Transcript Evolution of Complex Dynamics and the Inverse Problem in

Using Logical Circuits to Analyze
and Model Genetic Networks
Leon Glass
Isadore Rosenfeld Chair in
Cardiology,
McGill University
• Introduction to logical models – history and
application
• Mathematics and applications – evolution
of electronic circuits and the inverse
problem
Puzzle: Is there a simple way to
think about genetic networks?
• If no, we are in trouble.
• If yes, then might ideas developed using
logical networks be relevant?
“From molecular to modular cell biology”
Hartwell, Hopfield, Leibler, Murray
(Nature, 1999)
“The next generation of students should
learn how to look for amplifiers and logic
circuits, as well as to describe and look for
molecules and genes.”
Logical Network Models
• Neural Networks – A logical calculus of the ideas
immanent in nervous activity. McCulloch and
Pitts (1943). See also Kling & Szekely, Cowan,
Sejnowski, Grossberg, Hopfield and many
others
• Genetic Networks – Teleonomic mechanisms in
cellular metabolism, growth and differentiation.
Jacob and Monod (1961). See also Sugita,
Kauffman, Thomas, Bray and many others
Kling and Szekely, Kybernetik, 1968
Logical Models - Positive
• Many superb papers identify logical
functions as key controllers in biological
systems and have developed models
based on this concept.
(Bull. Math. Biol. 1995)
(Science, 1998)
(J. Theor. Biol. 1998)
Logical Models - Negative
• Logical models are not well known to experimental
biologists. Typical models often consist of complex
networks without an analytical context. If logical
models really worked, people would use them.
Logical Models - Positive
• Some experimental systems show clear
evidence of discrete genetic expression
patterns in time and space.
Slide from John Reinitz
Logical Models - Negative
• Some data does not
show any obvious
evidence of the
operation of discrete
expression levels
(Circ Res 2007)
Logical Models - Positive
Biobricks Website, MIT – Many parts
are based on logical models. Synthetic
biology competition.
Logical Models - Negative
(Science, 2002)
Logical Models - Positive
• Beautiful mathematical formulation for
analyzing such networks (I will describe
this in a minute).
Logical Models- Negative
• Logical formulations are not easily derived
from mass action kinetics unless one has
special features such as cooperativity or
cascades to achieve threshold-like control.
Many analytic problems may arise due to
important factors such as time delays,
stochasticity, spatial structure that have
not yet been carefully addressed.
Synthetic Biology Uses Ideas from
Logical Models
• Toggle switch
• Inhibitory loops (repressilator)
Construction of a genetic toggle switch in
Escherichia coli
Repressor A
GFP
“On”
Gene B off
Gene A on
Reporter
Repressor B
“Off”
Gene B on
Gene A off
Reporter
Gardner, Cantor & Collins (2000)
Construction of the plasmid
Gardner, Cantor & Collins (2000)
Two stable steady states
Gardner, Cantor & Collins (2000)
A synthetic oscillatory network of transcriptional
regulators
TetR
l cI
PC
gene A
mRNA A
protein A
PA
LacI
gene B
mRNA B
protein B
PB
gene C
mRNA C
protein C
Elowitz and Leibler, 2000
Plasmids
Repressilator
Reporter
PLtetO1
kanR
TetR
l cI
TetR
LacI
gfp-aav
GFP
ColE1
Elowitz and Leibler, 2000
Observation in Individual Cells
60
140
250
300
390
450
550
600
GFP
Fluorescence
Bright-Field
Fluorescence (a.u.)
120
100
80
60
40
20
0
0
100
200
300
400
500
600
time (min)
Elowitz and Leibler, 2000
Problem: How can we develop mathematical
models that represent the dynamics in real
networks?
A Boolean Switching Network
Xi is either 1 or 0
Bi is a Boolean function
Random boolean networks as gene models (Kauffman, 1969)
A differential equation
Glass, Kauffman, Pasternack, 1970s
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)
The Repressilator
The Hypercube Representation
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
Fixed Points
• A vertex that only has in arrows represents
a stable fixed point. It is robust under
changes in parameter values
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)
Evolving Rare Dynamics
• Long cycle
• Chaotic dynamics - increased complexity
using topological entropy as a measure of
complexity
The number of different networks
in N dimensions
Glass, 1975; Edwards and Glass, 2000
An Evolvable Circuit
(J. Mason, J. Collins, P. Linsay, LG, Chaos, 2004)
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
The Hybrid Analog-Digital Circuit
Circuit Elements
Distribution of Cycle Lengths in
Electronic Circuit (300 random
circuits with stable oscillations)
Choose a target period of 80 ms
Sample Evolutionary Run
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%
Prediction of Optimal Mutation Rate
Compares favorably with experimentally determined
value of ~5-10%
The Inverse Problem. Compute the number of
logical states needed to determine connectivity
diagram
Perkins, Hallett, Glass (2004)
Compute the number of switches needed to
determine the entire network
Gene expression in Drosophila
Perkins, Jaeger, Reinitz, Glass
PLOS Computational Biology 2006
(Perkins, Jaeger, Reinitz, Glass, PLOS Computational Biology 2006)
(Perkins, Jaeger, Reinitz, Glass, PLOS Computational Biology 2006)
Proposed network for gene control
Comparison with different models
Some Important Ideas About
Logical Network Models
• They do not require discrete time or states
• Logical networks can be embedded in
differential equations (that’s the main idea
of this talk)
• Qualitative features of networks are often
preserved by changing step function
control to sigmoidal function control
• Neural network models are a subclass of
the differential equations I described
Mathematical Models of Neural and
Gene Networks are Closely Related
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
Conclusions
• Logical models do provide a rich class of
models appropriate for many real
biological systems
• The limitations of this class of models is
not known
Thanks
• Stuart Kauffman, Joel Pasternack, Rod
Edwards, Jonathan Mason, Paul Linsay,
James Collins, Ted Perkins, Yogi Jaeger,
John Reintiz.
• NSERC, MITACS