse and classify the dynamics based on their logical structure. We also developed techniques to solve the inverse problem. In the current work, we extend these earlier methods to analyse a model equation for a genetic network proposed by Cummins and colleagues. For a simple negative feedback system in which there is a cyclic interaction diagram with an odd number of inhibitory links, if the data is sampled at a sufficiently fine time scale with sufficient accuracy that maxima and minima can be determined, the structure can be deduced by considering sequences of maxima and minima. Alternatively, one can use the sequence of logical states found by discretizing the dynamics based on the first derivative of the variables as a function of time. The most useful technique for determining the interactions involves assessing the dependence of the rate of change of each variable as a function of the other variables, taken one at a time.

Glass networks model systems of variables that interact via sharp switching. A body of theory has been developed over several decades that, in principle, allows rigorous proof of dynamical properties in high dimensions that is not normally feasible in nonlinear dynamical systems. Previous work has, however, used examples of dimensions no higher than 6 to illustrate the methods. Here, we show that the same tools can be applied in dimensions at least as high as 20. An important application of Glass networks is to a recently proposed design of a true random number generator that is based on an intrinsically chaotic electronic circuit. In order for analysis to be meaningful for the application, the dimension must be at least 20. Bifurcation diagrams show what appear to be periodic and chaotic bands. Here, we demonstrate that the analytic tools for Glass networks can be used to rigorously show where periodic orbits are lost and the types of bifurcations that occur there. The main tools are linear algebra and the stability theory of Poincaré maps. All main steps can be automated, and we provide computer code. The methods reviewed here have the potential for many other applications involving sharply switching interactions, such as artificial neural networks.

In a recent paper by Guglielmi and Hairer (SIADS 2015), an analysis in the $\varepsilon\to 0$ limit was proposed of regularized discontinuous ODEs in codimension-2 switching domains; this was obtained by studying a certain 2-dimensional system describing the so-called hidden dynamics. In particular, the existence of a unique limit solution was not proved in all cases, a few of which were labeled as ambiguous, and it was not clear whether or not the ambiguity could be resolved. In this paper, we show that it cannot be resolved in general. A first contribution of this paper is an illustration of the dependence of the limit solution on the form of the switching function. Considering the parameter dependence in the ambiguous class of discontinuous systems, a second contribution is a bifurcation analysis, revealing a range of possible behaviors. Finally, we investigate the sensitivity of solutions in the transition from codimension-2 domains to codimension-3 when there is a limit cycle in the hidden dynamics.

In this work, we studied the stability of radially symmetric growth in tumor spheroids using a reaction-diffusion model. In this model, nutrient concentration and internal pressure are local variables that implicitly relate the proliferation of cells to the growth of the tumor. The analytical solution of the governing model was presented in an orthonormal spherical harmonic basis. It was shown that the radially symmetric steady-state solution to the growth of tumor spheroids, under symmetric growth conditions, was unstable with respect to small asymmetric perturbations. Such perturbations excited the asymmetric modes of growth, which could grow in time and change the spherical configuration of the tumor. The number of such modes and their rates of growth depended on parameters such as surface tension, external energy and the rate of nutrient consumption. This analysis indicated that the spherical configuration of tumor spheroids, even under experimentally controlled symmetric growth conditions, were naturally unstable. This was confirmed by a comparison between the shapes of in vitro human glioblastoma (hGB) spheroids and the configuration of the first few asymmetric modes predicted by the model.

In 2019, Adams, Ehlting, and Edwards showed that in a model of plant phenylalanine metabolism following Michaelis–Menten kinetics, there are two mechanisms by which primary metabolism is prioritized over secondary metabolism when synthesis rates of shikimate, a precursor to phenylalanine, are low: the Precursor Shutoff Valve (PSV), a form of metabolic regulation effected by a series of reactions called the Shikimate Ester Loop (SEL), and threshold separation. They found that the SEL is completely effective in prioritizing primary metabolism when shikimate production is low, but when the SEL is absent, this prioritization is only seen when the threshold constant associated with the secondary pathway is sufficiently larger than that of the primary pathway. Since nonlinear terms can make analysis difficult, here we replace the Michaelis–Menten terms in Adams and colleagues’ model with piecewise approximations we call ramp functions. We show that the ramp function model behaves the same as the original model under low shikimate conditions; the SEL effects PSV type regulation regardless of threshold constants, while without the SEL, primary metabolism is only prioritized when the threshold constant of the secondary pathway is sufficiently large. This is in contrast to a step function version of the model studied by Edwards and Wood in 2021 where the SEL did not effect PSV type regulation on its own.

The Precursor Shutoff Valve (PSV) has been proposed as a motif in biochemical networks, active for example in prioritization of primary over secondary metabolism in plants in low-input conditions. Another branch prioritization mechanism in a biochemical network is a difference in thresholds for activation of the two pathways from the branch point. It has been shown by Adams and colleagues that both mechanisms can play a part in a model of plant metabolism involving Michaelis-Menten kinetics [1]. Here we investigate the potential role of these two mechanisms in systems with steeper activation functions, such as those involving highly cooperative reactions, by considering the limit of infinitely steep activation functions, as is done in Glass networks as models of gene regulation. We find that the Threshold Separation mechanism is completely effective in pathway prioritization in such a model framework, while the PSV adds no additional benefit, and is ineffective on its own. This makes clear that the PSV uses the gradual nature of activation functions to help shut off one branch at low input levels, and has no effect if activation is sharp. The analysis also serves as a case study in assessing behaviour of sharply-switching open systems without degradation of species.

Mathematical modeling has significant potential for understanding of biological models of cancer and to accelerate the progress in cross-disciplinary approaches of cancer treatment. In mathematical biology, solid tumor spheroids are often studied as preliminary in vitro models of avascular tumors. The size of spheroids and their cell number are easy to track, making them a simple in vitro model to investigate tumor behavior, quantitatively. The growth of solid tumors is comprised of three main stages: transient formation, monotonic growth and a plateau phase. The last two stages are extensively studied. However, the initial transient formation phase is typically missing from the literature. This stage is important in the early dynamics of growth, formation of clonal sub-populations, etc. In the current work, this transient formation is modeled by a reaction–diffusion partial differential equation (PDE) for cell concentration, coupled with an ordinary differential equation (ODE) for the spheroid radius. Analytical and numerical solutions of the coupled equations were obtained for the change in the radius of tumor spheroids over time. Human glioblastoma (hGB) cancer cells (U251 and U87) were spheroid cultured to validate the model prediction. Results of this study provide insight into the mechanism of development of solid tumors at their early stage of formation.

In models for networks of regulatory interactions of biological molecules, the sigmoid relationship between concentration of regulating bodies and the production rates they control has led to the use of continuous-time 'switching' systems, sometimes referred to as Glass networks, which result from a simplifying assumption that the switching behaviour occurs instantaneously at particular threshold values. Though this assumption produces highly tractable models, it also causes analytic difficulties, such as non-uniqueness, in certain cases, due to the discontinuities of the system. Here, the use of ramp functions is explored as an alternative approximation to the sigmoid, which restores continuity to the vector field and removes the assumption of infinitely steep switching by linearly interpolating the focal point values used in a corresponding Glass network. A general framework for describing a ramp system using the 'focal points' of the corresponding Glass network is given. Solutions of twodimensional networks are explored, and then higher-dimensional networks under certain restrictions. Periodic behaviour is explored using mappings between threshold boundaries. Limitations in these methods are explored, and a general proof of the existence of periodic solutions in negative feedback loops with ramp interactions is given. (C) 2021 Elsevier B.V. All rights reserved.

Purpose Cancer occurs when signaling pathways become unregulated or constitutively activated inside a cell. For example, deregulation of the mitogen activated protein kinase (MAPK) pathway often leads to cancer by promoting uncontrolled cellular proliferation. Chimeric proteins can rewire these signal transduction pathways active in cancer cells by linking activation of the MAPK pathway to activation of the Fas apoptosis pathway, causing the input signal for cell proliferation to be redirected to induce cell death. Methods We present here a kinetic model demonstrating how these chimeric proteins can trigger apoptosis upon stimulation of the MAPK pathway. This model consists of ordinary differential equations using rate constants found in literature along with experimental data from previously published work. At a concentration of 1500 nM, the chimeric protein caused a 60% decrease in MAPK activation, causing the cell to transition from a proliferative state to an apoptotic state, validating previous experimental observations. Even at much lower concentrations (e.g. 24 nM), the apoptosis pathway is activated, so the model suggests that cell death may occur even without a direct suppression of the proliferation pathway. Results and Conclusions We have developed a quantitative model of caspase activation and its effect on the MAPK pathway in the presence of a chimeric protein, providing insight into a potential mechanism for reprogramming cancer cells.

The oscillator made of a negative loop of two genes is one of the most classical motifs of genetic networks. We give solutions to control such an oscillator by modifying the synthesis rates. Our models are given by piecewise affine systems, and the control is qualitative, taking only two values. Thus, the necessary measurements for implementing this control only depend on the fact that some gene is expressed or not. Our first goal is to obtain sustained oscillations. Then we study the control by a sliding mode for negative ODE loops in general, to suppress sustained or damped oscillations. Finally, we introduced a general idea for creating sustained oscillations in systems with damped oscillations following a particular cycle of domains.

We introduce and explore a type of discrete dynamic system inheriting some properties of both cellular automata (CA) and L-systems. Originally suggested by Jean Della Dora, and thus called DEM-systems after him and the two current authors, these systems can have the structural flexibility of an L-system as well as algebraic properties of CA. They are defined as sequences on a one-dimensional loop with rules governing dynamics in which new sites can be created, depending on the states of a neighbourhood of sites, and complex behaviour can be generated. Although the definition of DEM-systems is quite broad, we define some subclasses, for which more complete results can be obtained. For example, we define an additive subclass, for which algebraic results on asymptotic growth are possible, and an elementary class of particularly simple rules, for which nevertheless impressive complexity is achievable. Unlike for CA, finite initial sequences can produce positive spatial entropy over time. However, even in cases where the entropy is zero, considerable complexity is possible, especially when the sequence length grows to infinity, and we demonstrate and study behaviours of DEM-systems including fragmentation of sequences, self-reproducing patterns, self-similar but irregular patterns, patterns that not only produce new sites but produce producers of new sites, and sequences whose growth rate is sublinear, linear, quadratic, cubic, or exponential. The most complex behaviour from small finite initial conditions and the simplest class of rules appear to have positive entropy, a suggestion for which we have so far only stong numerical evidence, though we present a proof for these ‘elementary’ DEM-systems that entropy cannot reach the theoretical maximum of 1.

Zeno hybrid systems exhibit an infinite number of switches between different configurations in finite time. This is the result of the abstraction of modeling and the assumption that the switching is instantaneous. Simulation of such systems will generally stop or give false results after a finite time, called the Zeno time. In reality, physical systems do not exhibit such behavior, because the switching is not in reality instantaneous, but also often because they are subject to a small amount of noise. This noise makes possible the continuous evolution of the real systems even if extremely fast switching arises. Thus, the problem is how to predict the behavior of the system after the Zeno time. In this paper a new technique is proposed to extend simulations beyond the Zeno time. It consists of a probabilistic regularization by a convolution method that includes noise in the system. For example, low-amplitude Gaussian noise can be used to transform discontinuous vector fields to continuous vector fields. This convolution approach makes the value observed for the continuous evolution over time slightly different from what would be predicted without the noise, but in a way that should correspond to what happens in reality. An example of a Zeno hybrid system is employed to illustrate the result. The evolution beyond the Zeno time varies depending on the noise that exists in physical systems and on how the noise is included in the regularized model. However, a useful idealization is obtained in the limit of low noise amplitude.

A ring-shaped logic circuit is proposed here as a robust design for a True Random Number Generator (TRNG). Most existing TRNGs rely on physical noise as a source of randomness, where the underlying idealized deterministic system is simply oscillatory. The design proposed here is based on chaotic dynamics and therefore intrinsically displays random behavior, even in the ideal noise-free situation. The paper presents several mathematical models for the circuit having different levels of detail. They take the form of differential equations using steep sigmoid terms for the transfer functions of logic gates. A large part of the analysis is concerned with the hard step-function limit, leading to a model known in mathematical biology as a Glass network. In this framework, an underlying discrete structure (a state space diagram) is used to describe the likely structure of the global attractor for this system. The latter takes the form of intertwined periodic paths, along which trajectories alternate unpredictably. It is also invariant under the action of the cyclic group. A combination of analytical results and numerical investigations confirms the occurrence of symmetric chaos in this system, which when implemented in (noisy) hardware, should therefore serve as a robust TRNG.

Continuous-time systems with switch-like behaviour occur in chemical kinetics, gene regulatory networks and neural networks. Networks with hard switching, as a limiting case of smooth sigmoidal switching, retain the richest possible range of behaviors but are mathematically more tractable. The form of an underlying discrete (fractional-linear) map encodes information on existence, stability and exact periods of periodic orbits. In richly connected structures with four or more variables, aperiodic behaviour can occur. We investigate a simple 4-dimensional example with Boolean interaction terms in which a Smale horseshoe-like object reveals chaotic dynamics.

In higher plants, the amino acid phenylalanine is a substrate of both primary and secondary metabolic pathways. The primary pathway that consumes phenylalanine, protein biosynthesis, is essential for the viability of all cells. Meanwhile, the secondary pathways are not necessary for the survival of individual cells, but benefit of the plant as a whole. Here we focus on the monolignol pathway, a secondary metabolic pathway in the cytosol that rapidly consumes phenylalanine to produce the precursors of lignin during wood formation. In planta monolignol biosynthesis involves a series of seemingly redundant steps wherein shikimate, a precursor of phenylalanine synthesized in the plastid, is transiently ligated to the main substrate of the pathway. However, shikimate is not catalytically involved in the reactions of the monolignol pathway, and is only needed for pathway enzymes to recognize their main substrates. After some steps the shikimate moiety is removed unaltered, and the main substrate continues along the pathway. It has been suggested that this portion of the monolignol pathway fulfills a regulatory role in the following way. Low phenylalanine concentrations (viz. availability) correlate with low shikimate concentrations. When shikimate concentratios are low, flux into the monolignol pathway will be limited by means of the steps requiring shikimate. Thus, when the concentration of phenylalanine is low it will be reserved for protein biosynthesis. Here we employ a theoretical approach to test this hypothesis. Simplified versions of plant phenylalanine metabolism are modelled as systems of ordinary differential equations. Our analysis shows that the seemingly redundant steps can be sufficient for the prioritization of protein biosynthesis over the monolignol pathway when the availability of phenylalanine is low, depending on system parameters. Thus, the phenylalanine precursor shikimate may signal low phenylalanine availability to secondary pathways. Because our models have been abstracted from plant phenylalanine metabolism, this mechanism of metabolic signalling, which we call the Precursor Shutoff Valve (PSV), may also be present in other biochemical networks comprised of two pathways that share a common substrate. (C) 2018 Elsevier Ltd. All rights reserved.

We review results concerning dynamics in a class of hybrid ordinary differential equations which incorporates logical control to yield piecewise linear equations. These equations relate qualitative features of the structure of networks to qualitative properties of the dynamics. Because of their simple structure, they have been studied using techniques from discrete mathematics and nonlinear dynamics. Initially developed as a qualitataive description of gene regulatory networks, many generalizations of the basic approach have been developed. In particular, we show how this qualitative approach may be adapted to switching biochemical systems without degradation, illustrated by an example of a motif in which two branches of a pathway may be regulated differently when the thresholds for the two pathways are separated.

Non-smooth approximations of steep sigmoidal switching networks, such as those used as qualitative models of gene regulation, lead to analytic and computational challenges that arise as a result of the discontinuities in the vector fields. In order to highlight the need for care in dealing with such systems, several particular phenomena are presented here through illustrative examples, including 'Zeno breaking', or computing beyond the finite time convergence of an infinite sequence of threshold transitions; the 'Contact' effect, in which in the discontinuous limit, trajectories can pass through a 'saddle point' without stopping, though these solutions are not unique and other solutions stop for arbitrary time intervals; and sensitive behaviour that arises from exotic dynamics within switching regions.

Human induced pluripotent stem cells (hiPSCs) can form any tissue found in the body, making them attractive for regenerative medicine applications. Seeding hiPSC aggregates into biomaterial scaffolds can control their differentiation into specific tissue types. Here we develop and analyze a mathematical model of hiPSC aggregate behavior when seeded on melt electrospun scaffolds with defined topography.