1
Modular Cell Biology:Modular Cell Biology: Retroactivity and Insulation Retroactivity and Insulation
Domitilla Del Vecchio
EECS, University of Michigan at Ann Arbor
MechE, MIT
Oxford, September 2009
2
Modularity: A fundamental propertyModularity: A fundamental property
Internal circuitry of an OPAMP: It is composed of well defined modules
The Emergent integrated circuit of the cell[Hanahan & Weinberg (2000)]
Modularity guarantees that the input/output behavior of a component (a module) does not change upon interconnection.Electronics and Control Systems Engineering rely on modularity to predict the behavior of a complex network by the behavior of the composing subsystems.
Result: Computers, Videos, cell phones…
Functional modules seem to recur also in biological networks (e.g. Alon (2007)). But…
But can they be interconnected and still maintain their behavior unchanged?
If not, what mechanism can be used to interconnect modules without altering their behavior?
Does nature already employ such mechanisms?
Z
X
Courtesy of Elowitz Lab at Caltech
(Elowitz and Leibler, Nature 2000)
Repressilator(Experimental Results)
WORKING “MODULES”NOT WORKING INTERCONNECTIONS !
Modularity: A grand challenge in synthetic Modularity: A grand challenge in synthetic biologybiology
Courtesy of Ninfa Lab at Umich
Experimental data
4
Modularity is Modularity is notnot a natural property of a natural property of bio-molecular circuitsbio-molecular circuits
How do we model these effects? How do we prevent them?
Retroactivity!
glnG
IPTGlacI
LacI-repNRI-act
glnKp
(Atkinson et al, Cell 2003)
LOAD
5
OutlineOutline
• A modeling framework for systems with retroactivity
• Retroactivity in transcriptional networks
• A lesson from OPAMPs: Insulation devices
• Design of a bio-molecular insulation device based on phosphorylation/dephosphorylation
• Fast time-scales as a key mechanism for insulation
6
A systems theory with retroactivityA systems theory with retroactivity
Basic Idea: u y
u y’
The interconnectionchanges the behavior
of the upstream system
FamiliarExamples:
7
A systems theory with retroactivityA systems theory with retroactivity
u y
srRetroactivity to the outputRetroactivity to the input
Def: The I/O model of the isolated system is obtained when s=0 and when r isnot an additional output
The interconnection of two systems is possible only when the internal state variablesets are disjoint:
y1
s1
u2
r2
u2=y1
s1=r2
8
OutlineOutline
• A modeling framework for systems with retroactivity
• Retroactivity in transcriptional networks
• A lesson from OPAMPs: Insulation devices
• Design of a bio-molecular insulation device based on phosphorylation/dephosphorylation
• Fast time-scales as a key mechanism for insulation
9
Gene regulatory circuitry: A network of Gene regulatory circuitry: A network of transcriptional modulestranscriptional modules
ZX
A transcriptionalcomponent is typically
viewed as an input/outputmodule
But, is its input/output response unchanged upon interconnection?
10
Retroactivity in transcriptional networks has Retroactivity in transcriptional networks has dramatic effects on the dynamics dramatic effects on the dynamics
(isolated)
s
(connected)
Downstreamcomponent
11
Isolated system (1D) Connected system (2D) s
We seek to quantify the difference in the dynamics of the state X betweenthe connected and isolated system
Measure of the retroactivity Measure of the retroactivity
To compare the X dynamics we seek a 1D approximation for the connected system:
Measure of retroactivity will be given by
12
We exploit the time-scale separation between the output X dynamics and the dynamics of the input stage of the downstream component
Calculation of s Calculation of s
13
Meaning and value of Meaning and value of
Isolated system dynamics:
Approximate connected system dynamics:
percentage differencebetween the isolated system
dynamics and the approximateconnected system dynamics
The value of the retroactivity measure for the interconnection through transcriptional regulation
Del Vecchio et al., Nature Molecular Systems Biology 2008
14
Effect of R(X) on the dynamicsEffect of R(X) on the dynamics
(isolated) (connected)
Downstreamcomponent
Retroactivity shifts the polesof the transfer function of the linearized system toward low frequency
15
OutlineOutline
• A modeling framework for systems with retroactivity
• Retroactivity in transcriptional networks
• A lesson from OPAMPs: Insulation devices
• Design of a bio-molecular insulation device based on phosphorylation/dephosphorylation
• Fast time-scales as a key mechanism for insulation
16
Dealing with retroactivity: Insulation devicesDealing with retroactivity: Insulation devices
In general, we cannot design the downstream system (the load) such that it has low retroactivity. But, we can design an insulation system to be placed between the upstream and downstream systems.
s
u y
r≈ 0
1. The retroactivity to the input is approx zero: r≈0
2. The retroactivity to the output s is attenuated
3. The output is proportional to the input: y=c u
Non-inverting amplifier:
17
Reaching small retroactivity to the input rReaching small retroactivity to the input r
because the input stage of an OPAMPabsorbs almost zero current
For example:
Choose the biochemical parameters of theinput stage to allow a small value of
18
Dealing with retroactivity: Insulation devicesDealing with retroactivity: Insulation devices
In general, we cannot design the downstream system (the load) such that it has low retroactivity. But, we can design an insulation system to be placed between the upstream and downstream systems.
s
u y
r≈ 0
1. The retroactivity to the input is approx zero: r≈0
2. The retroactivity to the output s is attenuated
3. The output is proportional to the input: y=c u
19
Attenuation of the retroactivity to the output Attenuation of the retroactivity to the output “s”: Large feedback and large amplification“s”: Large feedback and large amplification
For G large enough:
Non-inverting amplifier:
Conceptually:
20
Attenuation of the retroactivity to the output Attenuation of the retroactivity to the output “s” in the transcriptional component“s” in the transcriptional component
Apply large input amplification G and large output feedback G’
Connected system approximated dynamics Isolated system
How do we realize a large input amplification and a large negative feedback?
21
OutlineOutline
• A modeling framework for systems with retroactivity
• Retroactivity in transcriptional networks
• A lesson from OPAMPs: Insulation devices
• Design of a bio-molecular insulation device based on phosphorylation/dephosphorylation
• Fast time-scales as a key mechanism for insulation
22
A phosphorylation-based design for a bio-A phosphorylation-based design for a bio-molecular insulation devicemolecular insulation device
Amplification throughphosphorylation
Feedback throughdephosphorylation
Full ODE Model
Phospho/Dephospho Reactions:
rs
23
Simplified analysis: Why should it attenuate Simplified analysis: Why should it attenuate “s”?“s”?
s
Del Vecchio et al., Nature Molecular Systems Biology 2008
24
OutlineOutline
• A modeling framework for systems with retroactivity
• Retroactivity in transcriptional networks
• A lesson from OPAMPs: Insulation devices
• Design of a bio-molecular insulation device based on phosphorylation/dephosphorylation
• Fast time-scales as a key mechanism for insulation
25
Full system: The fast time-scale of the Full system: The fast time-scale of the device is a key feature for attenuating “s”device is a key feature for attenuating “s”
Phosphorylation and dephosphorylation reactions are often much faster than protein production and decay:
26
Fast time scales: A key mechanism for Fast time scales: A key mechanism for insulationinsulation
Basic Idea:
Large
Interconnectionthrough binding/unbinding(possibly large)
Claim: if G is large enough, signal x at the QSS is not affected by y.
27
Fast time scales: A key mechanism for Fast time scales: A key mechanism for insulationinsulation
Why would it work?
x(t) does not depend on y on the slowmanifold
Del Vecchio and Jayanthi, CDC 2008
28
Simulation results for the pho/depho Simulation results for the pho/depho insulation deviceinsulation device
Slow time-scale
Fast time-scale
Xp for the isolated systemXp for the connected system
The fast time-scale of thephosphorylation cycle allowsto reach insensitivity to very large loads (p=100)
29
ConclusionsConclusions
We have proposed a systems theory with retroactivity
We have provided a measure of retroactivity in transcriptionalnetworks
r=0We have introduced the notion of insulation device
We have presented a general (very well known in control systems engineering) mechanism to attenuate retroactivity to the output
Futile cycles, which are ubiquitous in natural signal transduction systems are excellent insulation device:they use time-scale separation as an insulation mechanism
30
Thanks to:Thanks to:
• Alexander J. Ninfa (University of Michigan Medical School, Prof. of Biological Chemistry)
• Eduardo D. Sontag (Rutgers University, Prof. of Mathematics)
• Sofia Merajver (University of Michigan Medical School, Cancer Center, Medical Doctor)
• Shridhar Jayanthi (EE: Systems, University of Michigan, graduate student)• Hamid Ossareh (EE: Systems, University of Michigan, graduate student)• Prasanna Varadarajan (ME, University of Michigan, graduate student) • Polina Mlynarzh (BME, University of Michigan, graduate student)
• Rackham Graduate School at University of Michigan/ CCMB/AFOSR
High gains improve signal-to-noise ratio…High gains improve signal-to-noise ratio…
31
Bio-molecular processes are intrinsically stochastic
How do high gains (required for retroactivity attenuation) impact noise?
Downstreamcomponent
calculated the Fokker-Planck Equationderiving from the Master Equation
calculated by linearizing the systemabout its equilibrium corresponding to
Courtesy of Elowitz lab (Caltech)
32
……but they also increase intrinsic noise at but they also increase intrinsic noise at higher frequencyhigher frequency
Downstreamcomponent
Use linearized Langevin approximation
Jayanthi and Del Vecchio, CDC 2009
33
The parts of the insulation device can be The parts of the insulation device can be designed so as to have small “r”designed so as to have small “r”
Retroactivity r after a fast transientis small if:
Approx linear input/output relationship:
Resulting input/output gain
34
Ongoing and Future WorkOngoing and Future Work
Experimental demonstration of genetic retroactivity in living cells(with Shridhar Jayanthi (EECS) and Alexander J Ninfa (Med School) at Umich)
lacZlac promoter
LacIrepressor
lacY lacA
IPTG
LacI binding sites(lacOp operators)
Periodic and step injection routines
35
Ongoing and Future WorkOngoing and Future Work
Construction of a phosphorylation-based insulation device (with Shridhar Jayanthi (EECS) and Alexander J Ninfa (Med School) at Umich)
NRI NRI~P
phosphatase(NRII H139N)
glnKpromoter
Amplifier/Insulator
Output
IPTG
Lac Repressor
kinase (NRII L16R)
RFP lacZ
NRI~P binding sites(enhancers)
lacpromoter
36
Ongoing and Future WorkOngoing and Future Work
Computation of a retroactivity measure in signaling pathways and of the“dampening” factor across stages(with Alejandra Ventura and Sofia Merajver in the Cancer Center at Umich)
DownstreamPerturbation
Upstream Effect
Does nature uses insulation devices by accident?Can we show that some natural systems would not work they way they workif the phosphorylation/dephosphorylation signaling cascades did not enjoyinsulation properties?
How general/descriptive is the system modeling with retroactivity?(with Eduardo D. Sontag at Rutgers)
37
Higher gains may contribute to higher Higher gains may contribute to higher biological noisebiological noise
Faster phosphorylation and dephosphorylation reactions lead to higher amplification and feedback gains (“higher
OPAMP amplification”), which lead to higher coefficient of variation.
38
Effect of R(X) on the dynamicsEffect of R(X) on the dynamics
(isolated) (connected)
Downstreamcomponent
Retroactivity shifts the polesof the transfer function of the linearized system toward low frequency