Design and Analysis of Gene Switches

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Design and Analysis of Gene Switches
Luonan Chen
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What is a gene network and its mathematical model?
What is a gene network ?
Why mathematical model ?

• A network which consists of mutual interactions
  between products of genes and other chemicals in
  a cell.
• Nonlinearity, delay and noise.

• Understanding dynamical aspects of inner cellular
  phenomena.
• Modeling some gene networks realistically as a
  huge amount of experimental data has been
  accumulated.

Ex. Functional differential equations ( FDE )
model
Eqn. (1)
The concentrations of chemicals
Rate of synthesis of chemicals
The degradation rates of chemicals
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Central Dogma of Molecular Biology
Poly A Site
Termination Site
Exon
DNA
Exon
Intron
Transcription
5Cap
RNA processing (Cleavage Polyadenylation)
Poly A
Primary transcript
Splicing
Functional mRNA
Translation
Degradation
Protein
Degradation
Function
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Modeling genetic regulatory networks

• Nodes are mRNAs, proteins
• Arrows represent regulatory interactions
• (These run through the proteins and metabolites!)

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An artificial gene network and a synthetic gene
switch
What is an artificial gene network and a gene
switch ?

• A state
• Switching

• A gene network which is designed artificially to
  have a certain function and installed into
  E.colis or yeasts

A mathematical interpretation
An artificial gene network which has
multistable states !

• A stable equilibrium point.
• Transition between two stable equilibrium points

• Applications
• Medicine ( gene therapy )
• Biotech
• Bio-computer

A state of the switch
( gene 1OFF, gene 2 ON)
Mathematical model can be used to design an
artificial gene network which has desirable
functions.
A state of the switch
( gene 1ON, gene 2 OFF)
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Toggle Switch two states
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The problems on designing of a gene switch
Some features of a gene network model

• There exist time delays
• e.g. transcription, translation, transportation
• The model gets higher dimensional as the number
  of the target genes increases.
• Noise e.g. uncertain facts or interactions

What are the problems ?
Failing to converge into equilibrium points
The problems with a gene switch

• It is hard task to analyze a high dimensional
  network with delays.
• It gets more difficult to ascertain that the
  network has only equilibrium points as
  attractors, for a high dimension system.

We can avoid this problems by constructing a gene
switch with only positive feedback loops
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A mathematical definition of types of an
interaction

• Types of an interaction
• Promoting the increase of synthesis of a
  chemical( the derivative of the synthesis is
  always non-negative. ) with time delay.
• ?Promoting the decrease of synthesis of a
  chemical ( the derivative of the synthesis is
  always non-positive. ) with time delay.

Types of a feedback loop The loop consists of
even number of negative interactions. ?The loop
consists of odd number of negative interactions
Sometime, delayed
Almost all reactions incorporated ever into a
gene network model satisfy this monotonisity.
Ex. Normal chemical reaction, Hill type reaction,
transcription, translation, transportation,
phosphorylation
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Our Model

• A network with only positive feedback loops

A negative feedfback loop
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Monotone Dynamical Systems

• Genetic network dx/dt f(x(t-tau)) D x

Translation, transcription, enzyme reaction,
transportation, phyophralation, dimerization,
chemical reaction
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Transformation

• Transform all edges of positive feedback loops
  into positive edges by coordinate P
• g P f P
• Genetic network dy/dt g(y(t-tau))-D y(t)

P
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The special features with only positive feedback
loops Part 1
1 The almost all trajectories of the network
fall into equilibria under several mild
conditions
The conditions

• It is automatically ascertained that the network
  has no attractor other than stable equilibrium
  points.
• This result is valid for a high dimensional
  network with delays.

• All feedback loops in a network are positive (
  Weaker condition is sufficient ).
• The Jacobian of Eqn (1) is always irreducible .
• f maps bounded subsets of C to bounded subsets
  of Rn.
• For each initial conditions fin C , the
  solution of Eqn. (1) is always defined for all t
  gt 0.
• The orbit of Eqn. (1) is bounded in C
• For each compact subet A of C There exists a
  bounded set B of C such that the omega limit
  set of Enq.(1) is included in B for every initial
  conditions in A.

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The special features with only positive feedback
loops Part 2
2 The equilibria of FDE correspond to those of
ODE one by one, and the stability is identical
with that of FDE.
The transient behavior of a switch is not
necessarily preserved.

• The gene switch can be analyzed and designed with
  ODE ( The asymptotic behavior of a switch is
  preserved ).

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The special features with only positive feedback
loops Part 3
3 It is possible to reduce the dimensions of
the gene switch model keeping the equilibria and
their stability.

• A switch can be reduced if a component has no
  auto regulatory direct feedback loop.

Reduction
Mathematically, this reduction is accomplished by
changing a ODE to an algebraic equation.
Simpler
minimal
Reduction
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Outline of Proof

• Coordinate Transformation by Interaction Graph
  each node has only positive interactions
• Proof of Bounded Flow
• Monotone Dynamical System strongly order
  preserved flow
• Stability of FDE ODE spectral radius
• Equivalence of Reduction local analysis

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Designing a complicated gene switch network.
Designing a switch by following the process
from 3 to 1
(2)Making the switch more realistic by adding
some components.
(1)Designing a minimal switch theoretically by
using ODE
Adding a component.
Adding a component.
(3)The time delays do not change the asymptotic
behavior of the switch unless designing it
abnormally.
Adding a component.
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Example--Four-state switch--
Theoretical prediction OK Three promoters
?
Extension of toggle switch
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Under experiment
Implementation
Tetramer
Dimer
Dimer
Operons avoid constructing logical gates PRM
binding site OR3 is mutated RBS 1 and 2 for
different RBS Toggle switch PLtet and Ptrc-2
with lacI and tetR
Artificially engineered
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Mutated PRM
Binding priority OR1 (0) gt OR2 () gt OR3 ( - )
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Necessary Conditions of 3 or 4 States Switch

• (1) Toggle switch has two stable equilibria
• (2) cI has low expression level

No problem for condition (1)
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Inefficiency of poly-cistronic transcription(tran
scriptional efficiency of the second gene at
downstream of promoter can be as low as 1/100 as
that of the first gene)
Problem Activation of cI is too strong !
cI is located at the second place in operons
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Parameter Design
FDE for operon model
Reduced to a two-dimensional ODE
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Equivalent Two-node Model
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Three-states Switch
Adjust RBS efficiency of each gene
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Implementation of 3-state gene switch
Mutual interactions are negative
Mutual interactions are positive
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Four-states Switch
Under experiment
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Conclusion

• Design genetic switch network with multiple genes
  and uncertain delays
• ---- complicated logical gates ----
• Develop a procedure of equivalent reduction
• Biological plausible example and numerical
  simulation
• General settings

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Future directions

• What type of switch can be constructed with only
  positive feedback loops.

• Designing of switching process.
• Designing switching process and signals based on
  stable and unstable manifolds.
• Design switching process and signals based on
  bifurcation of the network

This network has 4 ordered states
The switch can be flipped by appropriate
induction of change of the concentrations of the
chemicals
An equilibrium disappears due to change of a
parameter caused by switching signal and then the
switch falls into another equilibrium.
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References

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