Goal

1
Goal
Show the modeling process used by both Collins
(toggle switch) and Elowitz (repressilator) to
inform design of biological network necessary to
encode desired dynamical behavior (bi-stability
and oscillation, respectively).
2
Mathematical models predict qualitative behaviors
of biological systems.
Bi-stability in genetic toggle switch
1 Oscillation in genetic oscillator
2 Reversible flipping of an integrase-driven
bit 3 Counting cellular events 3
.
1 Collins, Cantor 2000 2 Elowitz 2000 3 Us
3
Mathematics can predict qualitative behaviors of
biological systems.
Bi-stability in genetic toggle switch
1 Oscillation in genetic oscillator
2 Reversible flipping of an integrase-driven
bit 3 Counting cellular events 3
.
1 Collins, Cantor 2000 2 Elowitz 2000 3 Us
4
Requirements
Bi-stable1 holds two states Inducible
switch between states
1 More than one attraction state two stable
equilibria in this case.
5
Design1
IPTG
LacI
Rep21
Thermal induction or atC
1 Two different designs. pTAK plasmids have
lacI repressor (IPTG inducible) and ptrc-2
promoter pair and PLs1con promoter with a
temperature-sensitive repressor (cIts). pIKE
plasmids have PLtetO-1 promoter in conjunction
with the Tet repressor (tetR). pTAK plasmids
switched by IPTG or thermal pulse. pIKE switched
by IPTG or atC.
6
State variables1
U repressor 1 V repressor 2
1 Repressor concentration are the continuous
dynamical state variables
7
Parameters
ODEs
Decay Repressor degradation /dilution
Repressor accumulation cooperative repression of
constitutively transcribed promoter

Beta 1 cooperativity in repression of
promoter
Alpha is the rate of protein synthesis
Parameters simplifications
U
RNA polymerase binding Open-complex
formation Transcript elongation Transcript
termination Repressor binding Ribosome
binding Polypeptide elongation
The dependence of transcription rate
Cooperativity Repression
Decay rates of protein, and messenger RNA
Cellular events complexity
1 The cooperativity arises from the
multimerization of the repressor proteins and the
cooperative binding of repressor multimers to
multiple operator sites in the promoter.
8
Possible outcomes
No steady state, mono stable1, bi-stable
1 One repressor always shuts down the other
9
Question
What parameter values yield bi-stability?
10
Coupled first-order ODEs
Accumulation cooperative repression of
constitutively transcribed promoters
Decay Degradation/dilution of the repressors
11
Find steady state1
0
0
1 Solution to both ODEs 0, for a given set of
parameter values
12
Solutions1
0
V
U0
0
V0
U
1 Across range of U, V given parameters
alpha1, by3
13
One equilibrium (intersection) point for the
system
0
V
U0
0
V0
U
14
Evaluate across the parameter space to find
bi-stability (gt 1 intersection)
Increasing cooperativity
by1
by2
by3
Alpha1
Increasing synthesis rate
Alpha2
Alpha3
V
U
15
Bi-stable1 when repressor expression rate and
cooperativity are high
by1
by2
by3
Alpha1
Alpha2
Alpha3
1 Multiple intersections arise from sigmoidal
shape, at b, y gt 1, and high rate of repressor
synthesis.
16
Similar1 to what Collins shows
U0
V
V0
U
1 Parameters alpha2, by3
17
Vector field shows system will move towards
steady state
Vector field
V
U
1 Parameters alpha2, by3
18
Approaches one steady state if initial condition
is high repressor 1
Initial condition high U
V
Repressor level
V0 (blue)
U2
U0
V0.3
Time
U
U
V
1 Parameters alpha2, by3
19
Alter dynamic balance with inducer, repressor 2
maximally expressed.
V
V0 (blue)
U0
U
U
V
1 Parameters alpha2, by3
20
New initial condition for the simulation settles
into new steady state.
Initial condition high V
V
Repressor level
V0 (blue)
V2
U0
U0.3
Time
U
U
V
1 Parameters alpha2, by3
21
Mono-stability1
Initial condition high V
Initial condition high U
1 Single steady-state for parameters alpha1,
by3
22
To achieve bi-stability
1. Balanced and high rate of repressor synthesis
2. High co-operativity of repression 3.
Induction to alters dynamic balance
23
Choose biological components (promoters / RBS /
repressors) that meet these requirements!
24
Mathematics can predict qualitative behaviors of
biological systems.
Bi-stability in genetic toggle switch
1 Oscillation in genetic oscillator
2 Reversible flipping of an integrase-driven
bit 3 Counting cellular events 3
.
1 Collins, Cantor 2000 2 Elowitz 2000 3 Us
25
Requirements
Oscillation
1 No settling into steady state
26
Design1
27
State variables1
3 mRNA 3 repressor proteins
1 Repressor and mRNA concentration are the
continuous dynamical state variables
28
Possible outcomes
Steady state, or oscillation
29
Question
What parameter values yield oscillation?
30
Six coupled first-order ODEs
Accumulation cooperative repression of
constitutively transcribed promoters
Decay Degradation/dilution of the repressors
Detailed discussion of parameters in appendix
31
Repressor logic embedded in equations
The appropriate protein represses the appropriate
mRNA synthesis and translation
32
Predict system behavior with respect to ODE
parameters
Linear algebra
Prediction of parameter values that yield steady
state and oscillation
33
Dynamic stability region with respect to
parameters
Unstable
Strength of repressors
Stable
1
Protein / mRNA degradation
34
Target similar protein and mRNA degradation,
minimal leakage (large drop in mRNA synthesis
when repressed)
Unstable
Strength of repressors
Stable
1
Protein / mRNA degradation
35
No leakage, high repressor expression
Parameters alpha50, alpha00, beta0.2, n2
36
Leakage causes steady state
Parameters alpha50, alpha01, beta0.2, n2
37
Low repressor expression causes steady state
Parameters alpha2, alpha00, beta0.2, n2
38
Process
Requirements Design Model state variables,
parameters Question the issue model needs to
resolve
Collins Steady state analysis Explore parameter
space Simulation
Elowitz Find stability region Set
parameters Simulation
Understand parameter settings that encode
desired dynamical behavior. Choose biological
parts that adhere to parameter settings.
39
Iterate
Requirements Design Model state variables,
parameters Question the issue model needs to
resolve
Collins Steady state analysis Explore parameter
space Simulation
Elowitz Find stability region Set
parameters Simulation
Understand parameter settings that encode
desired dynamical behavior. Choose biological
parts that adhere to parameter settings.
40
Appendix
41
Parameters mRNA model
ODEs continuous dynamical state
variables, (repressor concentration)
Decay Repressor degradation /dilution
mRNA accumulation cooperative repression of mRNA
synthesis 5

synthesis leaky 2 synthesis
cooperativity of repression of promoter
Parameters simplifications
U
RNA polymerase binding Open-complex
formation Transcript elongation Transcript
termination Repressor binding
The dependence of transcription rate
Cooperativity Repression
Decay rates of protein, and messenger RNA
Cellular events complexity
1 Number of protein copies per cell n the
presence of saturating amounts of repressor
(owing to the leakiness' of the promoter) 2
Here we consider only the symmetrical case in
which all three repressors are identical except
for their DNA-binding specificities. 3 Time is
rescaled in units of the mRNA lifetime 4 mRNA
concentrations are rescaled by their translation
efficiency, the average number of proteins
produced per mRNA molecule.
42
Parameters Repressor protein model
ODEs continuous dynamical state
variables, (repressor concentration)
Decay
Repressor accumulation cooperative repression of
proteins produced by mRNA 5

protein to mRNA ratio 1
protein to mRNA decay rate ratio 1
Parameters simplifications
Ribosome binding Polypeptide elongation
Degradation and dilution
Cellular events complexity
1 Time is rescaled in units of the mRNA
lifetime 2 Protein concentrations are written
in units of KM, the number of repressors
necessary to half-maximally repress a promoter