Evolution of Complex Dynamics in Mathematical and Electronic Models of Genetic Networks

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Evolution of Complex Dynamics in Mathematical and
Electronic Models of Genetic Networks

• Leon Glass
• Isadore Rosenfeld Chair in Cardiology,
• McGill University

2
A synthetic oscillatory network of
transcriptional regulators
TetR
LacI
l cI


gene B

gene C
gene A
PC
PA
PB
mRNA A
mRNA B
mRNA C
protein A
protein C
protein B
Elowitz and Leibler, 2000
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Plasmids
Repressilator
Reporter
PLlacO1
ampR
tetR-lite
PLtetO1
kanR
TetR
TetR
SC101 origin
gfp-aav
l PR
l cI
LacI
GFP
lacI-lite
ColE1
l cI-lite
Elowitz and Leibler, 2000
PLtetO1
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Observation in Individual Cells
60
140
250
300
390
450
550
600
GFP Fluorescence
Bright-Field
Fluorescence (a.u.)
time (min)
Elowitz and Leibler, 2000
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Problem How can we develop mathematical models
that represent the dynamics in real networks?
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A differential equation
Glass, Kauffman, Pasternack, 1970s
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Rationale for the equation

• A method was needed to relate the qualitative
  properties of networks (connectivity,
  interactions) to the qualitative properties of
  the dynamics
• The equations allow detailed mathematical
  analysis. Discrete math problems
  (classification), nonlinear dynamics (proof of
  limit cycles and chaos in high dimensions)

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The Repressilator
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The Hypercube Representation
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The Hypercube Representation for Dynamics (N
genes)

• 2N vertices each vertex represents an orthant
  of phase space
• N x 2N-1 edges each edge represents a
  transition between neighboring orthants
• For networks with no self-input, there is a
  corresponding directed N-cube in which each edge
  is oriented in a unique orientation

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Cyclic Attractors

• Any attracting cycle on the hypercube corresponds
  to either a stable limit cycle or a stable
  focus in the differential equation (Glass and
  Pasternack, 1978)

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Strategy of Proof of Limit Cycles in N dimensions
The Poincare map is a linear fractional map. a is
a (N-1)x(N-1) matrix, b and x(t) are (N-1)
vectors. The composition of 2 linear fractional
maps is a linear fractional map. From the
Frobenius-Perron theorem, it follows that if the
leading eigenvalue of a is gt1, then there is a
stable,unique limit cycle and if the leading
eigenvalue of a is less than 1, there is a
focus. Conjecture Cycles are robust when step
functions are changed to steep enough sigmoids.
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Chaotic dynamics in the model equations
A network that generates chaos
Mestl, Lemay, Glass, 1996
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A Chaotic System in 4D
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Projection of flows
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Evolution of dynamics
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The number of different networks in N dimensions
Glass, 1975 Edwards and Glass, 2000
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Evolving Rare Dynamics

• Long cycle
• Chaotic dynamics - increased complexity using
  topological entropy as a measure of complexity

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An Evolvable Circuit
(J. Mason, J. Collins, P. Linsay, LG, Chaos, 2004)
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Why study electronic circuits?

• It is real
• It leads us to think about issues in real
  circuits i.e. not all decay rates will be equal
• Circuits could be useful

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The Hybrid Analog-Digital Circuit
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Circuit Elements
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Distribution of Cycle Lengths in Electronic
Circuit (300 random circuits with stable
oscillations)
Choose a target period of 80 ms
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Sample Evolutionary Run
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Optimal Mutation Rate - Data

• Each trial starts with oscillating network
• 25 Trials at each mutation rate for 250
  generations
• Mutation rates of 2.5, 5, 10, 20, 100

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The Hypercube Representation for Fitness
Landscapes (N genes)

• 2N x 2N-1 vertices
• Each vertex represents a different genome
• Two vertices can have the same fitness, i.e.
  there can be neutral mutations

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Fitness Landscape
k is the Hamming distance of the truth table of a
network from a network with stable cycle
The mean improvement as a function of mutation
rate
where,
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Prediction of Optimal Mutation Rate

• Compares favorably with experimentally determined
  value of 5-10

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Chaotic Electronic Network
Glass, Perkins, Mason, Siegelmann, Edwards, J.
Stat. Phys. 2005
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Symbolic sequences of cycles that start and end
on vertex 00110

• 6 141343 4 1414 6 143143 6
  143413 14 14341523545245 14
  14345123545245 18 143453431352353254 8
  15421245 8 15421254 12 154351535343
  14 15435153535354 16 1543515423424252
  62 154512131512121212121212121214241241243242141
  24214124214124252 60 154512131512121212121212
  121214241241243242424214124254524252 56
  15451213151212121212121212121424124124324242424242
  545245 62 15451213151212121212121214241241243
  242141242141242141242545245 60
  15451213151212121212121214241241243242141242141242
  1545124252 58 1545121315121212121212121424124
  124324214124214124254524252 56
  15451213151212121212121214241241243242141242141242
  545245 52 15451213151212121212121214241241243
  24242424242545245 54 154512131512121212121212
  142432421412421412421545124252 46
  1545121315121212121212121424324214124214124252
  50 154512131512121212121212142432421412421412425
  45245 48 154512131512121212121212412412412432
  424242424252

Etc.- Can we use these to classify bifurcations
and to compare different types of chaos in
chaotic systems?
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Statistical Properties of Words in Return Map
Zipfs Law!
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Evolving Rare Dynamics

• Long cycle
• Chaotic dynamics - increased complexity using
  topological entropy as a measure of complexity

(An upper limit on this can be determined from
the underlying truth table of a network)
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Evolving Complex Dynamics
Wilds, Kauffman, Glass (2007)
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No neutral mutations
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With neutral mutations
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The theoretical limit on entropy increases during
evolution
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Conclusions

• These networks can model some real biological
  systems
• Electronic networks displaying rare dynamics can
  be evolved
• Systems with neutral mutations evolve faster than
  those that just do hill climbing
• During evolution, the neutral subspace decreases
  as evolution proceeds

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Properties of Networks Based on Logical Structure

• Extremal stable fixed points
• Limit cycles associated with cyclic attractors
  (stability and uniqueness)
• Necessary conditions for limit cycles and chaos
• Analysis of chaos in some networks
• Upper limit on topological entropy

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The inverse problem given the dynamics what is
the network?(Research with Ted Perkins, McGill)
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Fibrosis Pathway by STKE
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A simple feedback network
X1 X2(t) X1 X2(t1)
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Flows in phase space and the hypercube
representation for a simple feedback network
11
01
10
00
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Poincare map for the 2D cycle
, where a, b gt0
alt1
x(t1)
x(t)
x(t)
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Main Problems - To relate the interactions in
genetic networks to the dynamics.

• To predict the dynamics given the genetic network
• To determine the genetic network given the
  dynamics
• To understand how genetic networks evolved

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A differential equation
Glass, Kauffman, Pasternack, 1970s
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The Inverse Problem. Compute the number of
logical states needed to determine connectivity
diagram
Perkins, Hallett, Glass (2004)
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Compute the number of switches needed to
determine the entire network
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Chaotic Electronic Network
Glass, Perkins, Mason, Siegelmann, Edwards, J.
Stat. Phys. 2005
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Inverse problem for the electronic circuit
(parameter values)
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Simulation of differential equation with
parameters just determined
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Sequential recruitment and combinatorial
assembling of multiprotein complexes in
transcriptional activation.
Lemaire, Lee, Lei, Metivier, Glass (PRL,2006)
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Modeling the kinetics
A long sequence of biochemical transformations of
any kind taking place at the promoter

• complex formation
• enzymatic modification
• conformational modification
• ATP-dependent reaction
• etc.

• m unidentified complexes xi
• xi are composed of an arbitrary number of
  proteins

One constraint Period ? 40 min, the
experimental period
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Deterministic approach
In the limit of a large number of cells,
x1(t) full solution
x1(t) leading term
x1(t) sum of the 2 first terms
x1(t) sum of the 7 first terms
x1(t) sum of the 4 first terms
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Fitting results
Values of the ci,j (colored regions is 1)
Métivier et al 2003
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Study of gene expression in dogs who have rapid
pacing of the atria or ventricles.

• Experiment designed and carried out in the
  laboratory of Dr. S. Nattel, Sophie Cardin and
  Patricia Pelletier. Data analysis by Eric Libby
• (Circulation Research 2007)

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Affymetrix Gene Chips

• Surveys thousands of genes simultaneously
• Has 11-20 probes for each gene
• Probes are 25 base pairs long either PM or MM
• Species specific
• Millions of sequences in each 500,000 feature
  spots

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Chip Hybridization
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The Data

• CEL files
• Raw PM and MM intensities
• Few software packages can open files

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Ventricular Tachypacing

• Fibrotic remodeling
• Immune system response and apoptosis
• Pace right ventricle at 220-240 bpm

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Microarray Setup

• 5 dogs in each group
• 8 groups total
• Compare ATP and VTP in LA
• Compare 24 hr to 1-2 wk treatments
• Compare VTP LA to VTP LV

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B
A
ATP Expression Level (ln L)
ATP Expression Level (ln L)
ATP 24-h
ATP 1-w
xxxx
xxxx
xxxx
Sham Expression Level
Sham Expression Level
C
D
VTP Expression Level (ln L)
VTP Expression Level (ln L)
VTP 24-h
VTP 2-w
xxxx
Sham Expression Level (ln L)
Sham Expression Level (ln L)
Figure 1
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Upregulated
Downregulated
DNA/RNA syn/deg. Cell cycle Apoptosis Immunity ECM
Signal trans. Protein syn/deg. Ribosomes Hormones
Metabolism Cell structure Transport Mitochondria
14 12 10 8 6 4 2 0 2 4
Number of genes
Figure 2
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Hypothetical Network
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Conclusions

• For the model differential equations, and the
  electronic models of these equations
• The networks can evolve to find new very rare
  dynamics
• It is possible to determine the networks based on
  the observed dynamics in model equations, in
  electronic circuits, in drosophila
• Problems for complex physiological data
• Noise, biological variability, few time points
  (expense!), time delays

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The Inverse Problem of Genetic Networks

• Leon Glass
• Isadore Rosenfeld Chair in Cardiology, McGill
  University, Montreal, Quebec

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Plane of the Poincare map
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First analysis

• All reaction rates are assumed irreversible.
• All reaction rates are assumed identical ai a.

Choosing m 200, then a m/T0 5 min-1
Due to the stochastic nature of the system, the
chemical scheme can be interpreted as a
sequential Poisson process in which the duration
t before the next reaction takes place follows
the distribution
p(t) a e-at
After synchronization of the cells,

• the k-th cycle has mean starting time of km/a
• and variance kT02/m

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Fitting the model
Each protein is a component of different
complexes xi(t), but we do not know a priori
which ones.
xi(t)
xi1 (t)
xi2(t)
xm(t)
x1(t)
xi-2 (t)
xi-1 (t)
Po(t)
For the orange protein, co 0, ,0, 1, 1, 1, 0,
, 0
t
Pb(t)
For the blue protein, cb 0, ,0, 0, 0, 1, 0,
, 1
t
Pp(t)
For the purple protein, cp 0, ,0, 0, 0, 0, 1,
, 0
t
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Robustness of the results
We performed different numerical studies to test
the robustness of the fitted sequence

• Different values of m 100, 200, 300 and 400.
• Addition of 2 of noise to the data points.
• Changes in the vertical scaling of the data (up
  to 1.4).
• Selections of random reaction rates ai
  (provided that the period, for the selection of
  ai, is close to T0, and the solutions are not too
  damped).

Results Although there can be slight changes in
the sequence ci , the main pattern of the ci,j
remain unchanged.
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Biological meaning of the results

• Correlation between the sequences.
• Reconciling ChIP assays and Fluorescence
  microscopy-based methods.
• It is believed that 5 or 6 functional complexes
  are successively formed on the promoter of pS2.

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A differential equation
i(t)
i
t
Xi1
Xi0
Xi1
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(Side issue classification of complex dynamics)
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Mechanism Discovery

• Look up pathways of hormones and most
  significantly expressed genes

Fibrosis Pathway by STKE
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Global Results
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Decipher Mechanisms
9
36
533
503
24
4
4
357
14
12
172
1
5
47
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0
0
0
18 genes
1641 genes
91 genes
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Significant Genes
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Cumulative fraction of ranks
24-h VTP
2-w VTP
B
A
Fraction of ranks
Extracellular matrix
Cell structure/mobility
Immunity
Extracellular matrix
Cell structure/mobility
Metabolism
Ribosome
Metabolism
Others
Mitochondria
DNA/RNA syn/deg
Ribosome
Others
Gene groups deviating from overall VTP expression
pattern
Figure 4
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Gene expression in Drosophila Perkins, Jaeger,
Reinitz, GlassPLOS Computational Biology 2006
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(Perkins, Jaeger, Reinitz, Glass, PLOS
Computational Biology 2006)
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(Perkins, Jaeger, Reinitz, Glass, PLOS
Computational Biology 2006)
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Proposed network for gene control
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Comparison with different models