Computational Cell Biology

Transcription

1 Computational Cell Biology Course book: Fall, Marland, Wagner, Tyson: Computational Cell Biology, 2002 Springer, ISBN (can be found in the main library, prize at amazon.com $59.95). Synopsis: the purpose is to get a flavour of mathematical/numerical methods used in modelling dynamic processes in and between cells. The processes themselves hold an infinite reservoir of (boring?) details! -> best to concentrate on understanding the generic methods and use them to analyse some of the most important cell processes.

2 Methods Qualitative analysis of differential equation systems Phase plane (nullcline) analysis of two-dimensional flows, (classification of) bifurcations General reference: Strogatz: Nonlinear Dynamics and Chaos, 1994 Perseus Books Publishing, LLC, ISBN Additional (a bit more rigorous): Hochstadt: Differential Equations, Dover, ISBN Very basic Euler & Runge-Kutta - numerical methods Reference: Press, Teukolsky, Vetterling, Flannery: Numerical Recipes in C++/c/F90/F77, Cambridge University Press Equations for chemical and stochastic processes Rate equations, master equations, Schmoluchowski and Langevin equations References: Gardiner: Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, 1997 Springer, ISBN ; van Kampen: Stochastic Processes in Physics and Chemistry, 1992 Elsevier, ISBN ;Allen, Tildesley, Computer simulation of Liquids, Oxford University Press. Loads of pretty diagrams depicting the processes.

3 Topics Voltage Gated Ionic Currents and Ion Pumps Basis of the ionic battery, cell membrane model Ion-specific pumps and pores allow the transfer of charge up and down gradients e.g. a voltage-gated Potassium ion channel can be modelled simply by I = g V K K Transporters and pumps are based on a great variety of transport proteins for moving both ions and molecules from one cellular compartment to another The equilibrium ion distribution across the cell membrane set by the balance of osmotic force due to concentration gradient and the potential gradient caused by the charge gradient (selective ion channel); the so called Nernst potential across the cell membrane.

4 Topics The operation of these gated ion channels and the protein assisted ion pumps and their interactions are modelled by a set of differential equations which will necessarily have to be simplified to make them somewhat comprehendible Typically one separates the dynamics into slow and fast time scales and treats the slow variables as constants when analysing the fast ones (e.g. fast gating of the channels vs. restoration of the potential gradient by the protein-assisted ion pumps) or by assuming the fast processes to have attained an equilibrium

5 transmembrane ion Ion channels pumps potential -85 mv ion transport against the concentration gradient electrode

6 Excitation of a cardiac cell: The Action Potential plateau Ca ++ in repolarization Na + in threshold depolarization refractory period K + out

7 Topics Whole-Cell Models In a cell Ca++ binds to many proteins and modifies their enzymatic properties Thus Ca++ concentration is typically low save for brief and highly localised rises This is accomplished by two basic mechanisms: buffering and sequestration. Buffers are specialised Ca++ binding proteins that soak up % of the Ca++ in cytosol. Sequestering is done in internal stores the sarcoplasmic reticulum (SR) in muscle cells and the endoplasmic reticulum (ER) in other cells by proteins that hydrolyse ATP (split ATP and and water yielding ADP, phosphate and proton) to move Ca++ against steep concentration gradients. These proteins are called sarco/endoplasmic Ca++-ATPase (SERCA) pumps. Other pumps, plasma membrane Ca++- ATPase (PMCA) remove Ca++ from the cell.

8 Ca ++ waves in cytoplasm of Xenopus oocytes

9 Topics Intercellular communication E.g. heart cells communicate via electrical coupling: ionic current flows through channel proteins that span the plasma membranes of both communicating cells. The clusters of such proteins are called gap junctions.

10 Topics Spatial modelling Relax the constraint that the chemical concentrations are essentially uniform in space; crucial e.g. in the situation where a nerve cell fires so that a wave of membrane depolarisation is initiated at the base of the axon. Other examples of spatial phenomena in cell biology:

11 Topics the collective motion of amoeba is organised by waves of cyclic AMP that propagate through the extracellular medium Aggregation of Dictyostelium Discoidium amoebae by camp signalling

12 BZ chemical reaction

13 Topics The most general way of describing the evolving spatio-temporal patterns is the reaction-diffusionadvection equation c c( x, t) + ν c( x, t) D = f ( x, t, c) t x x where c(x,t) is rhe concentration, ν the velocity of the carrying fluid, D the diffusion constant, and f(x,t,c) includes the sources and sinks for c. The stoch.d.e. is then integrated in time (ranges from hard to impossible for realistic systems) clever simplifications and averages are called for.

14 Conclusion Systems to be studied are hard, exact solutions are rare The way to go is to develop an intuition on these systems based on which one builds models to play with General refs for the terminology and concepts: Alberts, et al.: Molecular Biology of the Cell, Garland Publishing; Nelson: Biological Physics, W.H. Freeman & Co.