Download Computational Cell Biology

Computational Cell Biology
• Course book: Fall, Marland, Wagner, Tyson:
Computational Cell Biology, 2002 Springer,
ISBN 0-387-95369-8 (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.
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 0-7382-0453-6
• Additional (a bit more rigorous): Hochstadt: Differential Equations, Dover, ISBN 0-48661941-9
– 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 3-540-61634-9; van Kampen: Stochastic
Processes in Physics and Chemistry, 1992 Elsevier, ISBN 0-444-89349-0;Allen,
Tildesley, Computer simulation of Liquids, Oxford University Press.
•
Loads of pretty diagrams depicting the processes.
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 K = g K ∆V
• 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.
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
transmembrane
ion
Ionchannels
pumps
potential
-85 mV
ion transport against the
concentration gradient
electrode
Excitation of a cardiac cell: The Action Potential
plateau
Ca++ in
repolarization
K+ out
Na+
threshold
in
depolarization
refractory period
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 95-99 % 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.
Ca++ waves in
cytoplasm of
Xenopus
oocytes
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.
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:
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
BZ chemical reaction
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.
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.