Subgroup Contributed Talks

Fiona Macfarlane

University of St. Andrews, Scotland, frm3@st-andrews.ac.uk

"Bridging the gap between individual-based and continuum models of growing cell populations"

Stochastic individual-based modelling approaches allow for the description of single cells in a biological system. These models generally include rules that each cell follows independently of other cells in the population. Various mechanisms and biological phenomena can be described using these simplistic mathematical models. However, these models cannot be analysed mathematically. Therefore, it can be beneficial to derive the corresponding deterministic model from the underlying random walk of the stochastic model. The resulting deterministic models, usually partial differential equations (PDEs), can then be analysed to provide further information about the biological systems studied. We have developed a range of simple IB models that describe biological systems with various properties of interest, such as, volume-filling and chemotaxis and pressure-dependent growth and proliferation. For each model, we were able to derive PDEs from the underlying random walk. We carried out comparisons between the stochastic and deterministic model highlighting situations where there is agreement in the models, and situations where they do not agree. Ultimately, the results illustrate how the simple rules governing the dynamics of single cells in our individual-based model can lead to the emergence of complex spatial patterns of population growth observed in continuum models. These models can be applied to a variety of biological situations such as tumour growth, tumour invasion and wound healing.

Alexey Penenko

ICM&MG SB RAS, Russian Federation, a.penenko@yandex.ru

"Inverse Modeling of Biological Processes with Adjoint Ensemble Methods"

In order to study complicated natural processes through mathematical modeling, it is necessary to take into account both mathematical models of the processes and the available measurement data collected about these processes. We refer to this modeling approach as inverse modeling. In biological studies, the models can change very rapidly, so the unified algorithms that do not take much effort to apply them to different models are preferable. The general inverse modeling framework based on the ensembles of the adjoint equation solutions is considered in the context of biomedical applications with the advection-diffusion-reaction models. The sensitivity relations and adjoin equations allow constructing the approximation of the inverse modeling problem stated a system of ordinary or partial differential equations in the form of a parametric family of quasi-linear operator equations with the sensitivity operators. This approximation allows both solving, analyzing, and comparing a wide range of biomedical inverse problems in a unified fashion.

Karina Islas Rios

Monash University Australia karina.islasrios@monash.edu

"NetScan: a computational tool for discovering and visualizing biochemical networks with defined topological structures."

Our quantitative understanding of biochemical networks empowered by computational modelling have shown that the topology (or structure) of a network often have determining roles in shaping the network’s dynamic and steady state behaviours. For examples, negative feedback can give rise to oscillation while positive feedback can bring about bistability to the host network. Thus, being able to systematically identify sub-networks with defined topological structures within the human protein interactome is critical for the discovery of biochemical networks with desired behavioural properties. However, this is a non-trivial task given the enormous complexity of the human protein-protein interaction network. Structural principles of biochemical networks can be discovered by focusing on small sub-networks. Finding those sub-networks in the assembly of complex biochemical networks can be achieved by implementing graph theory-based algorithms. Here, we develop NetScan, an open source web-based application capable of ‘scanning’ the large and complex human signalling interactome within the Signor 2.0 and STRING databases to identify all sub-networks with given structural topologies, e.g. those with a specific negative feedback, positive feedback or feed-forward loop wiring. NetScan allows users to specify the specific input topologies and the interactome network within which it will explore, and return all the smallest sub-networks with the desired topologies. The resulting sub-networks are displayed in two forms: a detailed version which includes all interaction links, and a simplified version presenting the net effects between the nodes. In summary, NetScan is a web application that provides unprecedented ability to systematically identify and visualise sub-networks within the human protein-protein interaction network with specific topological wiring.

Hyukpyo Hong

Korea Advanced Institute of Science and Technology, Republic of Korea, hphong@kaist.ac.kr

"Derivation of stationary distributions of biochemical reaction networks via structure transformation"

Long-term behaviors of biochemical reaction networks are described by steady states in deterministic models and stationary distributions in stochastic models. Unlike deterministic steady states, stationary distributions capturing inherent fluctuations of reactions are extremely difficult to derive analytically due to the curse of dimensionality. Here, we develop a method for deriving stationary distributions from deterministic steady states by transforming a network to have a special dynamic property. Specifically, we merge nodes and edges of a network to make the steady states complex balanced, i.e., the in- and out- flows of each node are equal. By applying our approach to networks that model autophosphorylation of EGFR, PAK1, and Aurora B kinase and a multi-timescale toggle switch, we identify robustness, sensitivity, and multi-modality of their stationary distributions. Our method provides an effective tool to understand long-term behaviors of stochastic biochemical systems.

Linard Hoessly

University of Copenhagen, Denmark, hoessly@math.ku.dk

"Stationary distributions of stochastic reaction networks via decomposition"

Stochastic reaction networks (CRNs) are often used to describe systems with small molecular counts, which applies to many processes in living systems. They are usually modelled through continuous-time Markov processes. Studying dynamics of stochastic CRNs is in general hard, both analytically and by simulation. In particular stationary distributions of stochastic reaction networks are only known in some cases like, e.g., complex balanced or autocatalytic CRNs. I will review some results on form of stationary distribution and convergence to stationary distribution. Then, I am going to analyse CRNs under the operation of join and examine conditions such that the form of the stationary distributions of a CRN is derived from the parts of the decomposed CRNs, which allows recursive application. To illustrate the theory I present examples of stochastic reaction networks of interest in order to highlight the decomposition.