DYNAMICS OF A BISTABLE FRUSTRATED UNIT

Hildegard Meyer-Ortmanns

Jacobs University Bremen

• The bistable frustrated unit in its deterministic description

• Frustration in excitable units, effect on the attractor landscape

• Stochastic description of a single bistable frustrated unit

Talk presented at the LAFNES seminar in Dresden, July 4-15, 2011

Work in collaboration with P. Kaluza, A. Garai, and B, WaclawP. Kaluza, and H MO: On the role of frustration in excitable systems; Chaos (2010)

20:043111, & Virtual Journal of Biological Physics Research -- November 15, 20 (10) (2010).A. Garai, B. Waclaw and HMO, Stochastic dynamics of a genetic circuit, in preparation

What is a bistable frustrated unit?

One bistable frustrated unit (S.Krishna, S. Semsey and M.H.Jensen, Phys.Biol.6 (2009))

A, B protein concentrationsγ ratio of half-life of A to that of BK strength of the repression (of B repressing A)α maximum rate of production of Aα b basal rate

• may serve as basic building block in larger systems• has its own rich phase structure• has an intrinsic time scale (fast and slow variable)• is “frustrated” on the basic level• is realized in natural systems whenever bistable units are coupled to negative

feedback loops

In the deterministic realization

• analyze the phase structure as function of one control parameterexcitable—oscillatory---excitable

• consider frustrated coupling of such units along simple geometries to study the effect of frustration on the attractor landscape

In the stochastic realization

• Search for qualitatively new effects: quasi-cycles or additional fixed points?

• How to disentangle limit cycles from quasi-cycles?

• Measure variances, autocorrelation functions and the power spectrum

Physics

Electrodynamics Field strength

Gauge theories

General relativity Curvature

Spin glassesOscillatory systems:phase oscillators, excitable systems

Frustration

Social systems Approach to balance Imbalance

Economics Financial markets Arbitrage

Excursion: Concept and impact of frustration

+

+-+

G. Mack, Commun.Math.Phys.219, 141 (2001).

in

in

anti

A

BC

A B

D C

Rough “energy” landscapes used for

• storing patterns, having many metastable states, exploiting the minima of such a landscape

• allowing for flexible dynamics

Conjecture:Tuning an appropriate degree of frustration:

• not too low so that the dynamics is flexible enough and therefore functionally stable• not too high, so that the dynamics is stable against noise

One of the driving forces in evolutionary processes

The impact of frustration

Frustration in systems of phase oscillators

Frustration in phase oscillators in larger systems to study synchronization behavior(H. Daido, PRL (1992); Progr. Theor. Phys. (1987); D. H. Zanette, EPL 72, 190 (2005); also seeE.Oh, C.Choi, B.Kahng and D.Kim, EPL 83, 68003 (2008).)

) Kuramoto dynamics drives the system in local minima of the frustration landscape, depending on the initial conditions

Quantitative measure

Qualitative measure:

+ in

+ in- antiw(i j) =+-1

Qualitative criterion for undirected couplings

Consider a loop with undirected interaction bonds and couplings that can be either

• attractive or repulsive• ferromagnetic or antiferromagnetic• excitatory or inhibitory

Consider a path from A to B along the shortest connection andalong the complementary path in the loop from B to A.

The bond from A to B is not frustrated if A acts upon B in the same way as B upon A (e.g. attractive), otherwise it is.

B

A

CA in phase with B, B with C ! C with A

Result of Daido: three Kuramoto oscillators coupled in a “frustrating way” lead to multistable behavior (Progr. Theor. Phys. 1987)

Frustration in excitable units , here the BFUa) One bistable frustrated unit (S.Krishna, S. Semsey and M.H.Jensen, Phys.Biol.6 (2009))

A, B protein concentrationsγ ratio of half-life of A to that of BK strength of the repression (of B repressing A)α maximum rate of production of Aα b basal rate

As function of α excitable, limit cycle, excitable behavior

zoom

In particular we see typical hysteresis effects for subcritical Hopf bifurcations at the transitions from excitable to limit cycle behavior and vice versa

b) How does a single bistable frustrated unit behave under noise?

internal noise external noise

For internal noise in B the oscillatory rangeis extended:

For external noise in α the oscillatory regime is almost the same.

In the oscillatory regime bistable frustrated units allow a rich variety of oscillatory behavior in frequency and amplitude, varying γ and α for given b and K

Typical choice of parameters: k=0.02, b=0.01, γ = 0.01

What happens if we add frustration on a second level, i.e. on the level of couplings of these units?

Consider a loop with directed interaction bonds and couplings that can be either

• repressing or activating• excitatory or inhibitory

Consider a path from A to B along the shortest connection andalong the complementary path in the loop from B to A. The bond from A to B is not frustrated if A acts upon B in the opposite way as B upon A (e.g. A to B activating, B to A via C and D repressing), otherwise it is frustrated

A

D

B

C

-

---

Different realizations of the frustration

• Via the number of couplings

• Via the type of couplings along with the number

C B

A

Criterion for frustration in case of directed couplings

c) Coupled bistable frustrated units

Adjacency matrix of repressing couplings Rij

Adjacency matrix of activating couplings Qij

Consider most simple motifs with and without frustration for which the frustration is implemented either:

• via the topology (even number of repressing couplings) or • via the type of coupling (replace one repressing by an activating one)

f: frustratedu: not frustrated

For the frustrated plaquette we obtain:

Frustration on two levels

Individual nodes in the oscillatory regime: α=80, βR =0.01

7 states depending on βR

3 patterns of phase-locked motion:

• 3 different phases out of four• 4 different phases• 2 different out of four

multistable behavior for βR =0.01, 0.1

Individual nodes in the excitatory regime α=110, βR =1.0

Results: • again multistable behavior: 1 fixed point solution, and 2 patterns of phase-

locked motion, depending on the choice of initial conditions

• no multistable behavior for the plaquette or triangle without frustration

Identification with concrete genetic units, relation to key regulators?Multistable behavior has been reported in repressilators coupled with cell-cell-communication according to the following motif: (Ullner et al., PRL 99 (2007))

A

B2D

B1

E

E

u

f

repressiveactivating

repressilator

Cell-cell-communication

u unfrustrated loopf frustrated loop

Is the deeper reason for the multistable behavior frustration ?

From Ullner et al. PRL99, 148103 (2007)

C

Summary so far (of the deterministic realization)

Qualitative criteria for frustration in systems without Hamiltonian can be defined

Phase space of attractors gets enriched as an effect of frustration• in phase oscillators • in genetic units: multistable behavior in small motifs

Study further the effect of defects and different realizations

) As alternative of assigning complex dynamics to individual nodes or links, design the combination of couplings with frustration

We expect robust functioning with respect to shortage on the individual level, due to frustration.

Identification with concrete genetic units or guiding principle for constructing synthetic units

Stochastic description of a bistable frustrated unit

Why at all?

• More realistic due to inherent stochasticity of various origin (finite number of species, biochemical reactions, decay and birth processes happen in a stochastic way)

• In general there may be such as• oscillations in space and time or additional fixed points

(pattern formation in ecological systems, Butler&Goldenfeld arXiv: 1011.0466, PRE (2009);for the brusselator see Boland, Galla& McKane J Stat Mech: Theory and Exp.(2008))

in contrast to limit cycles

If stochastic description goes, for example, along with a further zoom into the temporalresolution, there may be

as stable attractors (known from the toggle switch)(D.Schultz et al.PNAS(2008)H.Qian et al.PhysChemChemPhys (2009))

Here: quasi-cycles, later additional fixed points

Recall the deterministic set of equations:

2

002

00

1 1

( )

A

AA

BA

BA B

NbNNdN NNdt N

KN NdN N Ndt

α

γ

+ = − + +

= −

Corresponding reactions

A A+A

A φ

A B+A

B φ

f(NA,NB)

NA

γNA

γNB

Corresponding master equation

( , ) ( ( , ) ) ( , )

( 1) ( 1, ) ( 1, ) ( 1, )( 1) ( , 1) ( , 1)

A BA B A A B A B

A A B A B A B

B A B A A B

P N N f N N N N N P N Nt

N P N N f N N P N NN P N N N P N N

γ γ

γ γ

∂= − + + +

∂+ + + + − −+ + + + −

Gillespie simulations and histograms Numerical integration and van Kampenexpansion in N0-1/2

N0 parameterizes the system size,ranging from10 to 100000

Gillespie trajectories show a similar phase structure

excitable limit cycle excitable

N0=100

N0=1000

for k=0.02, b=0.01, γ=0.01

up to 3000 steps up to 100 000 steps up to 100 000 steps

up to 100 000 steps up to 100 000 stepsup to 100 000 steps

Quasi-cycles show up in Gillespie trajectories as “rare events” deeply in the former fixed point region

fixed point regionafter 4x105 time steps

close to the transition after 106 time steps

after 10000 time steps

for k=0.02, b=0.01, γ=0.01, N0=100

N0 # quasi-cycles50 3839100 15901000 2

α=15, k=0.02, γ=0.01, b=0.01, steps:107

γ # quasi-cycles0.001 1310.01 15890.5 7499

α=15, k=0.02, N0=100, b=0.01, steps:107

Number of quasi-cycles within 106 steps

On numbers and origin of quasi-cycles

(Large) fluctuations help to overcome the threshold of the excitable unit.

Comparison between histogram of the Gillespie simulations and the solution P(NA,NB) of the master equation

k=0.02, b=0.01, γ=0.01, α =50 Snapshot for the histogram at T=100

Van Kampen expansion in N0-1/2

To order N01/2

deterministic equations

Moment equations

To order N00:Fokker-Planck equation

√<< NA >>/ <NA >, √<< NB >>/ < NB > and << NANB >> / < NANB >

where << NA >> = < NA2> − < NA >2, << NB >> = < NB2> − < NB >2

and << NANB >> = < NA NB > − < NA >< NB >.

numerical integration of van Kampen expansionstochastic simulation via Gillespie algorithm

Calculate:• the variances in the stationary state, the autocorrelation functions and • the power spectrum both in the fixed point and the limit cycle regime

Variance in the limit cycle region in NB for different system sizestochastic (Gillespie), analytic (van Kampen)

T extends over 1.5 limit cycles, average over 100 Gillespie simulations

Autocorrelation functions in the stationary state

with and

Decay of the autocorrelation function for various values of α

Decay of autocorrelations faster for quasi-cycles than for limit cycles

Calculation of the power spectrum starting from the Langevin equation

Comparison of the power spectrum of fluctuations in NA from Gillespie (stochastic) and the van Kampen expansion (analytic)

fixed point region

transition region

Precursors of the transition

Enhanced coherence in time and stronger fluctuations close to the transition

γ=0.5

Summary of the stochastic part:

• We see quasi-cycles, the more the larger the fluctuations, the smaller N0 and larger γ

•Transition between phases with different stationary states can be localized via autocorrelations and power spectrum

•Histogram from Gillespie simulations can be checked via numerical integration of the master equations for P(NA,NB,t)

•Gillespie versus van Kampen expansion: comparison works well for both the fixed point and the limit cycle region as long as the generated fluctuations are small, that is for short times and outside the transition region

•Precursors of the transition are seen for larger γ as increased coherence in time and increased amplitudes of fluctuations

Conclusions

Bistable frustrated unit is a challenging motif with an intrinsic time scale

• it behaves as excitable or oscillatory unit with a variety of amplitudes and frequencies• focus on frustration on the level of couplings:

frustrated coupling leads to multistabilityfrustration may explain how functional stability and flexibility can go along

• (Large) fluctuations induce quasi-cycles even deeply in the former fixed point region• Quasi-cycles can be disentangled from limit cycles via the autocorrelation function• Quasi-cycles make the BFU even more flexible so that no fine-tuning is needed for

having oscillations, but are both oscillations really equivalent?

Challenge: Identify quasi-cycles in natural oscillatory genetic systems

What do they serve for? What is the “normal” mode of performance?

THANKS to my collaborators:

• Pablo Kaluza (now in Berlin) for the deterministic part

• Ashok Garai (Jacobs University Bremen) for the stochastic part

• Bartlomiej Waclaw (Edinburgh) “ “ “ “

Outlook

Zoom into the genetic level