Law of mass action and saturation in SIR model with application to Coronavirus modelling.

When using SIR and related models, it is common to assume that the infection rate is proportional to the product of susceptible and infected individuals. While this assumption works at the onset of the outbreak, the infection force saturates as the outbreak progresses, even in the absence of any interventions. We use a simple agent-based model to illustrate this saturation effect. Its continuum limit leads a modified SIR model with exponential saturation. The derivation is based on first principles incorporating the spread radius and population density. We use the data for coronavirus outbreak for the period from March to June, to show that using SIR model with saturation is sufficient to capture the disease dynamics for many jurstictions, including the overall world-wide disease curve progression. Our model suggests the R 0 value of above 8 at the onset of infection, but with infection quickly "flattening out", leading to a long-term sustained sub-exponential spread.

Localized outbreaks in an S-I-R model with diffusion.

We investigate an SIRS epidemic model with spatial diffusion and nonlinear incidence rates. We show that for small diffusion rate of the infected class [Formula: see text], the infected population tends to be highly localized at certain points inside the domain, forming K spikes. We then study three distinct destabilization mechanisms, as well as a transition from localized spikes to plateau solutions. Two of the instabilities are due to coarsening (spike death) and self-replication (spike birth), and have well-known analogues in other reaction-diffusion systems such as the Schnakenberg model. The third transition is when a single spike becomes unstable and moves to the boundary. This happens when the diffusion of the recovered class, [Formula: see text] becomes sufficiently small. In all cases, the stability thresholds are computed asymptotically and are verified by numerical experiments. We also show that the spike solution can transit into an plateau-type solution when the diffusion rates of recovered and susceptible class are sufficiently small. Implications for disease spread and control through quarantine are discussed.

Cortical geometry may influence placement of interface between Par protein domains in early Caenorhabditis elegans embryos.

During polarization, proteins and other polarity determinants segregate to the opposite ends of the cell (the poles) creating biochemically and dynamically distinct regions. Embryos of the nematode worm Caenorhabditis elegans (C. elegans) polarize shortly after fertilization, creating distinct regions of Par protein family members. These regions are maintained through to first cleavage when the embryo divides along the plane specified by the interface between regions, creating daughter cells with different protein content. In wild type single cell embryos the interface between these Par protein regions is reliably positioned at approximately 60% egg length, however, it is not known what mechanisms are responsible for specifying the position of the interface. In this investigation, we use two mathematical models to investigate the movement and positioning of the interface: a biologically based reaction-diffusion model of Par protein dynamics, and the analytically tractable perturbed Allen-Cahn equation. When we numerically simulate the models on a static 2D domain with constant thickness, both models exhibit a persistently moving interface that specifies the boundary between distinct regions. When we modify the simulation domain geometry, movement halts and the interface is stably positioned where the domain thickness increases. Using asymptotic analysis with the perturbed Allen-Cahn equation, we show that interface movement depends explicitly on domain geometry. Using a combination of analytic and numeric techniques, we demonstrate that domain geometry, a historically overlooked aspect of cellular simulations, may play a significant role in spatial protein patterning during polarization.

Dynamics and stability of a three-dimensional model of cell signal transduction.

In this paper, we consider a three-dimensional model of cell signal transduction. In this model, the deactivation of signalling proteins occur throughout the cytosol and activation is localized to specific sites in the cell. We use matched asymptotic expansions to construct the dynamic solutions of signalling protein concentrations. The result of the asymptotic analysis is a system of ordinary differential equations. This reduced system is compared to numerical simulations of the full three-dimensional system. As well, we consider the stability of equilibrium solutions. We find that the systems under consideration may undergo sustained oscillations, hysteresis and other complex behaviors. The simulations of the full three-dimensional system agree with simulations of the reduced ordinary differential equations.

Model of cell signal transduction in a three-dimensional domain.

Intracellular signalling molecules form pathways inside the cell. These pathways carry a signal to target proteins which results in cellular responses. We consider a spherical cell with two internal compartments containing localized activating enzymes where as deactivating enzymes are spread uniformly through out the cytosol. Two diffusible signalling molecules are activated at the compartments and later deactivated in the cytosol due to deactivating enzymes. The two signalling molecules are a single link in a cascade reaction and form a self regulated dynamical system involving positive and negative feedback. Using matched asymptotic expansions we obtain approximate solutions of the steady state diffusion equation with a linear decay rate. We obtain three-dimensional concentration profiles for the signalling molecules. We also investigate an extension of the above system which has multiple cascade reactions occurring between multiple signalling molecules. Numerically, we show that the speed of the signal is an increasing function of the number of links in the cascade.

The role of feedback in the formation of morphogen territories.

In this paper, we consider a mathematical model for the formation of spatial morphogen territories of two key morphogens: Wingless (Wg) and Decapentaplegic (DPP), involved in leg development of Drosophila. We define a gene regulatory network (GRN) that utilizes autoactivation and cros-sinhibition (modeled by Hill equations) to establish and maintain stable boundaries of gene expression. By computational analysis we find that in the presence of a general activator, neither autoactivation, nor cross-inhibition alone are sufficient to maintain stable sharp boundaries of morphogen production in the leg disc. The minimal requirements for a self-organizing system are a coupled system of two morphogens in which the autoactivation and cross-inhibition have Hill coefficients strictly greater than one. In addition, the GRN modeled here describes the regenerative responses to genetic manipulations of positional identity in the leg disc.

Wingless directly represses DPP morphogen expression via an armadillo/TCF/Brinker complex.

BACKGROUND: Spatially restricted morphogen expression drives many patterning and regeneration processes, but how is the pattern of morphogen expression established and maintained? Patterning of Drosophila leg imaginal discs requires expression of the DPP morphogen dorsally and the wingless (WG) morphogen ventrally. We have shown that these mutually exclusive patterns of expression are controlled by a self-organizing system of feedback loops that involve WG and DPP, but whether the feedback is direct or indirect is not known. METHODS/FINDINGS: By analyzing expression patterns of regulatory DNA driving reporter genes in different genetic backgrounds, we identify a key component of this system by showing that WG directly represses transcription of the dpp gene in the ventral leg disc. Repression of dpp requires a tri-partite complex of the WG mediators armadillo (ARM) and dTCF, and the co-repressor Brinker, (BRK), wherein ARM.dTCF and BRK bind to independent sites within the dpp locus. CONCLUSIONS/SIGNIFICANCE: Many examples of dTCF repression in the absence of WNT signaling have been described, but few examples of signal-driven repression requiring both ARM and dTCF binding have been reported. Thus, our findings represent a new mode of WG mediated repression and demonstrate that direct regulation between morphogen signaling pathways can contribute to a robust self-organizing system capable of dynamically maintaining territories of morphogen expression.

Model selection and optimal sampling in high-throughput experimentation.

The practical difficulties encountered in analyzing the kinetics of new reactions are considered from the viewpoint of the capabilities of state-of-the-art high-throughput systems. There are three problems. The first problem is that of model selection, i.e., choosing the correct reaction rate law. The second problem is how to obtain good estimates of the reaction parameters using only a small number of samples once a kinetic model is selected. The third problem is how to perform both functions using just one small set of measurements. To solve the first problem, we present an optimal sampling protocol to choose the correct kinetic model for a given reaction, based on T-optimal design. This protocol is then tested for the case of second-order and pseudo-first-order reactions using both experiments and computer simulations. To solve the second problem, we derive the information function for second-order reactions and use this function to find the optimal sampling points for estimating the kinetic constants. The third problem is further complicated by the fact that the optimal measurement times for determining the correct kinetic model differ from those needed to obtain good estimates of the kinetic constants. To solve this problem, we propose a Pareto optimal approach that can be tuned to give the set of best possible solutions for the two criteria. One important advantage of this approach is that it enables the integration of a priori knowledge into the workflow.

Stability analysis of Turing patterns generated by the Schnakenberg model.

We consider the following Schnakenberg model on the interval (-1,1): [formula see text] where D1 > 0, D2 > 0, B > 0. We rigorously show that the stability of symmetric N-peaked steady-states can be reduced to computing two matrices in terms of the diffusion coefficients D1, D2 and the number N of peaks. These matrices and their spectra are calculated explicitly and sharp conditions for linear stability are derived. The results are verified by some numerical simulations.

Tracking chemical kinetics in high-throughput systems.

Combinatorial chemistry and high-throughput experimentation (HTE) have revolutionized the pharmaceutical industry-but can chemists truly repeat this success in the fields of catalysis and materials science? We propose to bridge the traditional "discovery" and "optimization" stages in HTE by enabling parallel kinetic analysis of an array of chemical reactions. We present here the theoretical basis to extract concentration profiles from reaction arrays and derive the optimal criteria to follow (pseudo)first-order reactions in time in parallel systems. We use the information vector f and introduce in this context the information gain ratio, chi(r), to quantify the amount of useful information that can be obtained by measuring the extent of a specified reaction r in the array at any given time. Our method is general and independent of the analysis technique, but it is more effective if the analysis is performed on-line. The feasibility of this new approach is demonstrated in the fast kinetic analysis of the carbon-sulfur coupling between 3-chlorophenylhydrazonopropane dinitrile and beta-mercaptoethanol. The theory agrees well with the results obtained from 31 repeated C-S coupling experiments.

Kinetic studies of cascade reactions in high-throughput systems.

The application of robotic systems to the study of complex reaction kinetics is considered, using the cascade reaction A --> B --> C as a working example. Practical problems in calculating the rate constants k1 and k2 for the reactions A --> B and B --> C from concentration measurements of CA, CB, or CC are discussed in the light of the symmetry and invertability of the rate equations. A D-optimal analysis is used to determine the points in time and the species that will give the best (i.e., most accurate) results. When exact data are used, the most robust solution results from measuring the pair of concentrations (CA, CC). The system's information function is computed using numeric methods. This function is then used to estimate the amount of information obtainable from a given cascade reaction at any given time. The theoretical findings are compared with experimental results from a set of two-stage cascade experiments monitored using UV-visible spectroscopy. Finally, the pros and cons of using a single reaction sample to estimate both k1 and k2 are discussed.