Diversity%20and%20Design%20in%20Cellular%20Networks

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Diversity and Design in Cellular Networks

• Prediction, Control and Design of and with Biology

Adam Arkin, University of California,
Berkeley http//genomics.lbl.gov
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"Nothing in biology makes sense except in the
light of evolution." Theodosius Dobzhansky, The
American Biology Teacher, March 1973
A scientist
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The Advent of Molecular Biology
Genome
Macromolecules
Metabolites
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Biochemistry
Through RNA
Feedback Feedforward
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Myxococcus xanthus

• Even cells as simple as bacteria are highly
  social, differentiating, sensing/actuation systems

Images from Reichardt or D. Kaiser
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Immune cells

• They perform amazing engineering feats under the
  control of complex cellular networks

Onsum, Arkin, UCB
Mione, Redd, UCL
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1/50 of the known neutrophil chemotaxis network
c5a- receptor
Fc- receptor
PIP3 control
Calcium control
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Systems and Synthetic Biology

• Systems biology seeks to uncover the design and
  control principles of cellular systems through
• Biophysical characterization of macromolecules
  and other cellular structures
• Comparative genomic analysis
• Functional genomic and high-throughput
  phenotyping of cellular systems
• Mathematical modeling of regulatory networks and
  interacting cell populations.
• Synthetic biology seeks to develop new designs in
  the biological substrate for biotechnological,
  medical, and material science.
• Founded on the understanding garnered from
  systems biology
• New modalities for genetic engineering and
  directed evolution
• Scaling towards programmable biomaterials.

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Systems biology is necessary

• Because of the highly interconnected nature of
  cellular networks
• Because it is the best way to understand what is
  controllable and what is not in pathway dynamics
• Because it discovers what designs evolution has
  arrived at to solve cellular engineering problems
  that we emulate in our own designs.

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A broader overview

• Evolutionary Game Theory
• Ecological Modeling
• Population Biology
• Epidemiology
• Neuroscience
• Organ Physiology
• Immune Networks
• Cellular Networks
• Problems
• Static and Dynamic Representations
• Physical Picture for Representation (e.g.
  deterministic vs. stochastic)
• Mathematical Description of Physics (e.g.
  Langevin vs. Master Equation)
• Levels of abstraction Formal and ad hoc.
• Measurement High-throughput/broadbrush/imprecise
  vs. low-through/targetted/precise

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Chemical Kinetics The short course I.
Consider a collision between two hard
spheres In a small time interval, dt, sphere 1
will sweep out a small volume relative to sphere
2.
If the center of sphere 2 lies within this volume
at time t, then in the time small time interval
the spheres will collide. The probability that a
given sphere of type 2 is in that volume is
simply dVcol/V (where V is the containing
volume). All that remains is to average this
quantity over the velocity distributions of the
spheres.
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Chemical Kinetics The short course II.
Given that, at time t, there are X1 type-1
spheres and X2 type-2 spheres then the
probability that a 1-2 collision will occur on V
in the next time interval is
Now if each collision has a probability of
causing a reaction then in analogy to the last
equation, all we can say is
X1 X2 c1 dt average probability that an R1
reaction will occur somewhere in V within the
interval dt.
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Chemical Kinetics The Master Equation I.
If we wish to map trajectories of chemical
concentration, we want to know the probability
that there will be molecules of each
species in the chemical mechanism at time t in V.
We call that probability
This function gives complete knowledge of the
stochastic state of the system at time t. The
master equation is simply the time evolution of
this probability. To derive it we need to derive
which is simply done from our previous work.
It is the sum of two terms 1. The probability
that we were at X at time t and we stayed
there. 2. The probability that a reaction of type
m brought us to this state.
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Chemical Kinetics The Master Equation II
The first term is given by
Where
The probability that a reaction of type m will
occur given that the system is in a given state
at time t.
and where hm is a combinatorial function of the
number of molecules of each chemical species in
reaction type m.
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Chemical Kinetics The Master Equation III
The second term is given by
where Bm is the probability that the system is
one reaction m away from state at time t and
then undergoes a reaction of type m.
Plugging these terms into the equation for
and rearranging we arrive at the
master equation.
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Deterministic Kinetics I.
Deterministic kinetics may be derived with some
assumptions from the master equation. The end
result is simple a set of coupled ODEs
where s is the stoichiometric matrix and v is a
vector of rate laws. Example Enzyme kinetics
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Mathematical Representation
X 2 Y
2 Z
Z E
EZ
Enzymatic
EZ
EP
Very simplest Mass action representation
Flux Vector
Stoichiometric Matrix
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Mathematical Representation
X 2 Y
2 Z
X 2 Y
2 Z
E
Z E
EZ
Z
P
EZ
EP
Often timesthe enzyme isnt represented
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Enzyme Kinetics II.
But often times we make assumptions equivalent to
a singular perturbation. E.g. we assume that E,S
and ES are in rapid equilibrium
These forms are the common forms used in basic
analysis
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Stationary State Analysis
Clearly, the steady state fluxes are in the null
space of the stoichiometric matrix. But these
are only unique if significant constraints are
also applied (the system in under-determined). Al
so highly dependent on representation.
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The Stoichiometric Matrix

• This matrix is a description of the topology of
  the network.
• It is tricky to abstract into a simple incidence
  matrix, for example.
• Most experimental measurements can only capture a
  small fraction of the interactions that make up a
  network.
• However, it does put some limits on behavior

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Graph Theory Scale-Free networks?

• Nodes are protein domains
• Edges are interactions
• Statements are made about
• Robustness
• Signal Propagation (small world properties)
• Evolution

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Stability Analysis for Deterministic Systems

• ? a v m
• a?b v k a
• a 2 b ? 3 b v ab2
• b ?c v b
• da/dt m- ka ab2
• db/dt ka ab2 - b

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Stationary State

• da/dt m- ka ab20
• db/dt ka ab2 b0
• ass m/(m2k), bss m
• So for any given value of m or k we can calculate
  the steady-state. These are parameters

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Stability

• We calculate stability by figuring out if small
  perturbations around a stationary state grow away
  from the state or fall back towards the state.
• So we expand our differential equations around a
  steady state and ask how small pertubations in a
  and b grow.

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Stability
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Stability
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Stability
Thus the ? are the eigenvalues of the
perturbation matrix and will determine if the
perturbations grow or diminish.
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Why is quantitative analysis important?
B-p
Å
Ass
B-p
B-p
?
Å
E.g. Focal Adhesion Kinase Alternative Splice
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Quantitative Analysis
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Bistability
A simple model of the positive feedback
kC1.6
kc
Monostable
Stationary state FAK-I
Irreversibly Bistable
Weakly bistable
kc catalytic constant for the
trans-autophosphorylation.
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Signal Filtering
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Brief Digression Chemical Impedance
I?A?
So A is the signal inside the cell that I is
outside the cell. What if A signals to downstream
targets by reacting with them? AB?C
The rates and concentrations of downstream
processes degrade the signal from A.
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Brief Digression Chemical Impedance
I?A?
But what if reaction is by reversible
binding? AB?C
The rates and concentrations of downstream
processes dont affect the signal.
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But.what about the ME
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Error and ORDINARY DIFFERENTIAL EQUATIONS
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Ordinary Differential Equations

• A differential equation defines a relationship
  between an unknown function and one or more of
  its derivatives
• Physical problems using differential equations
• electrical circuits
• heat transfer
• motion

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Ordinary Differential Equations

• The derivatives are of the dependent variable
  with respect to the independent variable
• First order differential equation with y as the
  dependent variable and x as the independent
  variable would be

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Ordinary Differential Equations

• A second order differential equation would have
  the form

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Ordinary Differential Equations

• An ordinary differential equation is one with a
  single independent variable.
• Thus, the previous two equations are ordinary
  differential equations
• The following is not

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Partial Differential Equations
d
d
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Ordinary Differential Equations

• The analytical solution of ordinary differential
  equation as well as partial differential
  equations is called the closed form solution
• This solution requires that the constants of
  integration be evaluated using prescribed values
  of the independent variable(s).

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Ordinary Differential Equations

• At best, only a few differential equations can be
  solved analytically in a closed form.
• Solutions of most practical engineering problems
  involving differential equations require the use
  of numerical methods.

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One Step Methods

• Focus is on solving ODE in the form

h
y
yi
x
This is the same as saying new value old value
(slope) x (step size)
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One Step Methods

• Focus is on solving ODE in the form

This is the same as saying new value old value
(slope) x (step size)
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One Step Methods

• Focus is on solving ODE in the form

This is the same as saying new value old value
(slope) x (step size)
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Eulers Method

• The first derivative provides a direct estimate
  of the slope at xi
• The equation is applied iteratively, or one step
  at a time, over small distance in order to reduce
  the error
• Hence this is often referred to as Eulers
  One-Step Method

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Taylor Series
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EXAMPLE
For the initial condition y(1)1, determine y for
h 0.1 analytically and using Eulers method
given
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step size
dy/dx
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Recall the analytical solution was 1.4413 If we
instead reduced the step size to to 0.05
and apply Eulers twice
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If we instead reduced the step size to to 0.05
and apply Eulers twice
Recall the analytical solution was 1.4413
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Error Analysis of Eulers Method

• Truncation error - caused by the nature of the
  techniques employed to approximate values of y
• local truncation error (from Taylor Series)
• propagated truncation error
• sum of the two global truncation error
• Round off error - caused by the limited number of
  significant digits that can be retained by a
  computer or calculator

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Taylor Series
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Higher Order Taylor Series Methods
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Derivatives
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Modification of Eulers Methods

• A fundamental error in Eulers method is that the
  derivative at the beginning of the interval is
  assumed to apply across the entire interval
• Two simple modifications will be demonstrated
• These modification actually belong to a larger
  class of solution techniques called Runge-Kutta
  which we will explore later.

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Heuns Method

• Consider our Taylor expansion

Approximate f as a simple forward difference
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Heuns Method
Substituting into the expansion
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Heuns Method

• Determine the derivatives for the interval _at_
• the initial point
• end point (based on Euler step from initial
  point)
• Use the average to obtain an improved estimate of
  the slope for the entire interval
• We can think of the Euler step as a test step

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y
Take the slope at xi Project to get f(xi1
) based on the step size h
h
xi xi1
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y
h
xi xi1
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y
Now determine the slope at xi1
xi xi1
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y
xi xi1
Take the average of these two slopes
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y
xi xi1
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y
Use this average slope to predict yi1
xi xi1

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y
Use this average slope to predict yi1
xi xi1

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y
y
xi xi1
x
xi xi1
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y
x
xi xi1
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Improved Polygon Method

• Another modification of Eulers Method
• Uses Eulers to predict a value of y at the
  midpoint of the interval
• This predicted value is used to estimate the
  slope at the midpoint

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Improved Polygon Method

• We then assume that this slope represents a valid
  approximation of the average slope for the entire
  interval
• Use this slope to extrapolate linearly from xi to
  xi1 using Eulers algorithm

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Improved Polygon Method
We could also get this algorithm from
substituting a forward difference in f to i1/2
into the Taylor expansion for f, i.e.
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y
f(xi)
x
xi
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y
h/2
x
xi xi1/2
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y
h/2
x
xi xi1/2
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y
f(xi1/2)
x
xi xi1/2
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y
f(xi1/2)
x
xi xi1/2
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y
Extend your slope now to get f(x i1)
h
x
xi xi1/2 xi1
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y
f(xi1)
x
xi xi1/2 xi1
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Conclusions

• Algorithms can be more or less stable to
  truncation or round off error.
• Algorithms can be better or worse approximations
  to the math you want to do.
• Algorithms can be more or less complex

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Master Equation Simulation I

• (Based on Gillespie, D.T. (1977) JPC, 81(25)
  2340)
• We are given a system in the state (X1,...,XN)
  at time t.
• To move the system forward in time we must ask
  two questions
• When will the next reaction occur?
• What kind of reaction will it be?
• In order to answer these questions we introduce
• P(t,m)dt probability that, given the state
• (X1,...,XN) at time t, the next
• reaction in V will occur in the
• infinitesmal time interval
• (tt,ttdt) there will be a
• reaction of type Rm.

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Master Equation Simulation II
Now we can define the P(t,m) to be the
probability that no reaction occurs in the
interval (t,tt) (Po(t)) times the probability
that reaction Rm will occur in the infinitesmal
time dt following this interval
(aµdt) P(t,m)dt Po(t) aµd t Now aµ is
simply a term related to the rate equation for a
given reaction. In fact it is a transition
probability, cµ, times a combinatorial term which
enumerates the number of ways n-species can react
in volume V given the configuration (X1,...,XN),
hµ. Therefore 1-S aµd t ' probability
that no reaction will occur in time d t '
from the state (X1,...,XN). and Po(t ' d t
') Po(t ')1-S aµd t ' the solution of which
is Po(t ') exp-S aµ t
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Master Equation Simulation III
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Endogenous Noise
P
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t
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a
P

• One gene
• Growing cell, 45 minutes division time
• Average 60 seconds between transcripts
• Average 10 proteins/transcript

A

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A
A
A
2
S
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P
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A
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What happens when you have bistability and noise?
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Langevin equation
E
Å

• But what if there is external noise on E?
• Lets start with

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The compact Langevin

• Plug the conservation conditions into the
  equations for A-p (A)

Diffusion
Drift
Note that another term in 1/KA has been
introduced. There is now the possibility of a
cubic nullcline.
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The Fokker-Planck equivalent.
Which yields the stationary nullcline

• Compared to the deteriministic nullcline

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Depending on the noise type
Ass
p0 Normal Noise p1/2 Chi-square
noise p1 Log-normal noise
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Validation by ME simulation
It turns out this generates log-normal noise on E
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ME Simulation
With noise on E Without noise
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Stationary Distribution with Noise
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Stationary Distribution w/o Noise
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Summary

• Adding noise to a system (in this case external
  noise) can qualitatively change its dynamics.
• Interestingly we can predict the effect with a
  compact Langevin approach AND a MM approximation
  pretty well compared to whats observed in a full
  ME simulation.
• The implications for noise-induced bistability
  and switching havent been fully worked out.

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But an Ugly specter is raised.
Is this really a valid picture? Adding noise
changes the nullcline!
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Nonetheless Static noise can make things look
bistable
Linear
Switch
There is a relationship between the variance on E
and the slope of the response that determines
whether the stationary distribution will be
bimodal.
p(x)
p(x)
X
X
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Niches are Dynamic
abiotoic reservoir

• Characteristic times may be spent in each
  environment.
• Environments themselves are variable.

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Adaptability vs. Evolvability
Life Cycle

• Adaptability Adjustment on the time scale of
  the life cycle of the organism

• Evolvability Capacity for genetic changes to
  invade new life cycles

Chris Voigt
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Evolvability

• In a dynamic environment, the lineage that
  adapts first, wins
• Fewer mutations means faster evolution
• Are some biosystems constructed to minimize the
  mutations required to find improvements?

• Modularity
• Robustness / Neutral drift improves functional
  sampling
• Shape of functionality in parameter space
• Minimize null regions in parameter space
  (entropy of multiple mutations)

Chris Voigt
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Logic of B.subtilis stress response
Sporulation

• Network organization has a functional logic.
• There are different levels of abstraction to be
  found.

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Clustered Phylogenetic Profiles
species
2
1
4
3
5

• Clustered phylogenetic profile shows blocks of
  conserved genes
• methyl-processing receptors and chemotaxis genes
  in motile bacteria
• methyl-processing receptors and chemotaxis genes
  in motile Archaea
• flagellar genes in motile bacteria
• type III secretion system (virulence) in
  non-motile pathogenic bacteria
• motility genes in spore-forming bacteria
• late-stage sporulation genes in spore-forming
  bacteria
• spore coat and germination response genes in
  spore-forming bacteria that are not competent
• late-stage sporulation genes in spore-forming
  bacteria that are also competent
• DNA uptake genes in Gram positive bacteria
• DNA uptake genes in Gram negative bacteria

Chemotaxis
6
genes
Sporulation
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8
8
Competence
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10
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Consider Chemotaxis E. coli
Periplasm
Cytoplasm
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Consider Chemotaxis E. coli
Periplasm
Cytoplasm
Integral Feedback Controller
CheAWYZ
Flagella
receptors
cheB/cheR
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Clusters are functionally coherent
Receptors
Signal Transduction (che)
Hook and Flagellar Body
Flagellar export/Type III secretion
Flagellar length and motor control
Hypthothetical receptors
Cross-Regulation with Sporulation/Cell Cycle
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Different modules for different lives
Animal pathogens
Sporulators
Archeal Extremophiles
Plant pathogens
Endopathogens
Endopathogens
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What Ontology Recovers Modules?
Systems Ontology
Color legend sensor controller actuator
cross-talk between networks unknown
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Comparative analysis is especially important
Rao, CV, Kirby, J, Arkin, A,P. (2004) PLOS
Biology, 2(2), 239-252
These are the homologous chemotaxis pathways in
E.coli and B. subtilis They have the same
wild-type behavior. Different biochemical
mechanisms. Different robustnesses!
Chris Rao/John Kirby
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Two important features
Adaptation Time
Exact Adaptation
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Differences in robustness
E . Coli
Chris Rao/John Kirby
B . subtilis
Do these differences lead to differences in
actual fitness?
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Evolvability

• In a dynamic environment, the lineage that
  adapts first, wins
• Fewer mutations means faster evolution
• Are some biosystems constructed to minimize the
  mutations required to find improvements?

• Modularity
• Robustness / Neutral drift improves functional
  sampling
• Shape of functionality in parameter space
• Minimize null regions in parameter space
  (entropy of multiple mutations)

Chris Voigt
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Logic of B.subtilis stress response
Sporulation

• Network organization has a functional logic.
• There are different levels of abstraction to be
  found.

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Sporulation initiation
A Motif
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The SIN Operon A recurrent motif
Environmental Cellular Signals
Sporulation genes (stage II) spoIIG as model
Spo0A
P1
P3
sinR
sinI
SIN Operon

• Vegetative (healthy) growth Constitutive SinR
  expression from P3

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Feedback provides filtering
INPUT of Spo0AP
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Functional Regions in Parameter Space
Bistability
k1 DGS DGRNAP DGR AI gI KI
k3 AR gR KR
Parameter Space
Oscillations
Hopf points
k1 DGS DGRNAP DGR AI gI KI
k3 AR gR KR
SinR Activity
P3
P1
SinI Activity
Chris Voigt
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Full Bifurcation Analyses Evolvability?

• Tuning the expression of SinR (AR) with respect
  to SinI leads to dynamical plasticity
• Transcription from P3 (k3) strengthens
  bistability and damps oscillations

Bistability
Osc
Switch
Graded
Pulse
0A 10,000 nM
0A 10 nM
k3 (mRNA/s)
SinI (nM)
AR (protein/mRNA-s)
AR (protein/mRNA-s)
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Examples of Protein-Antagonist Operons

• How can complicated dynamical behavior arise
  from simple evolutionary events?
• What are the requirements to bias the operon to
  one function?
• Once established can one function evolve into
  another?

Chris Voigt
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Comparative analysis of SinI/SinR
region affecting k1
KI
Comparison of five strains of Bacillus anthracis
Across ALL sporulators Very variable.
In anthracis Mutations mostly affect KI and
k1 Threshold of the switch is most affected.
Voigt, CA, Wolf, DM, Arkin, AP, (2004) Genetics,
In press PMID 15466432
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Feedback induces stochastic bimodality
spo0Ap1nm
spo0Ap4nm
spo0Ap100nm
sinI
Though we must be careful since the addition of
noise itself changes the qualitative dynamics.
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Heterogeneity of Entry to Sporulation
A.
B.
Microscopic analysis of LF25 (amyEPspoIIE cm).
Observation by DIC X60 (A.) and fluorescence (B.)
of cells resuspended to induce sporulation and
incubated 3 hours at 37C. An example of cells
not showing fluorescence are circled in figure A.
Lisa Fontaine-Bodin, Denise Wolf, Jay Keasling
124
Summary 1
So this motif

• Has flexible function based on parameters
• Most parameters tune response
• A couple of parameters qualitatively change the
  response
• Is an example of a possible Evolvable Motif
• Sometimes exhibits stochastic effects
• Are they adaptive?

125
Stochastic Effects Are Ubiquitous
Images
Clones
Stochastic Gene Expression in HIV-1 Derived
Lentiviruses Stable Clones
No Positive Feedback
Tat Feedback Very Bright Sort
Tat Feedback Bright Sort
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Software

• MatLab
• Mathematica
• Berkeley Madonna
• GEPASI
• TerraNode
• JDesigner

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The game of life
E3
Environment
E1
E1
E4
E2
E2
noise
t
?1
?2
quorum
pi
S1
S2
SN
S3
Output signals
S5
S4
Organism 2
Organism 1
Beginning to link Game Theory to Dynamical
Cellular Strategies.
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Formal Model
?y
?y
sx1
a)
b)
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Example two environments, two moves, no sensor
e.g. xpili yno pili E1in host E2out
IF E1 selects for x, against y E2 selects
against x, for y
Denise Wolf, Vijay Vazirani
130
With no sensor, the options are
Denise Wolf, Vijay Vazirani

1. ALL cells in state x
2. ALL cells in state y
3. Statically mixed population (some x, some y)
4. Phase variation of individual cells between x and
   y

131
With no sensor, the options are
Denise Wolf, Vijay Vazirani
Extinction

1. ALL cells in state x
2. ALL cells in state y
3. Statically mixed population (some x, some y)
4. Phase variation of individual cells between x and
   y

132
With no sensor, the options are
Denise Wolf, Vijay Vazirani
Extinction

1. ALL cells in state x
2. ALL cells in state y
3. Statically mixed population (some x, some y)
4. Phase variation of individual cells between x and
   y

133
With no sensor, the options are
Denise Wolf, Vijay Vazirani
Extinction

1. ALL cells in state x
2. ALL cells in state y
3. Statically mixed population (some x, some y)
4. Phase variation of individual cells between x and
   y

134
With no sensor, the options are
Denise Wolf, Vijay Vazirani
Proliferation!

1. ALL cells in state x
2. ALL cells in state y
3. Statically mixed population (some x, some y)
4. Phase variation of individual cells between x and
   y

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Phase variation for survival
Rate of Y?X Switching
Rate of X?Y Switching
This is a Devils compromise Phase-variation
behaviors is not optimal in any one environment
but necessary for survival with noisy sensors in
a fluctuating environment.
Denise Wolf, Vijay Vazirani
136
Learning Environment from Cell State
Strategy Sensor profile Environmental profile
Random Phase Variation (RPV) No sensors Devils Compromise (DC) lifecycle time varying environment with different environmental states selecting for different cell states. Optimal switching rates a function of lifecycle asymmetries and environmental autocorrelation. Time variation required (spatial variation insufficient).
Random Phase Variation (RPV) OLow prob. observable transitions over DC or extinction set. Devils Compromise (DC) lifecycle time varying environment with different environmental states selecting for different cell states. Optimal switching rates a function of lifecycle asymmetries and environmental autocorrelation. Time variation required (spatial variation insufficient).
Random Phase Variation (RPV) DLong delays relative to env. transition times. Devils Compromise (DC) lifecycle time varying environment with different environmental states selecting for different cell states. Optimal switching rates a function of lifecycle asymmetries and environmental autocorrelation. Time variation required (spatial variation insufficient).
Random Phase Variation (RPV) Perfect sensors Frequency dependent growth curves with mixed ESS.
Sensor Based Mixed OHigh prob. observable transitions APoor accuracy Devils Compromise lifecycle. Asymmetric lifecycle required. Optimal mixing probabilities biased toward selected cell-states in dominant environmental states.
Sensor Based Mixed LPF OHigh prob. observable transitions APoor accuracy. NHigh additive noise. Devils Compromise lifecycle. Asymmetric lifecycle required. Optimal mixing probabilities biased toward selected cell-states in dominant environmental states.
Sensor Based Pure OHigh prob. observable transitions AHigh accuracy or moderate accuracy and low noise N. Temporally or spatially varying environment with each environmental state selecting for a single cell state.
Sensor Based Pure LPF OHigh prob. observable transitions AModerate accuracy. NHigh additive noise. Temporally or spatially varying environment with each environmental state selecting for a single cell state.
Denise Wolf, Vijay Vazirani
137
Robustness and Fragility

• The stratagems of a cell evolve in a given
  environment for robust survival.
• Evolution writes an internal model of the
  environment into the genome.
• But the system is fragile both
• to certain changes in the environment (though
  there are evolvable designs)
• And certain random changes in its process
  structure.
• One of the central questions has to be Robust on
  what time scale? Can evolution design for the
  future by learning from the past?

138
Summary

• The availability of large numbers of bacterial
  genomes and our ability to measure their
  expression opens a new field of Evolutionary
  Systems Biology or Regulatory Phylogenomics.
• Comparative genomics identifies particularly
  conserved motifs, parts of which are
  evolutionarily variable and select for different
  behaviors of the network.
• By understanding what evolution selects in a
  network context we better understand what the
  engineerable aspects of the network are.

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Acknowledgements

• Comparative Stress Response Amoolya Singh,
  Denise Wolf
• SinIR analysis Chris Voigt, Denise Wolf
• Chemotaxis Chris Rao, John Kirby
• HIV Leor Weinberger, David Schaffer
• Games Denise Wolf, Vijay V. Vazirani
• Funding
• NIGMS/NIH
• DOE Office of Science
• DARPA BioCOMP
• HHMI