Adaptive Connectivity Map Analysis

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Genetic networks from Computational Complexity
to System Biology
Hava Siegelmann Computer
Science UMass BioComputation lab,
Director
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Gene regulatory Network (GN)
Network of interaction between genes and gene
products in a cell
Reactor
Code
Chemicals
Gene regulatory systems as computers genetic
code-program protein levels-memory / affect
evolution
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Genetic Networks Computational Complexity
A. Ben-Hur and H. T. Siegelmann Chaos An
Interdisc. Journal of Nonlinear Science March
2004
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Computational Power
Nature2000 Gardner-Cantor-Collins
Elowitz-Leibler Successful fabrications of
synthetic networks programming GNs
Theoretical design of complex and
practical gene networks is a realistic
and achievable goal
We will prove the dynamics of a model gene
network is complex enough to be equivalent in its
expressive power to a Turing machine The
simulation is robust to perturbations
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Modeling Gene Network

• Variables concentrations of proteins real
  non-negatives
• No common timing device

differential equations
piecewise
protein yi production rate Gi constant
until threshold si dyi/dt
-kyi Gi(Y1, ,YN) ki-
degradation rate, Yi sgn(yi-si), Gi
production rate
Substitute xiyi-si, Xisgn(xi)
dxi/dt -xi Gi(X1, ,XN)
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The Dynamics

• Orthant of RN set corresponding to Boolean
  vector X

Dynamic crosses to another orthant or converges
to G(X) if at same orthant as x
Time for convergence/crossing tltln 2 (by
integrating to xi(t) gi (xi(0)-gi) e-t 0)
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Discrete Computation

• Symbolic Dynamic sequence of orthants X(1) x(2)
  
• Discrete network Zi ? sgn(Gi(Z)) Z in 0,1
• different dynamics than the continuous dxi/dt
  -xi Gi(X)
• Continuous typically chaotic, discrete not

N
How to relate continuous and discrete networks?
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The Design Principle Adjacent Nets

• Adjacent orthants differ in one coordinate
• G(X) adjacent if differs than X in one variable
  (or none)
• Lemma If G adjacent than discrete and continuous
  networks
• have the same symbolic dynamics
• Property Regularity Adjacent networks are not
  chaotic
• Property Robustness If G(X) adjacent and
  G(X)-G(X)lt dlt1
• then G adjacent and have same symbolic
  dynamics
• ? Design principle for robust GN (fabricated this
  way)

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TM by ODE/map?

• TM (1936) fixed control, unbounded tape
  countably infinite many configurations
• 1. in bounded set, two configurations encoded
  arbitrarily close
• 2. in unbounded set, minimal distance between
  encodings
• Both infeasible too sensitive to noise OR
  infinite realization
• Impossible to simulate TM by realizable
  analog machine
• Solution Turing-space(s(n)) for robust
  simulation

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Computing with Adjacent GN

• N variables 1,-1,1,..1, 0,0,0, A,
  H
• n inputs
  s(n) registers accepts halts

Language of Gn Ln -1,1n ? 0,1
Family of Uniform Networks There exist Turing
Machines M1, M2 that use constant memory only and
compute G and initial state for any input n
GN(s(n)) L in -1,1 uniform family Gn
n1?8 computes Ln
Adjacent-GN(s(n)) Turing-space(s(n))
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Conclusions

• 1. Adjacent nets design principle for fault
  tolerant
• programming in GN used in genetic toggle switch
  
• (In nature a gene can affect more than one other
  gene)
• 2. computational interpretation of dynamics of
  switch-like
• ODE Adjacent GN(s(n)) SPACE(s(n))
• Higher eukaryotes use more of their available
  memory
• (e.g. human vs c-elegance)

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PHYSICS NEWS UPDATE January 2004
The American Institute of Physics Bulletin of
Physics news
COMPUTATION IN GENE NETWORKS.  Searching for a
new way to produce a computational device, Asa
Ben-Hur (Stanford) and Hava Siegelmann (Amherst)
have developed a model which shows that the
functioning of a model gene network---genes
acting as a computer "program" and the gene
products in a cell (protein levels) acting as the
"memory"---is comparable in expressive power to
the workings of a Turing machine, the generic
idealized computer.  They compare a
hypothetical analog gene-network computer to
standard digital computers and suggest that
chemical reactions can be used to implement
Boolean logic and neural networks. 
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Circadian rhythm synchronicity and
jet-lag
with W. Bush and M. Harrington
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Circadian Rhythm
Orchestra of oscillations Endogenous daily clock
regulates sleep, temperature, endocrine
secretion, reproductive, immune
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SCN what do we know?

• 20,000 neurons, part of the hypothalamus
• Neurons have translation-transcription feedback
  loops (molecular clocks) of 24 hours
• SCN neuron spontaneous firing rates peak(4-10 Hz)
  during subjective day, decreases (0-2 Hz) at subj
  night
• Dissociated neurons from the same individual free
  run at slightly different periods.
• Neurons in intact SCN synchronize to global
  rhythm

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Questions

• What mechanism allows synchronicity of a
  population of oscillators with different periods?
• Could fast(8-10 Hz) neuronal firing mediate
  synchronization of 24 hr molecular clocks?
• What is efficient topology for synchronicity
• Experimental Studies
• Electrical silencing dissociates SCN neuron's
  molecular clocks(Yamaguchi et al., 2003.)
• (2) Exposure to neurotransmitters(GABA) causes
  phase shift of the molecular clock(Liu and
  Reppert, 2000)

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Our Oscillatory neural network model

• Firing rate modulated by the molecular clock
• Molecular clock can be shifted by
  neurotransmitter input in postsynaptic neuron

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Electrical Part of Neuron

• Leaky integratefire
• neuron Gerstner-Kistler2002
• Neurons fire at 10 Hz with no molecular clock
  input
• Firing rate inhibited by protein, P of molecular
  clock
• Fire when threshold reached
• Neurotransmitter released with neuronal firing

Gerstner, W., Kistler, W., Spiking neuron models
single neurons, populations,plasticity. Cambridge
University Press, New York, NY, 2002
Gerstner, W., Kistler, W., Spiking neuron models
single neurons, populations,plasticity. Cambridge
University Press, New York, NY, 2002
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Molecular Clock Model

• Model developed by Scheper et. al. 99
• Two state variables, mRNA (M) and protein (P),
  oscillate out of phase with one another.
• The sum of neurotransmitter input, IN, effects
  mRNA production rate, rM.

Scheper T, Klinkenberg D, Pennartz C, van Pelt
J., A mathematical model for the intracellular
circadian rhythm generator. J Neurosci. 1999
Jan 119(1)40-7.
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Simulation Results

• Fully connected 20 neurons
• Free running periods 23-26 hr
• Free initial phase
• Phase of the molecular clock measured by levels
  of mRNA.

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Results when cutting communication linksbetween
the clock and the firing

• Communication from clock to neuron is cut
• Communication from presynaptic neurons to clock
  is cut

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Measuring Sync

• Averages angles of the limit cycle around a unit
  circle
• Vector r represents synchronicity of population
• Value in 0-1
• Example, 20 neurons fully connected

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Small world networks

• Start with uniform connections
• of nearest neighbors
• Random rewire connections
• with probability p
• Experiments showing improvement in global
  synchrony using small world topology
• Different clusters become one

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Preliminary Results

• P 0
• P .1
• P .2

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Synchronization of Small World Network
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With Mary Harrington
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Contents
Summary
1. What is the computability of gene networks?
computational complexity and
design 2. What behavior can emerge by a group
of cells with GN?
self-constructing active agent 3. How genetic
oscillators in different cells talk
synchronize? circadian rhythm
computational complexity
self-construction
system biology
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T
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Thank you !
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Talk dedicated to Francis Crick
April 2004 Cricks home
For years of mentoring