Douglas A. Baxter

Mathematical modeling of gene networks

We thank A. Angers, J. Chin, L. Cleary, S. Candy, E. Robson, and K. Scholz for their comments on the manuscript. This work was supported by National Institutes of Health grants T32 NS07373, R01 RR11626, and P01 NS38310.

A Reduced Model Clarifies the Role of Feedback Loops and Time Delays in the Drosophila Circadian Oscillator

Although several detailed models of molecular processes essential for circadian oscillations have been developed, their complexity makes intuitive understanding of the oscillation mechanism difficult. The goal of the present study was to reduce a previously developed, detailed model to a minimal representation of the transcriptional regulation essential for circadian rhythmicity in Drosophila. The reduced model contains only two differential equations, each with time delays. A negative feedback loop is included, in which PER protein represses per transcription by binding the dCLOCK transcription factor. A positive feedback loop is also included, in which dCLOCK indirectly enhances its own formation. The model simulated circadian oscillations, light entrainment, and a phase-response curve with qualitative similarities to experiment. Time delays were found to be essential for simulation of circadian oscillations with this model. To examine the robustness of the simplified model to fluctuations in molecule numbers, a stochastic variant was constructed. Robust circadian oscillations and entrainment to light pulses were simulated with fewer than 80 molecules of each gene product present on average. Circadian oscillations persisted when the positive feedback loop was removed. Moreover, elimination of positive feedback did not decrease the robustness of oscillations to stochastic fluctuations or to variations in parameter values. Such reduced models can aid understanding of the oscillation mechanisms in Drosophila and in other organisms in which feedback regulation of transcription may play an important role.

Frequency selectivity, multistability, and oscillations emerge from models of genetic regulatory systems

To examine the capability of genetic regulatory systems for complex dynamic activity, we developed simple kinetic models that incorporate known features of these systems. These include autoregulation and stimulus-dependent phosphorylation of transcription factors (TFs), dimerization of TFs, crosstalk, and feedback. The simplest model manifested multiple stable steady states, and brief perturbations could switch the model between these states. Such transitions might explain, for example, how a brief pulse of hormone or neurotransmitter could elicit a long-lasting cellular response. In slightly more complex models, oscillatory regimes were identified. The addition of competition between activating and repressing TFs provided a plausible explanation for optimal stimulus frequencies that give maximal transcription. Such optimal frequencies are suggested by recent experiments comparing training paradigms for long-term memory formation and examining changes in mRNA levels in repetitively stimulated cultured cells. In general, the computational approach illustrated here, combined with appropriate experiments, provides a conceptual framework for investigating the function of genetic regulatory systems.To examine the capability of genetic regulatory systems for complex dynamic activity, we developed simple kinetic models that incorporate known features of these systems. These include autoregulation and stimulus-dependent phosphorylation of transcription factors (TFs), dimerization of TFs, crosstalk, and feedback. The simplest model manifested multiple stable steady states, and brief perturbations could switch the model between these states. Such transitions might explain, for example, how a brief pulse of hormone or neurotransmitter could elicit a long-lasting cellular response. In slightly more complex models, oscillatory regimes were identified. The addition of competition between activating and repressing TFs provided a plausible explanation for optimal stimulus frequencies that give maximal transcription. Such optimal frequencies are suggested by recent experiments comparing training paradigms for long-term memory formation and examining changes in mRNA levels in repetitively stimulated cultured cells. In general, the computational approach illustrated here, combined with appropriate experiments, provides a conceptual framework for investigating the function of genetic regulatory systems.

Phase response characteristics of model neurons determine which patterns are expressed in a ring circuit model of gait generation

Abstract. In order to assess the relative contributions to pattern-generation of the intrinsic properties of individual neurons and of their connectivity, we examined a ring circuit composed of four complex physiologically based oscillators. This circuit produced patterns that correspond to several quadrupedal gaits, including the walk, the bound, and the gallop. An analysis using the phase response curve (PRC) of an uncoupled oscillator accurately predicted all modes exhibited by this circuit and their phasic relationships – with the caveat that in certain parameter ranges, bistability in the individual oscillators added nongait patterns that were not amenable to PRC analysis, but further enriched the pattern-generating repertoire of the circuit. The key insights in the PRC analysis were that in a gait pattern, since all oscillators are entrained at the same frequency, the phase advance or delay caused by the action of each oscillator on its postsynaptic oscillator must be the same, and the sum of the normalized phase differences around the ring must equal to an integer. As suggested by several previous studies, our analysis showed that the capacity to exhibit a large number of patterns is inherent in the ring circuit configuration. In addition, our analysis revealed that the shape of the PRC for the individual oscillators determines which of the theoretically possible modes can be generated using these oscillators as circuit elements. PRCs that have a complex shape enable a circuit to produce a wider variety of patterns, and since complex neurons tend to have complex PRCs, enriching the repertoire of patterns exhibited by a circuit may be the function of some intrinsic neuronal complexity. Our analysis showed that gait transitions, or more generally, pattern transitions, in a ring circuit do not require rewiring the circuit or any changes in the strength of the connections. Instead, transitions can be achieved by using a control parameter, such as stimulus intensity, to sculpt the PRC so that it has the appropriate shape for the desired pattern(s). A transition can then be achieved simply by changing the value of the control parameter so that the first pattern either ceases to exist or loses stability, while a second pattern either comes into existence or gains stability. Our analysis illustrates the predictive value of PRCs in circuit analysis and can be extended to provide a design method for pattern-generating circuits.

In Vitro Analog of Operant Conditioning in Aplysia. II. Modifications of the Functional Dynamics of an Identified Neuron Contribute to Motor Pattern Selection

Previously, an analog of operant conditioning was developed using the buccal ganglia of Aplysia, the probabilistic occurrences of a specific motor pattern (i.e., pattern I), a contingent reinforcement (i.e., stimulation of the esophageal nerve), and monotonic stimulation of a peripheral nerve (i.e., n.2,3). This analog expressed a key feature of operant conditioning (i.e., selective enhancement of the probability of occurrence of a designated motor pattern by contingent reinforcement). In addition, the training induced changes in the dynamical properties of neuron B51, an element of the buccal central pattern generator. To gain insights into the neuronal mechanisms that mediate features of operant conditioning, the present study identified a neuronal element that was critically involved in the selective enhancement of pattern I. We found that bursting activity in cell B51 contributed significantly to the expression of pattern I and that changes in the dynamical properties of this cell were associated with the selective enhancement of pattern I. These changes could be induced by an explicit association of reinforcement with random depolarization of B51. No stimulation of n.2,3 was required. These results indicate that the selection of a designated motor pattern by contingent reinforcement and the underlying neuronal plasticity resulted from the association of reinforcement with a component of central neuronal activity that contributes to a specific motor pattern. The sensory stimulus that allows for occurrences of different motor acts may not be critical for induction of plasticity that mediates the selection of a motor output by contingent reinforcement in operant conditioning.

Simulation of Drosophila Circadian Oscillations, Mutations, and Light Responses by a Model with VRI, PDP-1, and CLK

A model of Drosophila circadian rhythm generation was developed to represent feedback loops based on transcriptional regulation of per, Clk (dclock), Pdp-1, and vri (vrille). The model postulates that histone acetylation kinetics make transcriptional activation a nonlinear function of [CLK]. Such a nonlinearity is essential to simulate robust circadian oscillations of transcription in our model and in previous models. Simulations suggest that two positive feedback loops involving Clk are not essential for oscillations, because oscillations of [PER] were preserved when Clk, vri, or Pdp-1 expression was fixed. However, eliminating positive feedback by fixing vri expression altered the oscillation period. Eliminating the negative feedback loop in which PER represses per expression abolished oscillations. Simulations of per or Clk null mutations, of per overexpression, and of vri, Clk, or Pdp-1 heterozygous null mutations altered model behavior in ways similar to experimental data. The model simulated a photic phase-response curve resembling experimental curves, and oscillations entrained to simulated light-dark cycles. Temperature compensation of oscillation period could be simulated if temperature elevation slowed PER nuclear entry or PER phosphorylation. The model makes experimental predictions, some of which could be tested in transgenic Drosophila.

Neural and Molecular Bases of Nonassociative and Associative Learning in Aplysia

A model that summarizes some of the neural and molecular mechanisms contributing to short- and long-term sensitization is shown in Figure 14. Sensitizing stimuli lead to the release of a modulatory transmitter such as 5-HT. Both serotonin and sensitizing stimuli lead to an increase in the synthesis of cAMP and the modulation of a number of K+ currents through protein phosphorylation. Closure of these K+ channels leads to membrane depolarization and the enhancement of excitability. An additional consequence of the modulation of the K+ currents is a reduction of current during the repolarization of the action potential, which leads to an increase in its duration. As a result, Ca2+ flows into the cell for a correspondingly longer period of time, and additional transmitter is released from the cell. Modulation of the pool of transmitter available for release (mobilization) also appears to occur as a result of sensitizing stimuli. Recent evidence indicates that the mobilization process can be activated by both cAMP-dependent protein kinase and protein kinase C. Thus, release of transmitter is enhanced not only because of the greater influx of Ca2+ but also because more transmitter is made available for release by mobilization. The enhanced release of transmitter leads to enhanced activation of motor neurons and an enhanced behavioral response. Just as the regulation of membrane currents is used as a read out of the memory for short-term sensitization, it also is used as a read out of the memory for long-term sensitization. But long-term sensitization differs from short-term sensitization in that morphological changes are associated with it, and long-term sensitization requires new protein synthesis. The mechanisms that induce and maintain the long-term changes are not yet fully understood (see the dashed lines in Fig. 14) although they are likely to be due to direct interactions with the translation apparatus and perhaps also to events occurring in the cell nucleus. Nevertheless, it appears that the same intracellular messenger, cAMP, that contributes to the expression of the short-term changes, also triggers cellular processes that lead to the long-term changes. One possible mechanism for the action of cAMP is through its regulation of the synthesis of membrane modulatory proteins or key effector proteins (for example, membrane channels). It is also possible that long-term changes in membrane currents could be due in part to enhanced activity of the cAMP-dependent protein kinase so that there is a persistent phosphorylation of target proteins.(ABSTRACT TRUNCATED AT 400 WORDS)

Classical Conditioning of Feeding in Aplysia: I. Behavioral Analysis

A training protocol was developed to classically condition feeding behavior in Aplysia californica using tactile stimulation of the lips as the conditional stimulus (CS) and food as the unconditional stimulus (US). Paired training induced a greater increase in the number of bites to the CS than unpaired training or US-only stimulation. Memory for classical conditioning was retained for at least 24 hr. The organization of the reinforcement pathway that supports classical conditioning was analyzed in additional behavioral experiments. No evidence was found for the contribution to appetitive reinforcement of US-mediating pathways originating in the lips of the animals. Bilateral lesions of the anterior branch of the esophageal nerve, which innervates parts of the foregut, however, were found to attenuate classical conditioning. Thus, it appears likely that reinforcement during appetitive classical conditioning of feeding was mediated by afferent pathways that originate in the foregut. The companion paper (Lechner et al., 2000) describes two neurophysiological correlates of the classical conditioning.

Classical and operant conditioning differentially modify the intrinsic properties of an identified neuron

A long-standing debate in neuroscience is whether classical and operant conditioning are mechanistically similar or distinct. The feeding behavior of Aplysia provides a model system suitable for addressing this question. Here we report that classical and operant conditioning of feeding behavior differentially modify the intrinsic excitability of neuron B51, a critical element for the expression of the feeding response, thus revealing that these two forms of associative learning differ at the cellular level.

Control of multistability in ring circuits of oscillators.

Abstract. The essential dynamics of some biological central pattern generators (CPGs) can be captured by a model consisting of N neurons connected in a ring. These circuits, like many oscillatory nonlinear circuits of sufficient complexity, are capable of multistability, that is, of generating different firing patterns distinguished by the phasic relationships between the firing in each circuit element (neuron). Moreover, a shift in firing pattern can be induced by a transient perturbation. A systematic approach, based on phase-response curve (PRC) theory, was used to determine the optimum timing for perturbations that induce a shift in the firing pattern. The first step was to visualize the solution space of the ring circuit, including the attractive basins for each stable firing pattern; this was possible using the relative phase of N−1 oscillators, with respect to an arbitrarily selected reference oscillator, as coordinate axes. The trajectories in this phase space were determined using an iterative mapping based only on the PRCs of the uncoupled component oscillators; this algorithm was called a circuit emulator. For an accurate mapping of the attractive basin of each pattern exhibited by the ring circuit, the emulator had to take into account the effect of a perturbation or input on the timing of two bursts following the onset of the perturbation, rather than just one. The visualization of the attractive basins for rings of two, three, and four oscillators enabled the accurate prediction of the amounts of phase resetting applied to up to N−1 oscillators within a cycle that would induce a transition from any pattern to any another pattern. Finally, the timing and synaptic characterization of an input called the switch signal was adjusted to produce the desired amount of phase resetting.