These are the abstracts of the papers that were presented in the First BIOMICS Summer Workshop (ordered according to the agenda). In most cases the full papers are available through the proceedings, downloadable from the Publications page.
Biological functionality arises from the complex interactions of simple components. Traditionally a reductionist approach has led to identification of cellular and molecular components. Detailed cartoons like wiring diagrams present the interactions in signalling pathways and gene network regulation. More recently the rise of high throughput technologies and increasingly sophisticated computational tools have revealed an increasing regulatory complexity, many more interacting signalling pathways and interdependencies between components. With the increasing complexity verbal reasoning and simple cartoons can no longer explain the dynamics of the interactions. A systems biology approach combining computing, mathematics and biology is therefore needed to analyse the complex interactions and dynamics. Here I focus on in silico modelling using Ordinary Differential Equations for the analysis of signalling and gene network regulation.
Systems Biology: Computing, Mathematics and Biology
Symmetry Analysis is a method which allows a systematic construction of exact solutions of differential equations. In this talk we will outline the underlying concepts and techniques utilised for solving first order ordinary differential equations (ODEs) and systems of first order ODEs. We will then proceed to exhibiting how these techniques can be applied in order to elucidate biological systems.
The paper uses the analysis of two cell regulatory/metabolic component systems by Lie groups, and otherwise, to begin a critical discussion of the Lie group method itself in terms of what it tells us about the problems it is able to solve.
In this talk the linearised symmetry condition for a system of ODEs will be presented. We also give a natural decomposition of the Lie algebra of infinitesimal symmetries into a direct sum of two components: the trivial symmetries (determined by the direction of the flow of the vector field associated to the system of ODEs), and the time-preserving infinitesimal symmetries (their action on the "time coordinate" is trivial). We show that the time-preserving infinitesimal symmetries can be characterised by the Lie bracket condition.
The general method of Lie-algebras is essential to many parts of mathematics and physics. This talk will give a brief review of the structures of finite-dimensional Lie algebras over the complex field. The theory of semisimple Lie algebras consists of very nice and attractive results utilizing various tools of linear algebra, abstract algebra and graph theory, including the adjoint representation, the Killing form, root space decomposition and Dynkin diagrams.
In this talk I would like to describe the algebra O of octonions or Cayley numbers. O is an 8-dimensional real algebra which can be constructed from the skew field of quaternions using an involutory anti-automorphism a → â, called conjugation (Cayley-Dickson process). O is a non-associative, but alternative, biassociative division algebra such that the Lie algebra of its derivations forms a 14-dimensional simple Lie algebra A. Using the classification of the simple Lie algebras, A is isomorphic to the Lie algebra of the exceptional compact Lie group G2.
Simple Lie algebras play a crucial role both in the theory of Lie groups and in finite group theory. On the one hand, simple Lie groups can generally be found as automorphism groups of simple Lie algebras. On the other hand, by the classification theorem of finite simple groups (CFSG), most of the finite simple groups are related to simple Lie algebras over finite fields. The goal of the talk is to highlight the relation between Lie algebras and groups.
Biochemical and biological dynamical systems are frequently described by directed graphs with labelled arrows and nodes. Often no strict convention is followed for the semantics of these graphs, but nevertheless such graphs are often given rigorous mathematical interpretations as specifications of dynamical systems, e.g. giving detailed stoichiometric specification of chemical reactions or other transformations. These directed graphs are typically labelled with kinetic rate constants, such as those involved in, e.g., Michaelis-Menten kinetics or rates of genetic transcription, and these 'constants' are often subject to modulation by many factors. Often graphical representations of these dynamical systems do not uniquely determine them (even for the same author within the same scientific paper), but require additional assumptions that systems biologists can deduce from the text to further constrain the "data flow" expressed in the graphs and processes behind this flow (from which one can, e.g., set up a well-defined system of ordinary differential equations(ODEs)). Systems biologists construct differential equation models, or Petri-net and automata models, based on such directed graphs, augmented by additional knowledge. Conversely, automatic tools exist that produce graphical networks based on ODE descriptions, and increasingly journals and scientific databases include and require SMBL (Systems Biology Markup Language) formulations of biological models. For such graphical models, computational tools support selection of subsets of variables, extension via new reactions, and Petri-net simulation or numerical integration of solutions, etc. To what extent can these constructions be made functorial so that mathematical structure can be used to relate solutions and dynamics in one kind of model to other kinds? For the systems biologist, inclusion, restriction, and joining of models are very simple, natural operations. Their functorial properties in this domain have apparently not been studied, although it is well-known that they can fundamentally affect the character of model dynamics. Directed graphs also have spectral properties associated to the canonical adjacency matrices describing them, as well as automata and transformation semigroups canonically associated to them (whose subgroups may correspond to "natural subsystems"). This raises the questions of what are categories appropriate for achieving functorial correspondences between these domains of interest - what are appropriate choices for defining their objects and morphisms, and what are the functors and adjoint relationships between these categories? While the heuristic use of simplified models in specific contexts of systems biology has certainly been fruitful, one must recognize that the model is a cartoon version of events and, perhaps, not the best basis upon which to draw conclusions concerning general principles of the fundamentals of biological systems. Nevertheless, constructions of combining, merging, and restricting models is carried out routinely, and functorial understanding of these operations would clarify their significance, and might provide a framework for thinking about scaling up complex systems and computation based on interaction. We discuss some initial explorations of these questions and consider some of their implications for the mathematical and biological foundations of interaction computing.
Finite discrete automata models of biological systems such as the lac operon, Krebs cycle, and p53-mdm2 genetic regulatory system constructed directly or from Boolean networks or Petri nets have canonically associated algebraic structures -- their transformation semigroups. These often contain permutation groups, which are local substructures exhibiting symmetry, and correspond to "pools of reversibility" or "natural subsystems". These natural subsystems are related to one another in a hierarchically structured manner. This relation is captured by the notion of "weak control" which hierarchically relates these natural subsystems (under which equivalent subsystems are in fact isomorphic as permutation groups). We exhibit examples of natural subsystems arising from several biological examples and describe their weak control hierarchies in detail.
Control-Flow Integrity (CFI) is a basic safety property of systems. Its enforcement is simple and practical, prevents attackers from arbitrarily controlling program behaviour, and has been addressed for stand-alone machines using inlined reference monitors and binary rewriting. However, CFI enforcements are thought to deal with the property on a stand-alone level. Nowadays, web applications are an increasing trend, and CFI violations can be observed at the level of HTTP request/responses initiated by the user of the web application, so CFI enforcement needs to be addressed at such level. In this work, we present a formal framework for the definition and enforcement of CFI policies for web applications, and we roughly cover the areas of modular definition of policies, their composition, and their enforcement using cascaded enforcement monitors.
This paper is motivated by our desire to understand how Krohn-Rhodes theory can be used for solving classical problems in computer science, and to identify the advantages and difficulties of applying such theory. For this purpose, we study the 3-Queens puzzle, which consists of finding a configuration of as many queens as possible on an 3 x 3 chessboard where no queen threatens another. We represent the 3-Queens puzzle as a graph-search problem, and formalize this graph-search as a transformation semigroup, so that it is possible to decompose it using holonomy decomposition, and we provide some insights into how variations in modelling choices yield different decomposition results.