Concept: Differential equation
Considering the recent experimental discovery of Green et al that present-day non-Africans have 1 to [Formula: see text] of their nuclear DNA of Neanderthal origin, we propose here a model which is able to quantify the genetic interbreeding between two subpopulations with equal fitness, living in the same geographic region. The model consists of a solvable system of deterministic ordinary differential equations containing as a stochastic ingredient a realization of the neutral Wright-Fisher process. By simulating the stochastic part of the model we are able to apply it to the interbreeding ofthe African ancestors of Eurasians and Middle Eastern Neanderthal subpopulations and estimate the only parameter of the model, which is the number of individuals per generation exchanged between subpopulations. Our results indicate that the amount of Neanderthal DNA in living non-Africans can be explained with maximum probability by the exchange of a single pair of individuals between the subpopulations at each 77 generations, but larger exchange frequencies are also allowed with sizeable probability. The results are compatible with a long coexistence time of 130,000 years, a total interbreeding population of order [Formula: see text] individuals, and with all living humans being descendants of Africans both for mitochondrial DNA and Y chromosome.
Predicting rates of cell state change caused by stochastic fluctuations using a data-driven landscape model.
- Proceedings of the National Academy of Sciences of the United States of America
- Published about 5 years ago
We develop a potential landscape approach to quantitatively describe experimental data from a fibroblast cell line that exhibits a wide range of GFP expression levels under the control of the promoter for tenascin-C. Time-lapse live-cell microscopy provides data about short-term fluctuations in promoter activity, and flow cytometry measurements provide data about the long-term kinetics, because isolated subpopulations of cells relax from a relatively narrow distribution of GFP expression back to the original broad distribution of responses. The landscape is obtained from the steady state distribution of GFP expression and connected to a potential-like function using a stochastic differential equation description (Langevin/Fokker-Planck). The range of cell states is constrained by a force that is proportional to the gradient of the potential, and biochemical noise causes movement of cells within the landscape. Analyzing the mean square displacement of GFP intensity changes in live cells indicates that these fluctuations are described by a single diffusion constant in log GFP space. This finding allows application of the Kramers' model to calculate rates of switching between two attractor states and enables an accurate simulation of the dynamics of relaxation back to the steady state with no adjustable parameters. With this approach, it is possible to use the steady state distribution of phenotypes and a quantitative description of the short-term fluctuations in individual cells to accurately predict the rates at which different phenotypes will arise from an isolated subpopulation of cells.
Recent evidence indicates that toll-like receptor (TLR) 2 and 4 are involved in the pathogenesis of dilated cardiomyopathy (DCM), but the exact mechanisms of their actions have not been elucidated. We explored the therapeutic potential of blocking TLRs in mice with established cardiomyopathy. Cardiomyopathy was generated by a single intraperitoneal injection of doxorubicin (10 mg/kg). Two weeks later, the mice were treated with TLR2 or TLR4 neutralizing antibody. Blocking TLR2, but not TLR4, activity not only reduced mortality, but also attenuated doxorubicin-induced cardiac dysfunction by 20% and inhibited myocardial fibrosis. To determine the differential effects of blocking TLR2 and TLR4 in chronic cardiomyopathy, mice were injected with doxorubicin (3.5 mg/kg) once a week for 8 weeks, followed by treatment with TLR2 or TLR4 neutralizing antibody for 40 days. Blocking TLR2 activity blunted cardiac dysfunction by 13% and inhibited cardiac fibrosis, which was associated with a significant suppression of myocardial inflammation. The underlying mechanism involved interrupting the interaction of TLR2 with its endogenous ligands, resulting in attenuation of inflammation and fibrosis. In contrast, blocking TLR4 exacerbated cardiac dysfunction and fibrosis by amplifying inflammation and suppressing autophagy. Our studies demonstrate that TLR2 and TLR4 play distinct roles in the progression of doxorubicin-induced DCM. TLR4 activity is crucial for the resolution of inflammation and cardiac fibrosis, while blocking TLR2 activity has therapeutic potential for the treatment of DCM.
Recent studies have shown that while psychopathy and non-psychopathic antisociality overlap, they differ in the extent to which cognitive impairments are present. Specifically, psychopathy has been related to abnormal allocation of attention, a function that is traditionally believed to be indexed by event-related potentials (ERPs) of the P3-family. Previous research examining psychophysiological correlates of attention in psychopathic individuals has mainly focused on the parietally distributed P3b component to rare targets. In contrast, very little is known about the frontocentral P3a to infrequent novel events in psychopathy. Thus, findings on the P3 components in psychopathy are inconclusive, while results in non-psychopathic antisocial populations are clearer and point toward an inverse relationship between antisociality and P3 amplitudes. The present study adds to extant literature on the P3a and P3b in psychopathy by investigating component amplitudes in psychopathic offenders (N = 20), matched non-psychopathic offenders (N = 23) and healthy controls (N = 16). Also, it was assessed how well each offender group was able to differentially process rare novel and target events. The offender groups showed general amplitude reductions compared to healthy controls, but did not differ mutually on overall P3a/P3b amplitudes. However, the psychopathic group still exhibited normal neurophysiological differentiation when allocating attention to rare novel and target events, unlike the non-psychopathic sample. The results highlight differences between psychopathic and non-psychopathic offenders regarding the integrity of the neurocognitive processes driving attentional allocation, as well as the usefulness of alternative psychophysiological measures in differentiating psychopathy from general antisociality.
In this study, the steady forced convection flow and heat transfer due to an impermeable stretching surface in a porous medium saturated with a nanofluid are investigated numerically. The Brinkman-Forchheimer model is used for the momentum equations (porous medium), whereas, Bongiorno’s model is used for the nanofluid. Uniform temperature and nanofluid volume fraction are assumed at the surface. The boundary layer equations are transformed to ordinary differential equations in terms of the governing parameters including Prandtl and Lewis numbers, viscosity ratio, porous medium, Brownian motion and thermophoresis parameters. Numerical results for the velocity, temperature and concentration profiles, as well as for the reduced Nusselt and Sherwood numbers are obtained and presented graphically.
MOTIVATION: Biochemical reaction networks in the form of coupled ordinary differential equations (ODEs) provide a powerful modeling tool for understanding the dynamics of biochemical processes. During the early phase of modeling, scientists have to deal with a large pool of competing nonlinear models. At this point, discrimination experiments can be designed and conducted to obtain optimal data for selecting the most plausible model. Since biological ODE models have widely distributed parameters due to, e.g., biologic variability or experimental variations, model responses become distributed. Therefore, a robust optimal experimental design (OED) for model discrimination can be used to discriminate models based on their response probability distribution functions (PDFs). RESULTS: In this work we present an optimal control based methodology for designing optimal stimulus experiments aimed at robust model discrimination. For estimating the time-varying model response PDF, which results from the nonlinear propagation of the parameter PDF under the ODE dynamics, we suggest using the Sigma-Point approach. Using the model overlap (expected likelihood) as a robust discrimination criterion to measure dissimilarities between expected model response PDFs, we benchmark the proposed nonlinear design approach against linearization with respect to prediction accuracy and design quality for two nonlinear biological reaction networks. As we show, the Sigma-Point outperforms the linearization approach in the case of widely distributed parameter sets and/or existing multiple steady states. Since the Sigma-Point approach scales linearly with the number of model parameter, it can be applied to large systems for robust experimental planing. AVAILABILITY: An implementation of the method in MATLAB/AMPL is available at http://www.uni-magdeburg.de/ivt/svt/person/rf/roed.html. CONTACT: email@example.com SUPPLEMENTARY INFORMATION: Supplementary details and discussions are available at Bioinformatics online.
A mathematical model is proposed for interpreting the love story between Elizabeth and Darcy portrayed by Jane Austen in the popular novel Pride and Prejudice. The analysis shows that the story is characterized by a sudden explosion of sentimental involvements, revealed by the existence of a saddle-node bifurcation in the model. The paper is interesting not only because it deals for the first time with catastrophic bifurcations in romantic relation-ships, but also because it enriches the list of examples in which love stories are described through ordinary differential equations.
During an infection, HIV experiences strong selection by immune system T cells. Recent experimental work has shown that MHC escape mutations form an important pathway for HIV to avoid such selection. In this paper, we study a model of MHC escape mutation. The model is a predator-prey model with two prey, composed of two HIV variants, and one predator, the immune system CD8 cells. We assume that one HIV variant is visible to CD8 cells and one is not. The model takes the form of a system of stochastic differential equations. Motivated by well-known results concerning the short life-cycle of HIV intrahost, we assume that HIV population dynamics occur on a faster time scale then CD8 population dynamics. This separation of time scales allows us to analyze our model using an asymptotic approach. Using this model we study the impact of an MHC escape mutation on the population dynamics and genetic evolution of the intrahost HIV population. From the perspective of population dynamics, we show that the competition between the visible and invisible HIV variants can reach steady states in which either a single variant exists or in which coexistence occurs depending on the parameter regime. We show that in some parameter regimes the end state of the system is stochastic. From a genetics perspective, we study the impact of the population dynamics on the lineages of an HIV sample taken after an escape mutation occurs. We show that the lineages go through severe bottlenecks and that in certain parameter regimes the lineage distribution can be characterized by a Kingman coalescent. Our results depend on methods from diffusion theory and coalescent theory.
Neuroprosthetic devices, such as cochlear and retinal implants, work by directly stimulating neurons with extracellular electrodes. This is commonly modeled using the cable equation with an applied extracellular voltage. In this paper a framework for modeling extracellular electrical stimulation is presented. To this end, a cylindrical neurite with confined extracellular space in the subthreshold regime is modeled in three-dimensional space. Through cylindrical harmonic expansion of Laplace’s equation, we derive the spatio-temporal equations governing different modes of stimulation, referred to as longitudinal and transverse modes, under types of boundary conditions. The longitudinal mode is described by the well-known cable equation, however, the transverse modes are described by a novel ordinary differential equation. For the longitudinal mode, we find that different electrotonic length constants apply under the two different boundary conditions. Equations connecting current density to voltage boundary conditions are derived that are used to calculate the trans-impedance of the neurite-plus-thin-extracellular-sheath. A detailed explanation on depolarization mechanisms and the dominant current pathway under different modes of stimulation is provided. The analytic results derived here enable the estimation of a neurite’s membrane potential under extracellular stimulation, hence bypassing the heavy computational cost of using numerical methods.
Mathematical modeling is used as a Systems Biology tool to answer biological questions, and more precisely, to validate a network that describes biological observations and predict the effect of perturbations. This article presents an algorithm for modeling biological networks in a discrete framework with continuous time. BACKGROUND: There exist two major types of mathematical modeling approaches: (1) quantitative modeling, representing various chemical species concentrations by real numbers, mainly based on differential equations and chemical kinetics formalism; (2) and qualitative modeling, representing chemical species concentrations or activities by a finite set of discrete values. Both approaches answer particular (and often different) biological questions. Qualitative modeling approach permits a simple and less detailed description of the biological systems, efficiently describes stable state identification but remains inconvenient in describing the transient kinetics leading to these states. In this context, time is represented by discrete steps. Quantitative modeling, on the other hand, can describe more accurately the dynamical behavior of biological processes as it follows the evolution of concentration or activities of chemical species as a function of time, but requires an important amount of information on the parameters difficult to find in the literature. RESULTS: Here, we propose a modeling framework based on a qualitative approach that is intrinsically continuous in time. The algorithm presented in this article fills the gap between qualitative and quantitative modeling. It is based on continuous time Markov process applied on a Boolean state space. In order to describe the temporal evolution of the biological process we wish to model, we explicitly specify the transition rates for each node. For that purpose, we built a language that can be seen as a generalization of Boolean equations. Mathematically, this approach can be translated in a set of ordinary differential equations on probability distributions. We developed a C++ software, MaBoSS, that is able to simulate such a system by applying Kinetic Monte-Carlo (or Gillespie algorithm) on the Boolean state space. This software, parallelized and optimized, computes the temporal evolution of probability distributions and estimates stationary distributions. CONCLUSIONS: Applications of the Boolean Kinetic Monte-Carlo are demonstrated for three qualitative models: a toy model, a published model of p53/Mdm2 interaction and a published model of the mammalian cell cycle. Our approach allows to describe kinetic phenomena which were difficult to handle in the original models. In particular, transient effects are represented by time dependent probability distributions, interpretable in terms of cell populations.