What is noise? - Personal Homepages

Transcription

Stochastic simulations!
Application to biomolecular networks!
Didier Gonze!
Unité de Chronobiologie Théorique!
Service de Chimie Physique - CP 231!
Université Libre de Bruxelles!
Belgium!
What is noise?
Genetically identic cells/organisms
can display some variability in their
physiology/behavior. This is due to
noise (i.e. stochastic effects).!
Lahav (2004) Science STKE
What is noise?
Oscillations and variability
in the p53 system.
Geva-Zatorsky N, Rosenfeld N,
Itzkovitz S, Milo R, Sigal A, Dekel
E, Yarnitzky T, Liron Y, Polak P,
Lahav G, Alon U. Mol Syst Biol.
(2006) 2:2006.0033
Stochastic gene
expression in E. coli.
M. Elowitz
http://www.elowitz.caltech.edu/
Max Delbruck
1940
"One molecule of pepsin should be sufficient under favorable conditions
to convert in a few hours any weighable amount of pepsin-precursor. In
experiments designed to test this one must be prepared to encounter
very large statistical fluctuations in the amount of reaction taking place
in a given time, or vice versa in the time required to effect a given amount
of reaction"
Max Delbrück (1906-1981)
German–American biophysicist
Nobel Prize Physiology/Medicine 1969
(replication mechanism and the genetic
structure of viruses)
[...]
"A closer theoretical study of their finer details seems therefore desirable
as an aid for the design of experiments and to prepare the way for a
discussion of the possible importance of such fluctuations for cell
physiology."
Stochastic gene expression: pioneer works
§  Singh UN (1969) Polyribosomes and unstable messenger RNA: a stochastic model
of protein synthesis. J Theor Biol. 25:444-60.
§  Blum H (1974) Stochastic processes in messenger RNA turnover. J Theor Biol.
48:161-71
§  Hiernaux J (1974) On some stochastic models for protein biosynthesis. Biophys
Chem. 2:70-5.
§  Smeach SC (1975) Stochastic and deterministic models for the kinetic behavior of
certain structured enzyme systems I: one enzyme-one substrate systems. J Theor
Biol. 51:59-78.
§  Rigney DR (1979) Stochastic model of constitutive protein levels in growing and
dividing bacterial cells. J Theor Biol. 76:453-80.
§  Ko MS (1991) A stochastic model for gene induction. J Theor Biol. 153:181-94
§  McAdams HH, Arkin A (1997) Stochastic mechanisms in gene expression. Proc Natl
Acad Sci USA. 94:814-9.
Overview
§  Introduction: theory and simulation methods
- Definitions (intrinsic vs extrinsic noise, robustness,...)
- Deterministic vs stochastic approaches
- Master equation, birth-and-death processes
- Gillespie and Langevin approaches
- Application to simple systems
§  Literature overview
- Measuring the noise, intrinsic vs extrinsic noise
- Determining the souces of noise
- Assessing the robustness of biological systems
§  Application to circadian clocks
- Molecular bases of circadian clocks
- Robustness of circadian rhythms to noise
Noise in biology
Multiple sources of noise:
§  Variablity of the
environment, of the
number of ribosomes, etc
§  Inequal partition of
molecules at cell division
§  Random collisions and
reactions
Noise in biology
Intrinsic noise
= Noise resulting form the probabilistic character of the (bio)chemical
reactions. It is particularly important when the number of reacting
molecules is low. It is inherent to the dynamics of any genetic or
biochemical systems. Intrinsic noise is generated by the circuit itself,
and it may depend on the architecture of the network.
Extrinsic noise
= Noise due to the random fluctuations in environmental parameters
(such as cell-to-cell variation in temperature, pH, kinetics parameters,
number of ribosomes,...). Extrinsic noise thus comes from the input.
Both Intrinsic and extrinsic sources of noise lead to fluctuations in
a single cell and result in cell-to-cell variability.
Noise in biology
How many molecules (of a specific protein) are there in a cell?
Volume of a mammalian cell:
V = 100-10000 µm3 ≡ 10-12 L
Volume of a yeast cell (S. cerevisiae):
V = 100 µm3 ≡ 10-13 L
Volume of a bacterial cell (e. coli):
V = 1 µm3 ≡ 10-15 L
Typical concentration of a specific protein:
~ 10 nM - 1µM
Source: BioNumbers (http://bionumbers.hms.harvard.edu/) !
§  A concentration of c = 0.1 µM is a cell
of V = 200 µm3 correspond to a
number of molecules N=10000.
Adding or removing 1 molecule causes
a 0.01 % change in the concentration.
§  A concentration of c = 0.1 µM is a cell
of V = 1 µm3 corresponds to a number
of molecules N=10.
Adding or removing 1 molecule causes
a 10 % change in the concentration.
The stochastic nature of the reactions
and translocation processes thus leads
to large fluctuations in protein
abundance.
Kaern et al (2005) Nature Rev Genet 6:451-64!
Noise in biology
Where does the intrinsic
noise come from?
Each steps is probabilistic
(stochastic). mRNA
transcription requires the
binding of RNAPol to the
DNA, protein synthesis
require the binding of
ribosome on the mRNA,
etc...
In addition, there are also
(probabilistic) interactions /
competition with repressors
and activators which also
influence the transcription
rate.
Fig. adapted from Swain & Longtin, Chaos 2006
Noise in biology
Intrinsic vs extrinsic noise
The yellow and the blue
genes are under the control
of the same promoter /
regulator / inducer; they are
in the same environnement.
Extrinsic noise only
Both extrinsic and
intrinsic noise
Figure from Kaern et al (2005) Nature Rev Genet 6:451-64; after Elowitz et al (2002) Science 297:1183-86!
Noise in biology
•  Regulation and binding to DNA!
•  Transcription (mRNA synthesis)!
•  Splicing of mRNA!
•  Transportation of mRNA to cytoplasm!
•  Translation (protein synthesis)!
•  Conformation of the protein !
•  Post-translational changes of protein!
•  Protein complexes formation !
•  Proteins and mRNA degradation!
•  Transportation of proteins to nucleus!
•  ...!
Noise in biology
Noise-producing steps and noise propagation
Promoter
state
mRNA
Protein
Kaufmann & van Oudenaarden (2007) Curr. Opin. Gen. Dev. , 17:107-12
Noise in biology
Noise propagation and feedback loops
Noise in biology
Noise propagation and feedback loops
Feedback loop
Without
feedback loop
With
feedback loop
Effects of noise
What is the effect
of noise?
Georges Seurat
Un dimanche après-midi à
la Grande Jatte
Fedoroff & Fontana (2002) Small numbers of big molecules, Science 297: 1129
Effects of noise
Effects of noise in biological systems include:
-  Fluctuation in the amount of proteins, which can
propagate in gene and signalling cascades
-  Imprecision in the timing of genetic events
destructive
effects
-  Imprecision in biological clocks
-  Noise-induced behaviors
-  Stochastic resonance
-  Stochastic focusing
-  Phenotypic variations
=> heterogeneous population that can adapt to
environment changes
=> cell differentiation
=> cells with various reproduction rates
constructive
effects
Noise-induced phenotypic variations
Stochastic kinetic analysis of a developmental pathway !
bifurcation in phage-λ Escherichia coli cell!
Arkin, Ross, McAdams (1998) Genetics 149: 1633-48!
λ phage
E. coli
Fluctuations in rates of gene
expression can produce highly
erratic time patterns of protein
production in individual cells and
wide diversity in instantaneous
protein concentrations across cell
populations.!
!
When two independently produced
regulatory proteins acting at low
cellular concentrations
competitively control a switch
point in a pathway, stochastic
variations in their concentrations
can produce probabilistic
pathway selection, so that an
initially homogeneous cell
population partitions into distinct
phenotypic subpopulations!
Imprecision in biological clocks
Circadian clocks limited by noise!
Barkai, Leibler (2000) Nature 403: 267-268
Time
In a previously studied model that depends on a time-delayed negative feedback, reliable oscillations
were found when reaction kinetics were approximated by continuous differential equations. However,
when the discrete nature of reaction events is taken into account, the oscillations persist but with
periods and amplitudes that fluctuate widely in time. Noise resistance should therefore be considered
in any postulated molecular mechanism of circadian rhythms.!
See more details in the presentation in circadian clocks
Noise-induced behaviors
Noise-induced behaviors include:!
§  Noise-induced oscillations!
§  Noise-induced synchronization!
§  Noise-induced excitability!
§  Noise-induced bistability (bimodality)!
§  Noise-induced sensitivity of regulations!
§  Noise-induced stabilization of an unstable state!
!
Noise-induced oscillations in an excitable system!
deteministic
stochastic
Vilar JM, Kueh HY, Barkai N, Leibler S (2002)
Mechanisms of noise-resistance in genetic
oscillators. Proc Natl Acad Sci USA 99:5988-92.
Noise-induced bistability (bimodality?)!
Genetic toggle switch without cooperative binding
Lipshtat A, Loinger A, Balaban NQ, Biham O.
Phys Rev Lett. 2006; 96:188101
A
B
Stochastic resonance
Nature (1999) 402:291-294
paddle fish
Stochastic resonance is the phenomenon whereby the addition
of an optimal level of noise to a weak information-carrying input
to certain nonlinear systems can enhance the information
content at their outputs. !
Here, we show that stochastic resonance enhances the normal
feeding behaviour of paddle fish (Polyodon spathula) which uses
passive electroreceptors to detect electrical signals from
planktonic prey (Daphnia).!
no noise:
no response
low level of
noise :
no response
intermediate
level of noise:
periodic
response
high level of
noise:
erratic
response
The maximal reponse is
obtained at intermediary
level of noise.
In absence of noise: (a) Threshold system
without noise. When a signal is applied at the
input, the output shows a pulse every time
the threshold is crossed from below. (b) If the
input is a subthreshold signal, no pulses can
be seen at the output.
http://www.accessscience.com/search.aspx?rootID=800548
In presence of noise: Threshold system with
(a–c) low, optimal, and high noise. (a) For
noise levels that are too low or (c) too high,
the pulses are observed at the output, but
their statistical distribution does not give
enough information on the input. (b) At the
optimal noise level, which is intermediate,
the probability of observing pulses is clearly
modulated by the input.
Noise, robustness and evolution
Kitano (2004) biological robustness. Nat. Rev. Genet. 5: 826-837!
Robustness is a property that
allows a system to maintain its
functions despite external and
internal noise.
It is commonly believed that
robust traits have been selected
by evolution.
However, in order to have
evolution, variability is required...
Kitano (2004) biological robustness. Nat. Rev. Genet. 5: 826-837!
In presence of noise, a bistable system (cell) may
spontaneously and randomly switch from one state to
another. These states may correspond to different
phenotypes.
Because of bistability and noise, a population of genetically
identical cells can thus spontaneously display
heterogeneity in their phenotypes.
stress +
selection
variability due
to stochasticity
Smits, Kuipers, Veening (2006) Phenotypic variations in bacteria:
the role of feedback regulation. Nature Rev. Microbiol. 4:259-271.!
Robustness to noise
Engineering stability in gene networks by autoregulation!
Becskei, Serrano (2000) Nature 405: 590-3!
Autoregulations (negative feedback loops) in gene circuits provide stability, thereby
limiting the range over which the concentrations of network components fluctuate.!
Robustness to noise
Design principles of a bacterial signalling network!
Kollmann, Lodvok, Bartholomé, Timmer, Sourjik (2005) Nature 438: 504-507!
Signalling network of chemotaxis
Among these different topologies the experimentally established chemotaxis network of
Escherichia coli (scheme c) has the smallest sufficiently robust network structure, allowing
accurate chemotactic response for almost all individuals within a population.!
Theory of stochastic systems!
Deterministic vs stochastic approaches
More abstract,
more qualitative
Statistical
correlation
Boolean
network
More specified,
more quantitative
Logical
approach
(Deterministic)
ODE model
Stochastic
approach
Deterministic vs stochastic approaches
Effect of noise
Variability in the
steady state.
Variability in the
amplitude and
period of the
osillations.
Deterministic formulation
Let's consider a single species (X) involved in a single reaction:
Deterministic description of its time evolution (ODE):
η = stoechiometric coefficient
v = reaction rate:
Mass action law
k = kinetic constant
Example
[A],[B] = concentration
This deterministic approach assumes that the time evolution of A, B and C is continuous
and obeys to the mass action law. The law of mass action stipulates that the rate of a
chemical reaction is proportional to probability that the reacting molecules meet (i.e. are
found together in a small volume). The kinetic rate is thus proportional to the product of the
reactants.
This deterministic description remains true for high concentrations of reactants. For the
case where the number of reactants is low compared to the total volume (low concentration)
this deterministic description can become wrong. For low concentrations, each reactant
has a low chance to collide with the other. The reactions process will not occur continuously
but discretely and stochastically (Markov process).
Let's now consider a several species (Xi) involved
in a couple of reactions:
Deterministic description of their time evolution (ODE):
Example
Evolution equations
Stochastic formulation
How does these equations translate
in the stochastic formulation?
In the stochastic approach:
§ we need to describe the system in term of the number of
molecules (and not in concentration).
The variables are XA=number of molecules of A, etc...
§ we describe the reaction as well as the state of the
system in terms of probabilities.
P(XA,XB,t) = probability to have XA molecules of A and XB
molecules of A at time t.
Let's start again with the reaction
In order to determine the time evolution of a quantity XA of
molecules A, we will consider the problem in term of probability of
reaction: what is the probability that reaction A+B->C occurs ?
Definition
P(XA,XB, t) = probability that the system is in the state (XA,XB) at
time t, i.e probability to have XA molecules of A and XB
molecules of B at time t.
P(Ri, [t; t + dt]), where i ∈ N = probability that a given reaction Ri
occurs in the time interval [t; t + dt].
Let's start again with the reaction
The probability that this reaction occurs in the time interval [t; t + dt] is
XA,XB = number of molecules
The function w1(XA,XB) = c1XA(t)XB(t) is called the propensity function
of the reaction.
More generally, for M reactions involving N different types of reactants,
we can define M propensity functions wi(X1, · · · ,XN), with i = 1, · · · ,M,
that depend on all the N reactants of the process.
Let's now consider the reaction
The reaction probability is
Indeed, we have
possible combinaisons of 2 molecules of A
Remember that the deterministic time evolution is:
Relation between c and k
Relation between the stochastic constant c and the reaction rate constant k can be
found be comparing the deterministic and stochastic time evolution in terms of the
number of molecules. We define V the volume in which the reaction takes place.
Rem: moles instead of number of
molecules are usually used for molecular
concentrations. So the Avogadro’s
number (denoted NA) has to be taken into
account. We thus have c1 = k1/(V.NA)
Relation between c and k
Relation between c and k: summary table
Stochastic formulation: master equation
Master Equation
The chemical master equation of a reaction Ri characterizes the time
evolution of the reaction state probability P(XA,XB,t).
Let's again consider the reaction
The probability to have XA molecules of A and XB molecules of B at time t+dt is:
Probability to have XA+1 molecules of A and XB+1 molecules of
B at time t and that the reaction R1 occurs during dt
Probability to have XA molecules of A and XB molecules of B at
time t and that the reaction R1 does not occur during dt
?
We are now interested by the evolution of the
probability P(XA,XB,t)
Master Equation
The first term of this equation describes the "gain" from another state and the
second, the "loss" of the given state.
Master Equation
Chemical master equation
Stochastic formulation: birth-and-death
Birth-and-death process (single species):
reaction propensities:
ω b = kb
(for the synthesis)
ωd (X)= kdX (for the degradation)
State transitions
Master equation for a birth-and-death process
Stochastic formulation: birth-and-death
Birth-and-death process (multiple reactions, multiple species):
Master equation for a birth-and-death process
Stochastic formulation: examples
Solving the master equation
Solving the Master Equation
Solving the master equation means to calculate/compute the probability of
any value of XA and XB at each time point t.
This required the knowledge of the initial state (XA and XB at time t=0).
Probability
Number of B molecules
t = 0 (initial condition)
t=τ
τ
Number of A molecules
t = 2τ
τ
Solving the Master Equation
To solve the master equation, we need to compute recursively, for t = t0, t1, ... tf:
Euler approximation:
Initial conditions (XA(0)=XA0 and XB(0)=XB0) :
After some rearrangements (see annexe), we find:
=> The evolution of the probability P(XA,XB,t) can be computed,
recursively from any given initial conditions (XA0,XB0).
See details in
lecture notes
Solving the Master Equation using sliding windows
§  This method computes an approximate solution of the CME by performing a sequence of local
analysis steps.
§  In each step, only a manageable subset of states is considered, representing a "window" into
the state space.
§  In subsequent steps, the window follows the direction in which the probability mass moves,
until the time period of interest has elapsed.
§  The window is constructed on the basis of the deterministic approximation of the future
behavior of the system by estimating upper and lower bounds on the populations of the
chemical species.
Wolf V, Goel R, Mateescu M, Henzinger TA (2010) Solving the chemical master equation using sliding windows.
BMC Syst Biol 4:42.
Solving the master equation by simulation
Numerical computation of the master equation, i.e. the
probability P(A,B,t) to find the system in state (A,B) at time t.
A+B -> C
(rate constant k1=1)
Initial conditions:
A -> D
B -> E
A (t=0) = 20
B (t=0) = 20
The plots have been obtained by running "in parallel" 400 Gillespie simulations
and by computing the state distribution of the system at different time steps.
Each plot thus gives the probability distribution in the phase space (A,B) at a
given time.
In our system, the level of A and B can only decrease. We thus observe that the
distribution progressively "moves" towards the state (0,0).
Initial
condition
P(A,B)
high
low
Initial condition
P(A,B)
high
low
Comparison of the different formalisms
Deterministic description
dX
= f (X)
dt
ODE
€
Stochastic description
(1 possible realization)
(10 possible realizations)
(probability distribution)
∂P(X,t)
= f (P(X,t))
∂t
Master Equation
€
Stochastic formulation: Fokker-Planck
Fokker-Planck equation
Master equation
Taylor expansion
Diffusion term
Time
Drift term
See demonstration in lecture notes
Concentration
Stochastic formulation: remark
Due to the large number of states, the master equation or the
Fokker-Planck equation is rarely solvable analytically.
Example
For N = 200 there are more than 1000000 possible
molecular combinations!
=> We can not solve the master equation
analytically (by hand).
=> We need to perform simulations (using
a computer and a good algorithm).
P(A,B,C)=?
P(200,0,0)=?
P(199,0,1)=?
P(199,1,0)=?
P(198,0,2)=?
P(198,1,1)=?
P(198,2,0)=?
...
P(0,0,200)=?
Numerical simulation
Numerical simulation
The Gillespie algorithm
Direct simulation of the master equation
P( production )
P(consumption )
""""
"→ X " " " ""→
The Langevin approach
€
Stochastic differential equation
dX
= f productoin (X) − f consumption (X) + f noise
dt
Gillespie algorithm
1976
1977
Gillespie algorithm
The Gillespie algorithm
A reaction rate wi is associated to each reaction step.
These probabilites (propensities) are related to the
kinetics constants.!
Initial number of molecules of each species are
specified.!
The time interval is computed stochastically
according the reation rates.!
At each time interval, the reaction that occurs is
chosen randomly according to the probabilities wi and
both the number of molecules and the reaction rates
are updated.!
t
Gillespie D.T. (1977) Exact stochastic simulation of coupled
chemical reactions. J. Phys. Chem. 81: 2340-2361.!
Gillespie D.T., (1976) A General Method for Numerically
Simulating the Stochastic Time Evolution of Coupled
Chemical Reactions. J. Comp. Phys., 22: 403-434.
time to the
next reaction?
which
reaction?
Gillespie algorithm
Principle of the Gillespie algorithm
Which reaction?
Probability that reaction r occurs
Reaction r occurs if
equiv to
with
z1 is a random number taken from a uniform
distribution between 0 and 1
Gillespie algorithm
Principle of the Gillespie algorithm
Time to the next reaction?
The time τ till reaction µ occurs follows the
following probability distribution:
In practive the time step can be
computed as:
z2
Δt=
z2 is a random number taken from a uniform
distribution between 0 and 1
See demonstrations
in lecture notes
Gillespie algorithm
In practice...
1.  Calculate the transition probability wi which are functions of the
kinetics parameters kr and the variables Xi .!
2.  Generate z1 and z2 and calculate the reaction that occurs as
well as the time till this reaction occurs.!
3.  Increase t by Δt and adjust X to take into account the
occurrence of the reaction that just occured.!
Gillespie algorithm
Specify the initial number of
molecules of each species
Define the propensity for each
reaction
Check if t > tend
yes
Exit the loop
no
Generate 2 random numbers (z1 and
z2) taken from a uniform distribution
between 0 and 1
Determine the time to the next
reaction as well as the reaction that
will take place
Update the number of molecules and
recompute all reaction propensities
save, plot, or analyse
the data
Remark: if you save all time
points, this might generate
huge amount of data. It is
thus recommended to save
time points every Δt time
steps (where Δt has to be
adjusted by the user).
Gillespie algorithm: Matlab (Brusselator)
omega=100;
a=2;b=6;
x=1; y=1;
% system size
% model parameters
% initial conditions
trans=0; tend=100;
tech=0.01;
t=0; told=0;
% time
% sampling time
% initilisation
R=[ ];
% results matrix
while (t<tend+trans)
% run simulation
w(1)=a*omega;
w(2)=b*x;
w(3)=x;
w(4)=x*(x-1)*y/omega^2;
c=cumsum(w);
ct=c(end);
z1=rand();
z2=rand();
tau=(-log(z1))/ct;
t=t+tau;
uct=z2*ct;
if (uct < c(1))
x=x+1;
elseif (uct < c(2))
x=x-1;
y=y+1;
elseif ( uct < c(3))
x=x-1;
elseif (uct < c(4))
x=x+1;
y=y-1;
end
if (t>trans) && (t>told+tech)
R=[R ; t x y];
told=t;
end
end % end of while
figure(1)
% plot time serie
plot(R(:,1),R(:,2),’b’,R(:,1),R(:,3),’r’);
xlabel(’Time’)
ylabel(’X (blue), Y (red)’)
Gillespie algorithm
System size
A key parameter in this approach is the system size Ω = V*NA. This parameter has
the units of a volume and is used to convert the reaction rate k into the stochastic
reaction propensity ω. It is in fact used to convert the concentration x into a number
of molecules X:!
X = Ω x!
For a given concentration (defined by the deterministic model), bigger is the system
size (Ω), larger is the number of molecules. Therefore, Ω allows us to control directly
the number of molecules present in the system (hence the noise). !
!
Typically, Ω appears in the reaction steps involving two (or more) molecular species
because these reactions require the collision between two (or more) molecules and
their rate thus depends on the number of molecules present in the system.!
!A → E
!
!A + B → C
!
!2A → D!
!c = k
!
!c = k / Ω c = 2 k / Ω
ω = c A
ω = c A B ω = c A (A-1) / 2
Gillespie algorithm
Remarks
§  The Gillespie algorithm requires to convert the ODE model
(usually in terms of concentrations) into a stochastic version (in
terms of numbers of molecules).!
§  The Gillespie algorithm is exact (it provides stochastic trajectories
which rigorously correspond to the master equation).!
§  The Gillespie algorithm is easy to implement (this is probably why
it remains widely used today).!
§  The Gillespie algorithm may be (very) slow, especially for large
systems (with many variables) and systems combining fast and
slow processes. For this reason, other algorithms have been
proposed.!
Gillespie algorithm: improvements
Next Reaction Method (Gibson & Bruck, 2000)
Gibson & Bruck's algorithm avoids calculation that is repeated in every
iteration of the computation. This adaptation improves the time
performance while maintaining exactness of the algorithm.
§  Reduce cost of calculating all aµ
§  Dependency graph
§  Reduce cost of summing all aµ to
calculate a0
§  Modify a0 by subtracting old values,
adding new
Gibson & Bruck (2000) J. Phys. Chem. A 104:1876-89
Source: Lecture slides by David Karig
Gillespie algorithm: improvements
Tau-Leap Method (Gillespie, 2001)
Instead of chosing which reaction occurs at which time step, the Tau-Leap
algorithm estimates how many of each reaction occur in a certain time interval.
We gain a substantial computation time, but this method is approximative and
its accuracy depends on the time interval chosen.
time interval Δt
t
R1 R3
R1 R2 R3
R1
t+Δt
R2
R1
Estimate the number of times each reaction takes place
during Δt and update the number of molecules accordingly
§ Gillespie DT (2001) Approximate accelerated stochastic simulation of chemically reacting systems, J Chem Phys
115:1716-11
§ Rathinam M, Petzold LR, Cao Y., Gillespie D (2003) Stiffness in stochastic chemically reacting systems: The implicit
tau-leaping method, J Chem Phys 119:12784-94.
§ Cao Y, Gillespie DT, Petzold LR (2006) Efficient step size selection for the tau-leaping simulation method, J Chem Phys
124:044109.
Gillespie algorithm: extensions
Delay Stochastic Simulation
Bratsun et al. (2005) and other authors have extended the Gillespie algorithm
to account for the delay in the kinetics. This adaptation can therefore be used
to simulate the stochastic model corresponding to delay differential equations.
t-δ
The state of the system
at time t-τ affect its
evolution in the future
t
t+Δt
time to the
next reaction?
which
reaction?
§ Bratsun D, Volfson D, Hasty J, Tsimring LS (2005) Delay-induced stochastic oscillations in gene regulation. PNAS
102: 14593-8.
§ Barrio M, Burrage K, Leier A, Tian T (2006) Oscillatory Regulation of hes1: Discrete Stochastic Delay Modelling
and Simulation. PLoS Comput Biol 2:1017.
§ Cai X (2007) Exact stochastic simulation of coupled chemical reactions with delays, J Chem Phys 126:124108.
Langevin stochastic equation
Langevin stochastic differential equation
If the noise is white (uncorrelated), we have:
mean of the noise
correlation of the noise
D measures the strength of the fluctuations.
NB: if the noise ξ should reflect only the intrinsic noise,
D should be appropriately chosen.
Langevin stochastic equation
A rigorous equivalence between the master equation and the Langevin equation
can be obtained by considering a multiplicative noise (Gillespie, 2000):
Function g(X) describes the stochasticity resulting from the internal dynamics of
the system (cf. the diffusion term in the Fokker-Planck equation) and should be
appropriately chosen. Gillespie demonstrated that in the limit of low noise the
chemical master equation is equivalent to the following equation
Chemical Langevin Equation
Gillespie DT (2000) The chemical Langevin equation, J Chem Phys 113:297-306.
See demonstration
in lecture notes
Gillespie vs Langevin modeling
Example
Deterministic model
Propensities table (Gillespie algorithm)
Stochastic Langevin Equation
Here the level of noise in
the Langevin equation (D)
is arbitrary and has been
manually adjusted to
"match" the Gillespie
simulations.
Here the level of noise in
the Langevin equation is
computed as a function of
the system size (Ω) as
defined in the Chemical
Langevin Equation. It
thus fits well the
corresponding Gillespie
simulations.
Hybrid models
Hybrid models
§  Genetic and biochemical systems often involve a combinaison of
slow and fast processes.
§  Slow processes contribute more to the (intrinsic) noise.
§  Fast processes may (sometimes drastically) slow down stochastic
simulations.
§  Hybrid models, combining deterministic description (for fast
processes) and stochastic description (for slow processes) have
been developed.
Pahle J (2009) Biochemical simulations: stochastic, approximate stochastic and hybrid
approaches. Briefings in Bioinformatics 10:53-64
Spatial stochastic modeling
Compartimentalization and diffusion
Illustration of the (very crowded)
interior of a cell (David Goodsell)
Dobson, Nature 432:444-5 (2004)
Space (incl. spatial effects and diffusion) can be taken into account
by discretizing the space and defining transport probabilities.
Compartimentalization
For many biological systems it is desirable to take into account the cell
compartimentalization because:
•  proteins may have different functions or activity in the various cell compartments
•  protein transport may induce delay and may itself be regulated
•  the number of proteins in the nucleus or in the membrane may be strongly reduced
compared to the cytosol and this may induce stochastic effects may also be important
The simplest way to account for such
compartmentalization is to treat nuclear,
cytosolic and membrane proteins as
different entities.
Transport (translocation) can then be
modeled in first approximation, by
standard mass action law (to be
converted into propensities, as the
chemical reaction rates).
Method for spatial stochastic approaches
off-lattice methods
•  Each particles in the system has
explicit spatial coordinates.
•  At each time step, molecules
with non-zero diffusion
coefficients are able to move,
randomly, to new positions.
•  When 2 reacting species are
sufficiently close to each other,
they may react, with a certain
probability
(microscopic) lattice
(mesoscopic) lattice
•  A computational grid (2 or 3 dimension) is used to represent a cellular
compartment, such as a membrane or the nucleus.
•  The lattice is then populated with particles of the different molecular species
(randomly or at chosen spatial locations).
•  Particles with non-null diffusion coefficient are able to diffuse throughout the
simulation domain by jumping to (empty) neighbouring sites.
•  Depending on specified reaction rules, appropriate chemical reactions can
take place in a given domain with a certain probability.
•  It is worth noting that the grid can represent microscopic (one molecule max/
domain) or mesoscopic domains.
Burrage et al (2011)
Andrews, Arkin (2006) Simulating cell biology. 16: R523-527.
Application to (bio)chemical and genetic systems!
Gene expression
Reactional scheme
Thattai - van Oudenaarden model
Thattai M, van Oudenaarden
A (2001) Intrinsic noise in gene
regulatory networks.
Proc Natl Acad Sci USA.
98:8614-9.
Gene expression
Protein
mRNA
Theoretical steady state:
Theoretical
steady state:
RSS = k1/k2
PSS = k1k3/k2k4
Poisson distribution
non Poisson
distribution
See demonstration
in annex
OK
theoretical
Poisson
distribution
≠
theoretical
Poisson
distribution
Gene expression
§  The "simple" gene expression
model is used to assess the
relative contribution of
transcription and translation to
the noise.
§  The predictions of the model are
compared to the experimental
data.
See more details in the next presentation...
Ozdudak, Thattai, Kurtser, Grossman, van
Oudenaarden (2002) Nat Genet 31: 69-73
Gene expression
Many factors regulate gene transcriptional "activity":
•  binding of transcription factors (activators/repressors)
•  chromatin remodeling
•  histone acetylation
•  DNA methylation
•  ...
It is therefore common to distinguish 2 forms of the gene, either
"active" (can be transcribed) or "inactive" (can not be transcribed):
Of course, detailed models with more
gene/promoter states (transcribed as
various levels) have also been proposed.
Gene expression
When the activation/
deactivation is fast compared
to the transcription rate, the
system behaves like in
the previous scheme, where
gene activity is constant (the
fluctuations are averaged out).
This is typically the case when
we consider the binding/
unbinding of a transcription
factor to the gene promoter
Gene expression
When the activation/
deactivation is slow compared
to the transcription rate, the
gene switches on
occasionnally and then remain
active for relatively long
periods of time. This results in
transcriptional bursts.
This is the case when the
gene activity is controlled by
slow processes such as
chromatin remodeling.
Recent genome-wide
analyses suggest that such
transcriptional bursting may
be the most common type of
gene expression kinetics.
Gene expression
§  Suter DM, Molina N, Gatfield D, Schneider K, Schibler U, Naef F (2011) Mammalian genes are transcribed with
widely different bursting kinetics. Science. 332:472-4
See also:
§  Molina N, Suter DM, Cannavo R, Zoller B, Gotic I, Naef F (2013) Stimulus-induced modulation of transcriptional
bursting in a single mammalian gene, Proc Natl Acad Sci USA 110:20563-8.
§  Dar RD, Razooky BS, Singh A, Trimeloni TV, McCollum JM, Cox CD, Simpson ML, Weinberger LS (2012)
Transcriptional burst frequency and burst size are equally modulated across the human genome. Proc Natl Acad
Sci USA 109:17454-9.
Gene expression
Conformational change (isomerization)
Reactional scheme
(e.g. isomerization)
A
k1
k2
B
The probability to find j molecules of A at steady state is
given by!
where n is the total
(constant) number
of molecules.!
As the number of molecules increases, the steady-state probability
density function becomes sharper (=> less fluctuations).!
Exercise: !
(1) Assuming that the (constant) total number of molecules is
N=A+B, write the master equation for this system, i.e.
dP(A,B;t)/dt.!
(2) Check that the solution given above is the solution of the
master equation at steady state.!
(3) Simulate the system using the Gillespie algorithm and
compare the distribution obtained with the analytical results.!
Rao, Wolf, Arkin (2002) Nature
Conformational change (isomerization)
Isomerization: Simulation vs Theory
A
k1
k2
B
N = A+B = 200
low noise
N = A+B = 40
larger noise
k1=k2
Michaelis-Menten
Reactional scheme
Deterministic
evolution equations
Michaelis-Menten
Reactional scheme
Stochastic
transition table
Master equation
Michaelis-Menten
Michaelis-Menten
Quasi-steady state assumption
If
Rate of production of P
Stochastic transition table
quasi-steady state
Michaelis-Menten
Is the quasi-steady state assumption still valid in
the stochastic description?
Rao CV, Arkin AP (2003) Stochastic chemical kinetics and the quasi-steady-state
assumption: Application to the Gillespie algorithm. J Chem Phys 118:4999-5010
The QSSA is a powerful tool for simplifying the reaction kinetics, and it has been
successfully applied to numerous problems in deterministic kinetics. We have
demonstrated how the QSSA may be applied to stochastic kinetics. Our experience to
date suggests that the conditions for the QSSA in stochastic kinetics are the same as
for deterministic kinetics.!
Grima R (2009) Noise-induced breakdown of the Michaelis-Menten equation in
steady-state conditions. Phys Rev Lett. 102:218103.
Using both theory and simulations, we show that intrinsic noise induces a breakdown
of the Michaelis-Menten equation even if steady-state metabolic conditions are
enforced.
Michaelis-Menten
Quasi-steady state assumption
Parameters:
k1=10
km1=10
k2=1
ET=0.1
S(0)=1
vmax=ETk2
KM=(km1+k2)/k1
System size:
Ω=100
Michaelis-Menten
Is the quasi-steady state assumption still valid in
the stochastic description?
The debate is still open...
§  Cao Y, Gillespie DT, Petzold LR (2005) Accelerated stochastic simulation of the stiff enzyme-substrate
reaction. J Chem Phys. 123:144917
§  Mastny EA, Haseltine EL, Rawlings JB (2007) Two classes of quasi-steady-state model reductions for
stochastic kinetics. J Chem Phys. 127:094106
§  Sanft KR, Gillespie DT, Petzold LR (2011) Legitimacy of the stochastic Michaelis-Menten approximation.
IET Syst Biol. 5:58
§  Thomas P, Straube AV, Grima R. (2011) Limitations of the stochastic quasi-steady-state approximation in
open biochemical reaction networks. J Chem Phys. 135:181103.
§  Smadbeck P, Kaznessis Y (2012) Stochastic model reduction using a modified Hill-type kinetic rate law. J
Chem Phys. 137:234109.
See also a (personal) commentary in:
§  Gonze D, Abou-Jaoudé W, Ouattara DA, Halloy J (2011) How molecular should your molecular model be?
On the level of molecular detail required to simulate biological networks in systems and synthetic biology.
Methods Enzymol. 487:171-215.
Negative auto-regulation
X
Does negative feedback loops (and, in particular negative
auto-regulation) increase the robustness of the steady state?
X
is known to increase the
stability of the steady
state.
X
dx
Kn
= ka n
− kb x
n
dt
K +x
dx
= ka − k b x
dt
k'bx
€
€
k'bx
kbx
kbx
ka
kaKn/(Kn+xn)
x'S
xS
x
x'S xS
In presence of a negative auto-regulation, the
steady state xs is less sensitive to parameter
variations (homeostasis).
x
This was experimentally demonstrated by:
Becskei & Serrano, Nature 2000.
Engineering stability in gene networks by
autoregulation
They used a slightly different model and run
stochastic simulations, but the conclusions
are the same.
X
X
dx
Kn
= ka n
− kb x
n
dt
K +x
dx
= ka − k b x
dt
€
Ω=1
€
The negative auto-regulation
increases the robustness of
the steady state.
Ω = 10
Brusselator
Brusselator
The Brusselator, sometimes called the trimolecular model is one
the simplest model demonstrating the emergence of self-sustained
oscillations in a chemical reaction scheme (Prigogine, 1968).
Reaction scheme
Deterministic
evolution equations
Brusselator
Stochastic
propensities
table
Master equation
Gillespie (1977) simulated this
model to illustrate the feasability
and the utility of his algorithm.
Brusselator
Brusselator
See demo
simulation
(in matlab)
Brusselator
The black trajectory is the results on a single
stochastic run (over 500 time units).
The red curve is the corresponding deterministic limit
cycle (note that the variables have been rescaled by
Ω to be expressed in number of molecules)
The two figures below show the probability to observe
the system in a given state (after some transients).
Here this probability distribution was computed
numerically but it should correspond to the solution of
the master equation at t -> infinity.
Brusselator
Quantification of the effect of noise on oscillations
§  Histogram of periods
p1
p2
p3
p4
...
standard deviation
of the period
maxima of the "green" time series
§  Auto-correlation function
half-life of the
auto-correlation
See illustration of
the principle of the
autocorrelation
function in annex
Brusselator
Brusselator
Standard deviation of the periods
The standard deviation of the periods
increases inversely proportionnally with the
square root of the system size, at least for
large system size (i.e. 1/sqrt(Ω) small).
Half-life of the decorrelation
The half-life of the decorrelation
increases proportionnally with the
system size Ω.
Gaspard P (2002) The correlation time of mesoscopic chemical clocks. J. Chem. Phys.117: 8905-8916.!
Lotka-Volterra
Predator-prey system
(Lotka-Volterra model)
α
β
γ
Deterministic
equations
δ
prey
predator
Lotka-Volterra
Fitzhugh-Nagumo
The Fitzhugh-Nagumo model is a example of a two-dimensional excitable
system. It was proposed as a simplication of the famous model by Hodgkin
and Huxley to describe the response of an excitable nerve membrane to
external current stimuli.!
The two non-dimensional variables x and y are !
!
x = voltage-like variable (activator) - slow variable!
y = recovery-like variable (inhibitor) - fast variable!
!
The nonlinear function f(x) (shaped like an inverted N, as shown in blue
in the next figure) is one of the nullclines of the deterministic system; a
common choice for this function is !
!
!
!
!
D (t) is a white Gaussian noise with intensity D. !
Fitzhugh-Nagumo
Deterministic
excitability
Stochastic
+ noise
oscillations
Fitzhugh-Nagumo
Stochastic coherence in the Fitzhugh-Nagumo model
Source: scholarpedia
Acknowledgements
I would like to thank:
José Halloy
Geneviève Dupont
Albert Goldbeter
Pierre Gaspard
Yannick De Decker
Laurence Rongy
Adama Ouattara
Wassim Abou-Jaoudé
for fruitful discussions
References
Reviews - Theory
§  De Jong H (2002) Modeling and Simulation of Genetic Regulatory Systems: A Literature Review
J Comput Biol 9: 67–103
§  Gillespie D and Petzold L (2006) Numerical Simulation for Biochemical Kinetics, In System Modelling
in Cellular Biology, Ed. Z. Szallasi, J. Stelling, V. Periwal MIT Press 2006
§  Gillespie DT (2007) Stochastic simulation of chemical kinetics. Annu Rev Phys Chem. 58:35-55.
§  Pahle J (2009) Biochemical simulations: stochastic, approximate stochastic and hybrid approaches.
Briefings in Bioinformatics 10:53-64.
§  Andrews SS, Dinh T, Arkin AP (2009) Stochastic models of biological processes, In Encyclopedia of
Complexity and System Science, Meyers, Robert (Ed.) Volume 9:8730-8749. Springer, NY, 2009.
§  Burrage K, Burrage P, Leier A, Marquez Lago T, Nicolau JrDV (2011) Stochastic Simulation for Spatial
Modeling of Biological Processes, In Design and Analysis of Bio-molecular Circuits, Koeppl H, Setti G,
di Bernardo M, Densmore D (eds.), Springer.
References
Reviews - Biology
§  Raser JM, O'Shea EK (2005) Noise in gene expression: origins, consequences, and control. Science.
309:2010-3.!
§  Kaern M, Elston TC, Blake WJ, Collins JJ (2005) Stochasticity in gene expression: from theories to
phenotypes. Nat Rev Genet. 6:451-64.!
§  Maheshri N, O'Shea EK (2007) Living with noisy genes: how cells function reliably with inherent
variability in gene expression. Annu Rev Biophys Biomol Struct. 36:413-34.!
§  Kaufmann BB, van Oudenaarden A (2007) Stochastic gene expression: from single molecules to the
proteome. Curr Opin Genet Dev. 17:107-12.
§  Raj A, van Oudenaarden A (2008) Nature, nurture, or chance: stochastic gene expression and its
consequences. Cell 135:216-26.
§  Eldar A, Elowitz MB (2010) Functional roles for noise in genetic circuits. Nature. 467:167-73.!
§  Sanchez A, Golding I (2013) Genetic determinants and cellular constraints in noisy gene expression.
Science. 342:1188-93.

What is noise? - Personal Homepages

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