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 Work in collaboration with P. Kaluza, A. Garai, and B, Waclaw P. 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 Talk presented at the LAFNES seminar in Dresden, July 4-15, 2011
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 concentrations γ ratio of half-life of A to that of B K strength of the repression (of B repressing A) α maximum rate of production of A α 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 parameter excitable—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
Excursion: Concept and impact of frustration Electrodynamics Field strength Gauge theories + General relativity Curvature Physics + - Spin glasses Frustration Oscillatory systems: + phase oscillators, excitable systems A Approach to balance Imbalance Social systems anti Economics Financial markets Arbitrage in B C in A B G. Mack, Commun.Math.Phys.219, 141 (2001). D C
The impact of frustration 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
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 see E.Oh, C.Choi, B.Kahng and D.Kim, EPL 83, 68003 (2008).) Quantitative measure Qualitative measure: - anti + in w(i j) =+-1 + in ) ) Kuramoto dynamics drives the system in local minima of the frustration landscape, depending on the initial conditions
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 and along 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. A A in phase with B, B with C ! C with A C B 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 BFU a) One bistable frustrated unit (S.Krishna, S. Semsey and M.H.Jensen, Phys.Biol.6 (2009)) A, B protein concentrations γ ratio of half-life of A to that of B K strength of the repression (of B repressing A) α maximum rate of production of A α 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? external noise internal noise
For internal noise in B the oscillatory range is 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?
Criterion for frustration in case of directed couplings 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 and along 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 B Different realizations of the frustration - - D C - • Via the number of couplings A • Via the type of couplings along with the number C B
c) Coupled bistable frustrated units Adjacency matrix of repressing couplings R ij Adjacency matrix of activating couplings Q ij 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)
Frustration on two levels f: frustrated u: not frustrated For the frustrated plaquette we obtain:
Individual nodes in the oscillatory regime : α =80, β R =0.01 7 states depending on β R 3 patterns of phase-locked motion: • 3 different phases out of four • 4 different phases • 2 different out of four multistable behavior for β R =0.01, 0.1
Individual nodes in the excitatory regime α= 110, β R =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 B1 u repressilator C Cell-cell- f communication E B2 D From Ullner et al. PRL99, 148103 (2007) E repressive Is the deeper reason for the multistable behavior frustration ? activating u unfrustrated loop f frustrated loop
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 temporal resolution, 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 N + A b α N dN N = − 0 A 0 N N0 parameterizes the system size, A 2 N dt + N B 1 + ranging from10 to 100000 A 1 KN N 0 0 dN = γ − B N N ( ) A B dt Corresponding reactions Corresponding master equation f(NA,NB) ∂ A A+A P N ( , N ) = − + + γ + γ A B ( ( f N , N ) N N N ) ( P N , N ) ∂ NA A B A A B A B t A φ + + + + − − ( N 1) ( P N 1, N ) f N ( 1, N ) ( P N 1, N ) γ NA A A B A B A B A B+A + γ + + + γ − N P N N N P N N ( 1) ( , 1) ( , 1) B A B A A B γNB B φ Numerical integration and van Kampen expansion in N0 -1/2 Gillespie simulations and histograms
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