9 Importance Applying mathematics to biology has a long history, but only recently has there been an explosion of interest inthe field. Some reasons for this include:the explosion of data-rich information sets, due to the genomics revolution, which are difficult to understand without the use of analytical tools,recent development of mathematical tools such as chaos theory to help understand complex, nonlinear mechanisms in biology,an increase in computing power which enables calculations and simulations to be performed that were not previously possible, andan increasing interest in in silico experimentation due to the complications involved in human and animal research.
10 Why to Model Biological Systems To help reveal possible underlying mechanisms involved in a biological processTo help interpret and reveal contradictions/incompleteness of dataTo help confirm/reject hypothesesTo predict system performance under untested conditionsTo supply information about the values of experimentally inaccessible parametersTo suggest new hypotheses and stimulate new experiments
11 Limitations of Mathematical Models Not necessarily a ‘correct’ modelUnrealistic models may fit data very well leading to incorrect conclusionsSimple models are easy to manage, but complexity is often requiredRealistic simulations require a large number of hard to obtain parametersModels are not explanations and can never alone provide a complete solution to a biological problem.
12 How Are Models Derived? Start with at problem of interest Make reasonable simplifying assumptionsTranslate the problem from words to mathematically/physically realistic statements of balance or conservation laws
13 What do you do with the model? Solutions—Analytical/NumericalInterpretation—What does the solution mean in terms of the original problem?Validation—Are results consistent with experimental observations?Predictions—What does the model suggest will happen as parameters change?
14 “Modelling could seem simple to an outsider: define your system, choose a modelling approach on the basis of what you know and want to know, download a simulation tool, input some parameters, run the simulation, and collect the results. However, as in the earlier days of protein design, what looks nice on a screen does not necessarily carry any biological meaning. There are many conceptual pitfalls for the modeler, which result in unrealistic predictions.”Ventura et al, Nature, Vol 443|5 October 2006
15 Modeling Examples - I ”Modeling the Heart-from Genes to Cells to the Whole Organ”Noble, D. Science 2002; 295:
16 Modeling Examples - II Tumer growth Mathematical models have been developed thatdescribe tumorprogression and helppredict response totherapy.
17 Modeling Examples - III HOG pathwayKlipp et al. 2005, Nature Biotech.
18 And other examples… Modelling of neurons Mechanics of biological tissuesTheoretical enzymology and enzyme kineticsCancer modelling and simulationModelling the movement of interacting cell populationsMathematical modelling of scar tissue formationMathematical modelling of intracellular dynamics
19 Modeling processSystematic modeling approach – an overview
20 Model focus what are we modeling and why? Modeling to see and understand a specific systemHypothesis driven
21 Mathematical framework A model of a biological system is converted into a system ofequations, although the word 'model' is often usedsynonymously with the system of corresponding equations.The solution of the equations, by either analytical or numericalmeans, describes how the biological system behaves eitherover time or at equilibrium.There are many different types of equations and the type of behaviorthat can occur is dependent on both the model and the equationsused. The model often makes assumptions about the system. Theequations may also make assumptions about the nature ofwhat may occur
22 Mathematical framework Deterministic processes (dynamical systems)A fixed mapping between an initial state and a final state. Starting from an initial condition and moving forward in time, a deterministic process will always generate the same trajectory and no two trajectories cross in state space.Ordinary differential equations (Continuous time. Continuous state space. No spatial derivatives.)Partial differential equations (Continuous time. Continuous state space. Spatial derivatives.)Maps (Discrete time. Continuous state space)
23 Mathematical framework Stochastic processes (random dynamical systems)A random mapping between an initial state and a final state, making the state of the system a random variable with a corresponding probability distribution.Non-Markovian processes -- Generalized master equation (Continuous time with memory of past events. Discrete state space. Waiting times of events (or transitions between states) discretely occur and have a generalized probability distribution.)Jump Markov process -- Master equation (Continuous time with no memory of past events. Discrete state space. Waiting times between events discretely occur and are exponentially distributed.) See also Monte Carlo method for numerical simulation methods, specifically Continuous-time Monte Carlo which is also called kinetic Monte Carlo or the stochastic simulation algorithm.Continuous Markov process -- stochastic differential equations or a Fokker-Planck equation (Continuous time. Continuous state space. Events occur continuously according to a random Wiener process.)
24 Constructing the network Using what we know about our system toconstruct a biochemical network. E.g whatare the entities involved? in what processes?Where does it all take place? Etc.
27 Components in Biochemical Networks The biochemical entities are involved in different processes :Reactions (catalysis, transfer, binding, assembly/disassembly)Translocations (e.g., transport – active/passive)Expression
28 Components in Biochemical Networks The processes themselves might be influenced by other players:Enzymes catalyze reactions (activator)Transcription factors facilitates expression (activator)Metabolites may inhibit reaction rates (inhibitor)
29 Components in Biochemical Networks The processes take place somewhere:Reactions of entities within a compartmentTransport of an entity over a membrane (between compartments)Recruitment of proteins to a membraneDimerization of receptors on a membrane
30 Rate expression, parameter estimation and validation Out of the scope of this lecture but we will see something in the example and more in the future…
32 StandardizationComputational, dynamical modeling is essential to achieving a deeper understanding of mechanisms underlying complex biological systemsThere is currently a proliferation of software toolsDifferent packages have complementary strengthsModel editing, simulation, analysis, displayNo single tool is able to do everythingNew techniques ( new tools) evolve all the timeResearchers need to use more than one toolThe field need agreed-upon standards enabling tools to be used cooperatively
33 SBMLA machine-readable format for representing computational models in systems biologyExpressed in XML (Extensible Markup Language)Intended for software tools— not for humans(Although it is text-based and therefore readable)Intended to be a tool-neutral exchange language for software applications in systems biologySimply an enabling technology