Approximations and their consequences for dynamic modelling of signal transduction pathways

Signal transduction is the process by which the cell converts one kind of signal or stimulus into another. This involves a sequence of biochemical reactions, carried out by proteins. The dynamic response of complex cell signalling networks can be modelled and simulated in the framework of chemical kinetics. The mathematical formulation of chemical kinetics results in a system of coupled differential equations. Simplifications can arise through assumptions and approximations. The paper provides a critical discussion of frequently employed approximations in dynamic modelling of signal transduction pathways. We discuss the requirements for conservation laws, steady state approximations, and the neglect of components. We show how these approximations simplify the mathematical treatment of biochemical networks but we also demonstrate differences between the complete system and its approximations with respect to the transient and steady state behavior

Numerical study of cancer cell invasion dynamics using adaptive mesh refinement: the urokinase model

In the present work we investigate the chemotactically and proteolytically driven tissue invasion by cancer cells. The model employed is a system of advection-reaction-diffusion equations that features the role of the serine protease urokinase-type plasminogen activator. The analytical and numerical study of this system constitutes a challenge due to the merging, emerging, and travelling concentrations that the solutions exhibit. Classical numerical methods applied to this system necessitate very fine discretization grids to resolve these dynamics in an accurate way. To reduce the computational cost without sacrificing the accuracy of the solution, we apply adaptive mesh refinement techniques, in particular h-refinement. Extended numerical experiments exhibit that this approach provides with a higher order, stable, and robust numerical method for this system. We elaborate on several mesh refinement criteria and compare the results with the ones in the literature. We prove, for a simpler version of this model, $L^\infty$ bounds for the solutions, we study the stability of its conditional steady states, and conclude that it can serve as a test case for further development of mesh refinement techniques for cancer invasion simulations

Ranking ligand affinity for the DNA minor groove by experiment and simulation

The structural and thermodynamic basis for the strength and selectivity of the interactions of minor-groove binders (MGBs) with DNA is not fully understood. In 2003 we reported the first example of a thiazole containing MGB that bound in a phase shifted pattern that spanned 6 base-pairs rather than the usual 4 (for tricyclic distamycin-like compounds). Since then, using DNA footprinting, nuclear magnetic resonance spectroscopy, isothermal titration calorimetry and molecular dynamics, we have established that the flanking bases around the central 4 being read by the ligand have subtle effects on recognition. We have investigated the effect of these flanking sequences on binding and the reasons for the differences and established a computational method to rank ligand affinity against varying DNA sequences

Quantifying the implicit process flow abstraction in SBGN-PD diagrams with Bio-PEPA

For a long time biologists have used visual representations of biochemical networks to gain a quick overview of important structural properties. Recently SBGN, the Systems Biology Graphical Notation, has been developed to standardise the way in which such graphical maps are drawn in order to facilitate the exchange of information. Its qualitative Process Diagrams (SBGN-PD) are based on an implicit Process Flow Abstraction (PFA) that can also be used to construct quantitative representations, which can be used for automated analyses of the system. Here we explicitly describe the PFA that underpins SBGN-PD and define attributes for SBGN-PD glyphs that make it possible to capture the quantitative details of a biochemical reaction network. We implemented SBGNtext2BioPEPA, a tool that demonstrates how such quantitative details can be used to automatically generate working Bio-PEPA code from a textual representation of SBGN-PD that we developed. Bio-PEPA is a process algebra that was designed for implementing quantitative models of concurrent biochemical reaction systems. We use this approach to compute the expected delay between input and output using deterministic and stochastic simulations of the MAPK signal transduction cascade. The scheme developed here is general and can be easily adapted to other output formalisms

Hybrid modelling of time-variant heterogeneous objects.

The physical world consists of a wide range of objects of a diverse constitution. Past research was mainly focussed on the modelling of simple homogeneous objects of a uniform constitution. Such research resulted in the development of a number of advanced theoretical concepts and practical techniques for describing such physical objects. As a result, the process of modelling and animating certain types of homogeneous objects became feasible. In fact most physical objects are not homogeneous but heterogeneous in their constitution and it is thus important that one is able to deal with such heterogeneous objects that are composed of diverse materials and may have complex internal structures. Heterogeneous object modelling is still a very new and evolving research area, which is likely to prove useful in a wide range of application areas. Despite its great promise, heterogeneous object modelling is still at an embryonic state of development and there is a dearth of extant tools that would allow one to work with static and dynamic heterogeneous objects. In addition, the heterogeneous nature of the modelled objects makes it appealing to employ a combination of different representations resulting in the creation of hybrid models. In this thesis we present a new dynamic Implicit Complexes (IC) framework incorporating a number of existing representations and animation techniques. This framework can be used for the modelling of dynamic multidimensional heterogeneous objects. We then introduce an Implicit Complexes Application Programming Interface (IC API). This IC API is designed to provide various applications with a unified set of tools allowing these to model time-variant heterogeneous objects. We also present a new Function Representation (FRep) API, which is used for the integration of FReps into complex time-variant hybrid models. This approach allows us to create a practical multilevel modelling system suited for complex multidimensional hybrid modelling of dynamic heterogeneous objects. We demonstrate the advantages of our approach through the introduction of a novel set of tools tailored to problems encountered in simulation applications, computer animation and computer games. These new tools empower users and amplify their creativity by allowing them to overcome a large number of extant modelling and animation problems, which were previously considered difficult or even impossible to solve

Stronger computational modelling of signalling pathways using both continuous and discrete-state methods

Starting from a biochemical signalling pathway model expresses in a process algebra enriched with quantitative information, we automatically derive both continuous-space and discrete-space representations suitable for numerical evaluation. We compare results obtained using approximate stochastic simulation thereby exposing a flaw in the use of the differentiation procedure producing misleading results