1 Modular Cell Biology: Retroactivity and Insulation Domitilla Del Vecchio EECS, University of Michigan at Ann Arbor MechE, MIT Oxford, September 2009

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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

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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

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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

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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

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A systems theory with retroactivityA systems theory with retroactivity

Basic Idea: u y

u y’

The interconnectionchanges the behavior

of the upstream system

FamiliarExamples:

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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

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OutlineOutline






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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?

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Retroactivity in transcriptional networks has Retroactivity in transcriptional networks has dramatic effects on the dynamics dramatic effects on the dynamics

(isolated)

s

(connected)

Downstreamcomponent

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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

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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

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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

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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

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OutlineOutline






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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:

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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

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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

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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:

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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?

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OutlineOutline






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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

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Simplified analysis: Why should it attenuate Simplified analysis: Why should it attenuate “s”?“s”?

s

Del Vecchio et al., Nature Molecular Systems Biology 2008

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OutlineOutline






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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:

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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.

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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

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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)

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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

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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…

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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)

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……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

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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

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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

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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

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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)

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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.

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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

Documents

1 Modular Cell Biology: Retroactivity and Insulation Domitilla Del Vecchio EECS, University of Michigan at Ann Arbor MechE, MIT Oxford, September 2009