Ivo Siekmann

A Kinetic Model for Type I and II IP3R Accounting for Mode Changes

Based upon an extensive single-channel data set, a Markov model for types I and II inositol trisphosphate receptors (IP(3)R) is developed. The model aims to represent accurately the kinetics of both receptor types of IP(3)R depending on the concentrations of inositol trisphosphate (IP(3)), adenosine trisphosphate (ATP), and intracellular calcium (Ca(2+)). In particular, the model takes into account that for some combinations of ligands the IP(3)R switches between extended periods of inactivity alternating with intervals of bursting activity (mode changes). In a first step, the inactive and active modes are modeled separately. It is found that, within modes, both receptor types are ligand-independent. In a second step, the submodels are connected by transition rates. Ligand-dependent regulation of the channel activity is achieved by modulating these transitions between active and inactive modes. As a result, a compact representation of the IP(3)R is obtained that accurately captures stochastic single-channel dynamics including mode changes in a model with six states and 10 rate constants, only two of which are ligand-dependent.

MCMC Estimation of Markov Models for Ion Channels

Ion channels are characterized by inherently stochastic behavior which can be represented by continuous-time Markov models (CTMM). Although methods for collecting data from single ion channels are available, translating a time series of open and closed channels to a CTMM remains a challenge. Bayesian statistics combined with Markov chain Monte Carlo (MCMC) sampling provide means for estimating the rate constants of a CTMM directly from single channel data. In this article, different approaches for the MCMC sampling of Markov models are combined. This method, new to our knowledge, detects overparameterizations and gives more accurate results than existing MCMC methods. It shows similar performance as QuB-MIL, which indicates that it also compares well with maximum likelihood estimators. Data collected from an inositol trisphosphate receptor is used to demonstrate how the best model for a given data set can be found in practice.

Examination of the Effects of Heterogeneous Organization of RyR Clusters, Myofibrils and Mitochondria on Ca2+ Release Patterns in Cardiomyocytes

Spatio-temporal dynamics of intracellular calcium, [Ca2+]i, regulate the contractile function of cardiac muscle cells. Measuring [Ca2+]i flux is central to the study of mechanisms that underlie both normal cardiac function and calcium-dependent etiologies in heart disease. However, current imaging techniques are limited in the spatial resolution to which changes in [Ca2+]i can be detected. Using spatial point process statistics techniques we developed a novel method to simulate the spatial distribution of RyR clusters, which act as the major mediators of contractile Ca2+ release, upon a physiologically-realistic cellular landscape composed of tightly-packed mitochondria and myofibrils. We applied this method to computationally combine confocal-scale (~ 200 nm) data of RyR clusters with 3D electron microscopy data (~ 30 nm) of myofibrils and mitochondria, both collected from adult rat left ventricular myocytes. Using this hybrid-scale spatial model, we simulated reaction-diffusion of [Ca2+]i during the rising phase of the transient (first 30 ms after initiation). At 30 ms, the average peak of the simulated [Ca2+]i transient and of the simulated fluorescence intensity signal, F/F0, reached values similar to that found in the literature ([Ca2+]i ≈1 μM; F/F0≈5.5). However, our model predicted the variation in [Ca2+]i to be between 0.3 and 12.7 μM (~3 to 100 fold from resting value of 0.1 μM) and the corresponding F/F0 signal ranging from 3 to 9.5. We demonstrate in this study that: (i) heterogeneities in the [Ca2+]i transient are due not only to heterogeneous distribution and clustering of mitochondria; (ii) but also to heterogeneous local densities of RyR clusters. Further, we show that: (iii) these structure-induced heterogeneities in [Ca2+]i can appear in line scan data. Finally, using our unique method for generating RyR cluster distributions, we demonstrate the robustness in the [Ca2+]i transient to differences in RyR cluster distributions measured between rat and human cardiomyocytes.

Statistical analysis of modal gating in ion channels

Ion channels regulate the concentrations of ions within cells. By stochastically opening and closing its pore, they enable or prevent ions from crossing the cell membrane. However, rather than opening with a constant probability, many ion channels switch between several different levels of activity even if the experimental conditions are unchanged. This phenomenon is known as modal gating: instead of directly adapting its activity, the channel seems to mix sojourns in active and inactive modes in order to exhibit intermediate open probabilities. Evidence is accumulating that modal gating rather than modulation of opening and closing at a faster time scale is the primary regulatory mechanism of ion channels. However, currently, no method is available for reliably calculating sojourns in different modes. In order to address this challenge, we develop a statistical framework for segmenting single-channel datasets into segments that are characteristic for particular modes. The algorithm finds the number of mode changes, detects their locations and infers the open probabilities of the modes. We apply our approach to data from the inositol-trisphosphate receptor. Based upon these results, we propose that mode changes originate from alternative conformational states of the channel protein that determine a certain level of channel activity.

Predation may defeat spatial spread of infection

A model of a phytoplankton–zooplankton prey-predator system with viral infection of phytoplankton is investigated. Virus particles (V) are taken into account by an explicit equation. Phytoplankton is split into a susceptible (S) and an infected (I) class. A lytic infection is considered, thus, infected phytoplankton cells stop reproducing as soon as the infection starts and die at an increased mortality rate. Zooplankton (Z) is grazing on both susceptible and infected phytoplankton following a Holling-type II functional response. After the local dynamics of the V−S−I−Z system is analysed, numerical solutions of a stochastic reaction–diffusion model of the four species are presented. These show a spatial competition between zooplankton and viruses, although these two species are not explicitly coupled by the model equations.

Bond graph modelling of chemoelectrical energy transduction

Energy-based bond graph modelling of biomolecular systems is extended to include chemoelectrical transduction thus enabling integrated thermodynamically-compliant modelling of chemoelectrical systems in general and excitable membranes in particular. Our general approach is illustrated by recreating a well-known model of an excitable membrane. This model is used to investigate the energy consumed during a membrane action potential thus contributing to the current debate on the trade-off between the speed of an action potential event and energy consumption. The influx of Na+ is often taken as a proxy for energy consumption; in contrast, this paper presents an energy based model of action potentials. As the energy based approach avoids the assumptions underlying the proxy approach it can be directly used to compute energy consumption in both healthy and diseased neurons. These results are illustrated by comparing the energy consumption of healthy and degenerative retinal ganglion cells using both simulated and in vitro data.

Modelling modal gating of ion channels with hierarchical Markov models

Many ion channels spontaneously switch between different levels of activity. Although this behaviour known as modal gating has been observed for a long time it is currently not well understood. Despite the fact that appropriately representing activity changes is essential for accurately capturing time course data from ion channels, systematic approaches for modelling modal gating are currently not available. In this paper, we develop a modular approach for building such a model in an iterative process. First, stochastic switching between modes and stochastic opening and closing within modes are represented in separate aggregated Markov models. Second, the continuous-time hierarchical Markov model, a new modelling framework proposed here, then enables us to combine these components so that in the integrated model both mode switching as well as the kinetics within modes are appropriately represented. A mathematical analysis reveals that the behaviour of the hierarchical Markov model naturally depends on the properties of its components. We also demonstrate how a hierarchical Markov model can be parametrized using experimental data and show that it provides a better representation than a previous model of the same dataset. Because evidence is increasing that modal gating reflects underlying molecular properties of the channel protein, it is likely that biophysical processes are better captured by our new approach than in earlier models.

Invasive competition with Fokker-Planck diffusion and noise

Defeat and success of the competitive invasion of a populated area is described with a standard Lotka-Volterra competition model. The resident is adapted to the heterogeneous living conditions, i.e., its motion is modelled as space-dependent, so-called Fokker-Planck diffusion. The invaders diffusion is taken as neutral Fickian. Furthermore, it is studied how multiplicative environmental noise fosters or hinders the invasion.

Mathematical Models of Pattern Formation in Planktonic Predation-Diffusion Systems: A Review

Plankton form the basis of aquatic food webs. The mathematical modelling of plankton dynamics was initiated by fisheries in the early 20th century. Today, the significant role of plankton in the global carbon cycle and, hence, in climate control has been recognized. The main aim of modelling is to improve understanding of the functioning of food chains and webs and their dependence on internal and external conditions. Population-dynamical models have not only to account for growth and interactions but also for spatial processes like random or directed and joint or relative motion of species as well as the variability of the environment. Early attempts began with exponential growth, Lotka-Volterra type interactions and physico-chemical diffusion. These approaches have been continuously refined to more realistic descriptions of the development of natural populations. The aim of this paper is to give an introduction to the subject of equation-based modelling and the corresponding bibliography, based on and extending previous reviews [1]–[5]. The fascinating variety of temporal, spatial and spatio-temporal patterns in such systems and the governing mechanisms of their generation and further evolution are described and related to plankton dynamics.

An applied mathematician's perspective on Rosennean complexity

Abstract The theoretical biologist Robert Rosen developed a highly original approach for investigating the question “What is life?”, the most fundamental problem of biology. Considering that Rosen made extensive use of mathematics it might seem surprising that his ideas have only rarely been implemented in mathematical models. On the one hand, Rosen propagates relational models that neglect underlying structural details of the components and focus on relationships between the elements of a biological system, according to the motto “throw away the physics, keep the organisation”. Rosens strong rejection of mechanistic models that he implicitly associates with a strong form of reductionism might have deterred mathematical modellers from adopting his ideas for their own work. On the other hand Rosens presentation of his modelling framework, (M, R) systems, is highly abstract which makes it hard to appreciate how this approach could be applied to concrete biological problems. In this article, both the mathematics as well as those aspects of Rosens work are analysed that relate to his philosophical ideas. It is shown that Rosens relational models are a particular type of mechanistic model with specific underlying assumptions rather than a fundamentally different approach that excludes mechanistic models. The strengths and weaknesses of relational models are investigated by comparison with current network biology literature. Finally, it is argued that Rosens definition of life, “organisms are closed to efficient causation”, should be considered as a hypothesis to be tested and ideas how this postulate could be implemented in mathematical models are presented.