Milos Dolnik

Designer gene networks: Towards fundamental cellular control

The engineered control of cellular function through the design of synthetic genetic networks is becoming plausible. Here we show how a naturally occurring network can be used as a parts list for artificial network design, and how model formulation leads to computational and analytical approaches relevant to nonlinear dynamics and statistical physics. We first review the relevant work on synthetic gene networks, highlighting the important experimental findings with regard to genetic switches and oscillators. We then present the derivation of a deterministic model describing the temporal evolution of the concentration of protein in a single-gene network. Bistability in the steady-state protein concentration arises naturally as a consequence of autoregulatory feedback, and we focus on the hysteretic properties of the protein concentration as a function of the degradation rate. We then formulate the effect of an external noise source which interacts with the protein degradation rate. We demonstrate the utility of such a formulation by constructing a protein switch, whereby external noise pulses are used to switch the protein concentration between two values. Following the lead of earlier work, we show how the addition of a second network component can be used to construct a relaxation oscillator, whereby the system is driven around the hysteresis loop. We highlight the frequency dependence on the tunable parameter values, and discuss design plausibility. We emphasize how the model equations can be used to develop design criteria for robust oscillations, and illustrate this point with parameter plots illuminating the oscillatory regions for given parameter values. We then turn to the utilization of an intrinsic cellular process as a means of controlling the oscillations. We consider a network design which exhibits self-sustained oscillations, and discuss the driving of the oscillator in the context of synchronization. Then, as a second design, we consider a synthetic network with parameter values near, but outside, the oscillatory boundary. In this case, we show how resonance can lead to the induction of oscillations and amplification of a cellular signal. Finally, we construct a toggle switch from positive regulatory elements, and compare the switching properties for this network with those of a network constructed using negative regulation. Our results demonstrate the utility of model analysis in the construction of synthetic gene regulatory networks. (c) 2001 American Institute of Physics.

Pattern formation arising from interactions between Turing and wave instabilities

We study pattern formation arising from the interaction of the stationary Turing and wave (oscillatory Turing) instabilities. Interaction and competition between these symmetry-breaking modes lead to the emergence of a large variety of spatiotemporal patterns, including modulated Turing structures, modulated standing waves, and combinations of Turing structures and spiral waves. Spatial resonances are obtained near codimension-two Turing-wave bifurcations. Far from bifurcation lines, we obtain inwardly propagating spiral waves with Turing spots at their tips. We demonstrate that the coexistence of Turing spots and traveling waves is a result of interaction between Turing and oscillatory modes, while the inwardly propagating waves (antispirals) do not require this interaction; they can arise from the wave instability combined with a negative group velocity.

Pattern formation arising from wave instability in a simple reaction‐diffusion system

Pattern formation is studied numerically in a three‐variable reaction‐diffusion model with onset of the oscillatory instability at a finite wavelength. Traveling and standing waves, asymmetric standing‐traveling wave patterns, and target patterns are found. With increasing overcriticality or system length, basins of attraction of more symmetric patterns shrink, while less symmetric patterns become stable. Interaction of a defect with an impermeable boundary results in displacement of the defect. Fusion and splitting of defects are observed.

Turing patterns beyond hexagons and stripes.

The best known Turing patterns are composed of stripes or simple hexagonal arrangements of spots. Until recently, Turing patterns with other geometries have been observed only rarely. Here we present experimental studies and mathematical modeling of the formation and stability of hexagonal and square Turing superlattice patterns in a photosensitive reaction-diffusion system. The superlattices develop from initial conditions created by illuminating the system through a mask consisting of a simple hexagonal or square lattice with a wavelength close to a multiple of the intrinsic Turing patterns wavelength. We show that interaction of the photochemical periodic forcing with the Turing instability generates multiple spatial harmonics of the forcing patterns. The harmonics situated within the Turing instability band survive after the illumination is switched off and form superlattices. The square superlattices are the first examples of time-independent square Turing patterns. We also demonstrate that in a system where the Turing band is slightly below criticality, spatially uniform internal or external oscillations can create oscillating square patterns.

Experimental observations of periodic and chaotic regimes in a forced chemical oscillator

Abstract Experiments showing periodic and chaotic behavior in a Belousov-Zhabotinski (BZ) reaction mixture in a stirred flow reactor with periodic addition of bromide ions are presented. An alternating sequence of periodic and aperiodic regimes as a function of the period of Br - pulses is observed in experiments. This behavior is well described by a simple model based on the results of single-pulse experiments.

Spatio-temporal patterns in a reaction–diffusion system with wave instability

An intraocular lens implantation forceps, including first and second arms having handles and lens engagement blades; the blades extending generally longitudinally in laterally spaced relation and at opposite sides of a plane bisecting the forceps. The arms have primary and secondary arm sections extending between the handles and blades. The arms primary sections extend in cross-over relation to define a cross-over locus, when the blades are in open position; the secondary arm sections extending generally longitudinally in substantially parallel relation and positioned such that when the blades are in the closed positions one secondary arm section extends at one side of the plane and the other secondary arm section extends at the other side of the plane, and when the blades are in the open positions the one secondary arm section extends at the other side of the plane and the other secondary arm section extends at the one side of the plane.

Dynamic regimes in a periodically forced reaction cell with oscillatory chemical reaction

Abstract Periodic and aperiodic regimes in a forced chemical system are studied experimentally and the observations are interpreted on the basis of phase transition curves evaluated both from the model equations and experimentally. The periodically oscillating system of the Belousov-Zhabotinski reaction in a flow-through stirred reaction cell exhibited phase transition curves both of the type 1 and 0 when a single pulse perturbation by bromide ions was used. This behaviour is only partly described by the mathematical models studied. Phase synchronization, intermittency and chaos were observed when the frequency and amplitude of concentration perturbations were varied in continuous forcing experiments. A one-dimensional deterministic model based on the experimental phase curves describes results of continuous forcing relatively well; better agreement was reached when effects of experimental noise were included in the model.

A coupled chemical burster: The chlorine dioxide-iodide reaction in two flow reactors

The dynamical behavior of the chlorine dioxide–iodide reaction has been studied in a system consisting of two continuous flow stirred tank reactors (CSTRs). The reactors are coupled by computer monitoring of the electrochemical potential in each reactor, which is then used to control the input into the other reactor. Two forms of coupling are employed: reciprocally triggered, exponentially decreasing stimulation, and alternating mass exchange. The reaction, which exhibits oscillatory and excitable behavior in a single CSTR, displays neuronlike bursting behavior with both forms of coupling. Reciprocal stimulation yields bursting in both reactors, while with alternating mass exchange, bursting is observed in one reactor and complex oscillation in the other. A simple model of the reaction gives good agreement between the experimental observations and numerical simulations.

Resonance behaviour in two-parameter families of periodically forced oscillators

Abstract We study resonance (periodic) behaviour in two-dimensional autonomous oscillators, periodically forced by discrete jumps in state space. Two different models are examined numerically using continuation methods. The results are qualitatively similar in both cases and show that regions of various resonances are bounded in the forcing-amplitude-forcing-period parameter plane and have a rich internal bifurcation structure.

Communication with chemical chaos in the presence of noise.

We use control of chaos to encode information into the oscillations of the Belousov-Zhabotinsky reaction. An arbitrary binary message is encoded by forcing the chaotic oscillations to follow a specified trajectory. The information manipulating control requires only small perturbations to vary the binary message. In this paper we extend our recent theoretical work [Bollt and Dolnik, Phys. Rev. E 64, 1196 (1990)] by introducing a new and simplified encoding technique which can be utilized in the presence of experimental noise. We numerically and theoretically study several practical aspects of controlling symbol dynamics including: modeling noisy time-series, learning underlying symbol dynamics, and evaluation of derivatives for control by observing system responses to an intelligent and deliberate sequence of input parameter variations. All of the modeling techniques incorporated here are ultimately designed to learn and control symbol dynamics of experimental data known only as an observed time-series; the simulation assumes no global model. We find that noise affects reliability of encoding information and may cause coding errors. But, if the level of noise is confined to relatively small values, which are achievable in experiments, the control mechanism is robust to the noise. Thus we can still produce a desired symbolic code. However, scarce errors in encoding may occur due to rare but large fluctuations. These errors may be corrected during the decoding process by a variation of the filtering technique suggested by Rosa et al. [Phys. Rev. Lett. 78, 1247 (1997)]. (c) 1998 American Institute of Physics.