Gene modelling Flashcards
state of a system
snapshot of the system at a given time that
contains enough information to predict the behavior of the system for all future times.
differential equation models
list of concentrations of each chemical type
boolean models
a list, for each gene involved, of whether the gene is expressed or not expressed
stochastic model configuration
list of the actual number of molecules of each type
- state is configuration or current probability distribution
molecular dynamics model
list of positions and momenta of each molecule
Each model does ______
- defines what it means by the state of the system
- model predicts which state or states can occur next (given current state)
- some states are equilibrium states in the sense that once in that state, the system stays in that state
kinetics
changes of state
equilibria
which states are equilibrium states
A state is said to be at equilibrium when
its state ceases to change
- forward reaction = back reaction
differential equation for [X*DNA]
d[XDNA]/dt = k1[X][DNA] - k-1[XDNA] = 0(at equilibrium)
Keq
k1/k-1 = [X*DNA]/[X][DNA]
What happens when applying the equilibrium condition to the differential equation of the change in [X*DNA]
the differential equation is reduced to an algebraic equation
(d[XDNA]/dt = k1[X][DNA] - k-1[XDNA] –> k1/k-1 = [X*DNA]/[X][DNA]
stochastic equation typically uses the _______ of molecules rather than the _______ of molecules
number
concentration
Real physical systems tend towards equilibrium unless
energy is added
equilibrium
the state of the system as time->∞
To simplify models
fast reactions assumed to be at equilibrium
If a and b are in equilibirum and b and c are in equilibrium, then __________-
a and c are in equilibrium
From an energy difference
one can predict the final state of the system,
cannot predict the time course of the state from initial to final
Gibbs free energy
G =
(total internal energy) - (absolute temperature) * entropy + pressure * volume = ∑(chemical potential) * (partical number)
for reactants and products with the free energy difference ∆G
Keq =
e^(-∆G/RT)
Paritition functions [DNA]3/[DNA]total = Z = K3 = [DNA]total
k3[P][Q]/Z
1 + K + K + K 1 2 3 [P] [Q] [P][Q].
[DNA]3/[P][Q][DNA]0
[DNA]0 + k1[P][DNA]0 + K2[Q][DNA]0 + K3[P][Q][DNA]0 = [DNA]0Z
Questions to consider when evaluating models
State of the model
How does the state change over time
What are the equilibrium states
Others What assumptions(biological and computational) For what time scale is model valid # molecules present Time complexity Amount of data available What can it expect to predict
If the # of molecules is small (_______) ______ must be used
Once very large, use _______
tens or low hundreds
stochastic models
differential equations
Boolean network models
- each gene fully expressed or not expressed at all
- state space in finite
- obtain a first representation of a complex system
- all genes change state at same time
kinetic logic models
state of each gene as a discrete value
- deal with rates at which systems change from one state to another
- genes change state at independent rates
continuous logical models
transition
from one state to another is governed by linear differential equations with constant coefficients
Differential equation models
provide a general framework in which to consider gene
regulation processes
One way to transform a system of chemical reactions and physical constraints into a system of non-ordinary differential equations
reaction/enzyme kinetics: considering the transition rates between all microscopic states
differential equations assume
changes of state are continuous and deterministic
Langevin approach
adding a noise term to a differential equation (useful for non-deterministic problems)
Fokker-plank method
- start from a probabilistic framework
- write equations that describe a change in probabilistic framework as a function of time
- assume continuity: probability distribution is a continuous function of concentration
Fully stochastic models consider
the individual molecules involved in gene regulation
Langevin and Fokker-Plank models are approximations of
a fully stochastic model
Truth tables
specify what the next state is for each current state
with n genes, there are 2^n states, exactly one next state
Ways to represent boolean models
Truth tables
finite-state machines
differential equations approach to modelling gene regulations
state is a list of concentrations of each chemical species
- concentrations continuous
Use a stochastic model when
assume solution is well mixed(a given molecule is equally likely to be anywhere in the solution)
- rate of molecular collisions is much greater than the rate of molecular reactions
Assumptions for differential equations framework
- # of molecules is sufficiently high that discrete changes of a single molecule can be approximated as continuous changes in concentration
- Fluctuations about the mean are small compared to the mean itself
To go from differential equations to boolean models
one assumes that the function ƒ in the differential equation (d/dt)v = ƒ(v) has saturating nonlinearities (when v is very small or very large, ƒ(v) tends towards a limit)
- one assumes that the nonsaturated region between two extremes is transient and can be ignored
