Using%20Logical%20Circuits%20to%20Analyze%20and%20Model%20Genetic%20Networks

1
Using Logical Circuits to Analyze and Model
Genetic Networks

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

2

• Introduction to logical models history and
  application
• Mathematics and applications evolution of
  electronic circuits and the inverse problem

3
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?

4
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.

5
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

6
Kling and Szekely, Kybernetik, 1968
7
Logical Models - Positive

• Many superb papers identify logical functions as
  key controllers in biological systems and have
  developed models based on this concept.

8
(Bull. Math. Biol. 1995)
9
(Science, 1998)
10
(J. Theor. Biol. 1998)
11
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.

12
Logical Models - Positive

• Some experimental systems show clear evidence of
  discrete genetic expression patterns in time and
  space.

13
Slide from John Reinitz
14
Logical Models - Negative

• Some data does not show any obvious evidence of
  the operation of discrete expression levels

(Circ Res 2007)
15
Logical Models - Positive

Biobricks Website, MIT Many parts are based on
logical models. Synthetic biology competition.
16
Logical Models - Negative
(Science, 2002)
17
Logical Models - Positive

• Beautiful mathematical formulation for analyzing
  such networks (I will describe this in a minute).

18
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.

19
Synthetic Biology Uses Ideas from Logical Models

• Toggle switch
• Inhibitory loops (repressilator)

20
Construction of a genetic toggle switch in
Escherichia coli
Gardner, Cantor Collins (2000)
21
Construction of the plasmid
Gardner, Cantor Collins (2000)
22
Two stable steady states
Gardner, Cantor Collins (2000)
23
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
24
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
25
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
26
Problem How can we develop mathematical models
that represent the dynamics in real networks?
27
A Boolean Switching Network
Xi is either 1 or 0
Bi is a Boolean function
Random boolean networks as gene models (Kauffman,
1969)
28
A differential equation
Glass, Kauffman, Pasternack, 1970s
29
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)

30
The Repressilator
31
The Hypercube Representation
32
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

33
Fixed Points

• A vertex that only has in arrows represents a
  stable fixed point. It is robust under changes in
  parameter values

34
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)

35
Evolving Rare Dynamics

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

36
The number of different networks in N dimensions
Glass, 1975 Edwards and Glass, 2000
37
An Evolvable Circuit
(J. Mason, J. Collins, P. Linsay, LG, Chaos, 2004)
38
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

39
The Hybrid Analog-Digital Circuit
40
Circuit Elements
41
Distribution of Cycle Lengths in Electronic
Circuit (300 random circuits with stable
oscillations)
Choose a target period of 80 ms
42
Sample Evolutionary Run
43
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

44
Prediction of Optimal Mutation Rate

• Compares favorably with experimentally determined
  value of 5-10

45
(No Transcript)
46
The Inverse Problem. Compute the number of
logical states needed to determine connectivity
diagram
Perkins, Hallett, Glass (2004)
47
Compute the number of switches needed to
determine the entire network
48
Gene expression in Drosophila Perkins, Jaeger,
Reinitz, GlassPLOS Computational Biology 2006
49
(No Transcript)
50
(No Transcript)
51
(No Transcript)
52
(Perkins, Jaeger, Reinitz, Glass, PLOS
Computational Biology 2006)
53
(Perkins, Jaeger, Reinitz, Glass, PLOS
Computational Biology 2006)
54
Proposed network for gene control
55
Comparison with different models
56
Some Important Ideas About Logical Network Models

• They do not require discrete time or states
• Logical networks can be embedded in differential
  equations (thats 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

57
Mathematical Models of Neural and Gene Networks
are Closely Related
58
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

59
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

60
Thanks

• Stuart Kauffman, Joel Pasternack, Rod Edwards,
  Jonathan Mason, Paul Linsay, James Collins, Ted
  Perkins, Yogi Jaeger, John Reintiz.
• NSERC, MITACS