power points on computational biology
TRANSCRIPT
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AMS/BIO 332
Spring 2014
Lectures 03-04
Mass-action kinetic models of generegulatory networks.
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Info and exhibits:
Melville Library Lobby
9AM3PM
Feb. 14
http://life.bio.sunysb.edu/darwinsbu/
Keynote speaker:
Dr. David Jablonski
University of Chicago
Evolution and Extinction:
Lessons from the Fossil Record
7:30 PM Earth & Space Sci 001Films and Discussion:Javits Conference Room 2nd floor Melville Library
"Great Transformations 9:30AM - 11:00AM
"Why Sex?" 11:00AM - 12:30PM
"Evolutionary Arms Race" 12:30PM - 2:00PM"Extinction" 2:00PM - 3:30PM
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Assignment #1
Due date shifted due to class cancellations:
New due date Wednesday, Feb. 19th.
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Quiz!
Answer on a blank sheet of paper
***DO NOT LOOK AT YOUR NOTES***
Explain the difference between lysis
and lysogeny in the life cycle of abacteriophage.
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Self-assessment!
Give yourself a score out of 10
based on your confidence in your
answer.
We will go over the answers next
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Quiz answers!
Lysis and lysogeny are the two possible outcomes from theinfection of a bacterial cell by a bacteriophage (virus).
In lysis, the bacteriophage reproduces quickly inside the bacterialcell, creating new phage; when many new phage have beensynthesized and assembled, the bacteria bursts (lyses), and the new
phage are free to infect new hosts. In lysogeny, the phage genome is inserted into the bacterial
genome, but no new phage are synthesized. However, as theinfected bacterium grows and divides, all daughter cells also containthe phage genome in a quiescent state.
Infected bacteria that are in a lysogenic state will remain stable (andnot undergo lysis) indefinitely (for many generations). However,when exposed to stress (such as UV radiation), the cell will switchinto a lytic state: new phage will be rapidly synthesized andreleased into the environment.
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Self-assessment!
Give yourself a second score (againout of 10) based on the answers
weve just discussed.
Turn in your graded papers for review
and recording of grades.
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Mechanism of lysis-lysogeny decision
making circuit.
cro and other genes for lysiscI and other genes for lysogeny
OR1 OR2 OR3
Phage genome is structured in two parts, with control elements between them.
The genes for lysogeny include the transcription factor cI.
The genes for lysis include the transcription factor cro.
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Mechanism of lysis-lysogeny decision
making circuit.
OR1 OR2 OR3
cro protein (dimer)
(plus other lytic proteins)
mRNA of cro (and other lytic genes)
cI protein (dimer)
(plus other lysogenic proteins)
mRNA of cI (and other lysogenic genes)
Transcription
Translation
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Mechanism of lysis-lysogeny decision
making circuit.
OR1 OR2 OR3
lysogeny
cI (dimer)
preferentially
binds to OR3
cro (dimer)
preferentially
binds to OR1
lysis
X X
cro binding to OR1 blocks
transcription of lysogenic genes
cI binding to OR3 blocks
trancsription of lytic genes(cro)2 (cI)2
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A double-negative feedback loop.
cI gene cI mRNA cI
cro gene cro mRNA cro
cI cro
Mutual inhibition:
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Examples of other types of gene-
regulatory structures.
gene (DNA) mRNA Protein
A
Autoregulatory (auto-induced):
Triple-negative feedback: Negative feedforward:
B
A
C
BA C
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Building a mathematical model of a
gene-regulatory circuit.
What are the fundamental processes
involved?
Transcription (synthesis of mRNA)
Translation (synthesis of protein)
Protein oligomerization (e.g. dimerization)
Protein binding to DNA
Degradation of mRNA and protein!
Without degradation, quantities can only increase.
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Building a mathematical model of a
gene-regulatory circuit.
What type of mathematical expression might
we use?
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Building a mathematical model of a
gene-regulatory circuit.
What type of mathematical expression might
we use?
Chemical (mass-action) kinetics describe the rates
of reactions as the rate of change of concentration
over time.
][][ 1 XkdtXd ]][[][ 2 BAk
dtABd
YX ABBA
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Building a mathematical model of a
gene-regulatory circuit.
What type of mathematical expression might
we use?
A system of Ordinary Differential Equations
(ODEs) describes the rates of change ofconcentrations.
To be contrasted with Partial Differential
Equations (PDEs) and Stochastic Master Equationsthat we will discuss later.
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Building a mathematical model of a
gene-regulatory circuit.
Building our equations.
At the most fundamental level we have two
species for each gene that we care about:
mRNA for gene A, protein for gene A, etc.
For each one, we will have a general rate
equation:
There is no single right answer to how we construct the equations; we
will consider one of the simplest possible models!
nRateDegradatioateSynthesisR][
dt
Xd
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Building a mathematical model of a
gene-regulatory circuit.
Degradation:
The simplest model of degradation is a
unimolecular process, much like radioactive
decay:
The degradation rate constant may be different for
each species.
][][
Xdt
XdX0X
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Building a mathematical model of a
gene-regulatory circuit.
Protein synthesis (translation):
The simplest model of translation assumes that
the rate of protein synthesis is directly
proportional to the amount of mRNA (of the givenprotein) that is present:
This assumes that the other materials involved in protein
synthesis (ribosomes, tRNA, etc) are present in abundance.
][][
mRNAX
proteinX
dt
XdproteinmRNAmRNA XXX
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Building a mathematical model of a
gene-regulatory circuit.
mRNA synthesis (transcription):
Our model of transcription needs to consider the
control elementsthis is more complicated!
There are two extremes:
If fully activated, transcription will occur at a constant
(maximum) rate.
If completely unactivated, the rate will be at a
minimum (typically zero).
XmRNA
dt
Xd
][max 0
][min
dt
Xd mRNA
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Building a mathematical model of a
gene-regulatory circuit.
mRNA synthesis (transcription): We might assume that the degree of activation is
proportional to the fraction of the time that a transcriptionfactor is bound to the regulatory site.
Enhancer: If the fraction bound is 0, we have the minimalrate (0), and if the fraction bound is 1, we have themaximal rate:
Repressor: If the fraction bound is 0, we have the maximalrate, and if the fraction bound is 1, we have the minimalrate (0):
XX
mRNA
dt
Xd )1(
][
XXmRNA
dt
Xd
][
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Building a mathematical model of a
gene-regulatory circuit.
mRNA synthesis (transcription):
In most cases, the transcription factor is a protein
encoded by some other gene:
22
2
proteinXY
protein
XYK
Y
XXmRNA
dt
Xd )1(
][
Repression
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Building a mathematical model of a
gene-regulatory circuit.
mRNA synthesis (transcription):
In some cases, the transcription factor is the
protein encoded by the very same gene:
22
2
proteinXX
protein
XXK
X
XXmRNA
dt
Xd
][
Auto-regulation
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Building a mathematical model of a
gene-regulatory circuit.
mRNA synthesis (transcription):
In networks, we need to keep track of multiple
genes, and their effects on each other:
22
2
proteinXY
protein
XYK
Y
XXmRNA
dt
Xd )1(
][
222
proteinYX
proteinY
XKX
YYmRNA
dtYd )1(][
Cross-regulation (repression) of two genes
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Building a mathematical model of a
gene-regulatory circuit.
Putting it all togetheran auto-regulatory
gene:
][][
22
2
mRNAXmRNAX
proteinXY
proteinmRNA X
XK
X
dt
Xd
][][][ proteinXproteinmRNAXprotein XXdt
Xd
Synthesis Degradation
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Building a mathematical model of a
gene-regulatory circuit.
Putting it all togethermutual repression:
][1][
22
2
mRNAXmRNAX
proteinXY
proteinmRNA XYK
Y
dt
Xd
][1][
22
2
mRNAYmRNAY
proteinYX
proteinmRNA YXK
X
dt
Yd
][][][
proteinXproteinmRNAX
proteinXX
dt
Xd
][][][ proteinYproteinmRNAYprotein YYdt
Yd
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Modeling the behavior of a gene-
regulatory circuit.
So now what?
We have been able to write down a system of
ordinary differential equations: the rate of
change in the concentrations of both mRNA andprotein of each species of interest.
How do we use this?
Are the rates of change what we really careabout?
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Modeling the behavior of a gene-
regulatory circuit.
What we really care about is the AMOUNT
(concentration) of each species at any given
point of time!
Our equations tell us how these amounts
change.
If we know how much we begin with, we can
compute the changes over time!
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Modeling the behavior of a gene-
regulatory circuit.
Boundary value problem:
We know the values of the variables at a certain
point in time (typically t=0).
We have a general expression for the rate ofchange of each variable.
We want to know the values of the variable at an
arbitrary time.),( yxf
dt
dx
),( yxgdt
dy
0)0( xx
0)0( yy
?)( tx
?)( ty
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Modeling the behavior of a gene-
regulatory circuit.
Solving the problem for a small time step:
The challenge is that the derivatives change over
time, because the variables themselves change.
If we only consider a small period of time, thechange will be small; we can approximate this as a
constant.
),( 00 yxft
x
tyxfxtx ),()( 000
),( 00 yxgt
y
tyxfx ),( 00
tyxgy ),( 00 tyxgyty ),()( 000
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Modeling the behavior of a gene-
regulatory circuit.
An iterative approach (Forward Euler):
We can find an approximate value of x(t) as
described; the smaller t is, the better.
Taking x(t) as our starting point, we can find anapproximate value of x(2t) the same way.
Repeat to find x(3t), x(4t), x(5t) etc!
ttytxftxttx )(),()()( )(),( tytxftx
)(),( tytxgt
y
ttytxgtytty )(),()()(
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Modeling the behavior of a gene-
regulatory circuit.
Forward Eulerin Matlab:
xedt
dx Integrate the equation:
for 100 s, with a time step of 0.1 s,
beginning at x(0) = 2.7.
Plot the results as x vs. t.
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Modeling the behavior of a gene-
regulatory circuit. Forward Eulerin Matlab:
numsteps = 1000;deltat = 0.1;x = zeros(numsteps,1);t = zeros(numsteps,1);x(1) = 2.7;t(1) = 0;for (i=1:numsteps-1)
dx_dt = exp(-x(i));x(i+1) = x(i) + dx_dt*deltat;t(i+1) = t(i) + deltat;
endplot(t,x,k-);xlabel(Time);ylabel(Value of x);title(exp(-x) vs Time, using Forward Euler)
1. Initialization of variables
2. Integration loop
3. Visualization of results
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Modeling the behavior of a gene-
regulatory circuit.
Add lines for additional variables:
% initialize variables (not shown)
% numsteps, x, y and t arrays, x(1) and y(1), etc
for (i=1:numsteps-1)% calculate derivatives using values at (i)
dx_dt = y(i)-x(i); % For example given
dy_dt = 3*x(i)^2; % For example given
x(i+1) = x(i) + dx_dt*deltat;
y(i+1) = y(i) + dy_dt*deltat;t(i+1) = t(i) + deltat;
end
23xdt
dy xydt
dx
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Modeling the behavior of a gene-
regulatory circuit.
The Forward Euler algorithm is simple, bothconceptually and to implement.
It gives reasonable answers for many systems, if thetime step is small enough.
If the time step is too big, the solution can be quiteinaccurate!
Many other algorithms exist that are based on similarideas, but that use a better approximation (e.g.Runge-
Kutta) Many of these are implemented as Matlab functions:
More discussion of these later
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Modeling the behavior of a gene-
regulatory circuit.
Our implementation of the Forward Euler algorithmwas built around a for loop:
for (i=1:numsteps-1)
x(i+1) = x(i) + dx_dt*deltat;
end
What is the right number to steps?
What is the right value of deltat? Knowing how to answer these questions is essential
to being able to model a system!
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Modeling the behavior of a gene-
regulatory circuit.
N = 10
T = 1
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Modeling the behavior of a gene-
regulatory circuit.
N = 50
T = 0.2
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Modeling the behavior of a gene-
regulatory circuit.
N = 100
T = 0.1
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Modeling the behavior of a gene-
regulatory circuit.
N = 500
T = 0.02
f
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Modeling the behavior of a gene-
regulatory circuit.
When the time step is particularly large, thetrajectory can be obviously coarse.
More subtle differences can occur at
moderate step sizes. Check the reasonableness of the time step by
repeating the simulation with a smaller value(divide by 2 or more):
Any differences should be very difficult to see.
But what about the length of the simulation?
d l h b h f
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Modeling the behavior of a gene-
regulatory circuit.
N = 500
T = 0.02
d li h b h i f
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Modeling the behavior of a gene-
regulatory circuit.
N = 1000
T = 0.02
M d li h b h i f
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Modeling the behavior of a gene-
regulatory circuit.
N = 2000
T = 0.05
M d li h b h i f
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Modeling the behavior of a gene-
regulatory circuit.
Does the simulation reach a steady-state, or appear asit may be getting close to one?
Confirm your guess by doubling the length of thesimulation:
There should be virtually no change in any value of thesecond half of the extended simulation.
Note: A system doesnt have to reach a steady-state;it could, for example, display persistent oscillations. How to assess a long enough simulation will be different
in these cases; we may want to make sure that oursimulation lasts multiple periods of oscillation to that anyvariations in the period and/or amplitude would benoticed.
S d i i d
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Steady-states, stationary points and
equilibrium.
What does is mean if we observe that the
system reaches a steady-state?
St d t t t ti i t d
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Steady-states, stationary points and
equilibrium.
What does is mean if we observe that thesystem reaches a steady-state?
It does NOT mean that no reactions are
occurring; but simply that the overallconcentrations of all species are not changingover time:
Synthesis and degradation rates are balance from
all species.
The system is in equilibrium.
St d t t t ti i t d
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Steady-states, stationary points and
equilibrium.
What does is mean if we observe that the systemreaches a steady-state?
It means that the rate of change of all species is
zero, e.g.:
We are at a root of the system of ODEs:
A stationary point (or critical point).
0][ dt
Xd mRNA0][ dt
Xd protein
0][
dt
Yd protein 0][
dt
Yd mRNA
St d t t t ti i t d
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Steady-states, stationary points and
equilibrium.
In some cases, we can solve directly for the existence
of stationary points.
][][
22
2
mRNAXmRNAX
proteinXY
proteinmRNA XXK
X
dt
Xd
][][][
proteinXproteinmRNAX
proteinXX
dt
Xd
][][ proteinXproteinmRNAX XX 0][ dt
Xd protein
0][
dt
Xd mRNA
][
22
2
mRNAXmRNAX
proteinXY
proteinX
XK
X
St d t t t ti i t d
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Steady-states, stationary points and
equilibrium.
The root of a single equation is called a null-cline;these are generally a line (or curve) through thestate space.
Stationary points occur at the intersection of thenull clines (one null cline per variable).
][][ proteinX
XproteinmRNA XX
0][
dt
Xdprotein
0][
dt
Xd mRNA
XmRNAX
proteinXY
protein
mRNAXK
XX
22
2
][
St d t t t ti i t d
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Steady-states, stationary points and
equilibrium.
][][ proteinX
Xprotein
mRNA XX
XmRNA
X
proteinXY
protein
mRNAXK
XX
22
2
][
St d t t t ti i t d
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Steady-states, stationary points and
equilibrium.
Null clines can indicate switching lines for the sign
of the rate of change for one variable.
The general direction of motion in a region can be
qualitatively sketched. Stationary points can be of different types:
Stable, unstable, saddle.
This analysis doesnt tell us the details of HOW a
system reaches the stationary state, just a
qualitative picture, simulation is still essential.
It CAN help a lot with interpretation!