NONLINEAR TIME SERIES/SYSTEM ANALYSIS (STATE-SPACE, DYNAMICS, INTERVENTION, RESILIENCE, KALMAN)

A discussion of nonlinear dynamics, demonstrated by the familiar automobile, is followed by the development of a systematic method of analysis of a possibly nonlinear time series using difference equations in the general state-space format. This format allows recursive state-dependent parameter estimation after each observation thereby revealing the dynamics inherent in the system in combination with random external perturbations.^ The one-step ahead prediction errors at each time period, transformed to have constant variance, and the estimated parametric sequences provide the information to (1) formally test whether time series observations y(,t) are some linear function of random errors (ELEM)(,s), for some t and s, or whether the series would more appropriately be described by a nonlinear model such as bilinear, exponential, threshold, etc., (2) formally test whether a statistically significant change has occurred in structure/level either historically or as it occurs, (3) forecast nonlinear system with a new and innovative (but very old numerical) technique utilizing rational functions to extrapolate individual parameters as smooth functions of time which are then combined to obtain the forecast of y and (4) suggest a measure of resilience, i.e. how much perturbation a structure/level can tolerate, whether internal or external to the system, and remain statistically unchanged. Although similar to one-step control, this provides a less rigid way to think about changes affecting social systems.^ Applications consisting of the analysis of some familiar and some simulated series demonstrate the procedure. Empirical results suggest that this state-space or modified augmented Kalman filter may provide interesting ways to identify particular kinds of nonlinearities as they occur in structural change via the state trajectory.^ A computational flow-chart detailing computations and software input and output is provided in the body of the text. IBM Advanced BASIC program listings to accomplish most of the analysis are provided in the appendix. ^

Dynamics of a minimal model of interlocked positive and negative feedback loops of transcriptional regulation by cAMP-response element binding proteins.

cAMP-response element binding (CREB) proteins are involved in transcriptional regulation in a number of cellular processes (e.g., neural plasticity and circadian rhythms). The CREB family contains activators and repressors that may interact through positive and negative feedback loops. These loops can be generated by auto- and cross-regulation of expression of CREB proteins, via CRE elements in or near their genes. Experiments suggest that such feedback loops may operate in several systems (e.g., Aplysia and rat). To understand the functional implications of such feedback loops, which are interlocked via cross-regulation of transcription, a minimal model with a positive and negative loop was developed and investigated using bifurcation analysis. Bifurcation analysis revealed diverse nonlinear dynamics (e.g., bistability and oscillations). The stability of steady states or oscillations could be changed by time delays in the synthesis of the activator (CREB1) or the repressor (CREB2). Investigation of stochastic fluctuations due to small numbers of molecules of CREB1 and CREB2 revealed a bimodal distribution of CREB molecules in the bistability region. The robustness of the stable HIGH and LOW states of CREB expression to stochastic noise differs, and a critical number of molecules was required to sustain the HIGH state for days or longer. Increasing positive feedback or decreasing negative feedback also increased the lifetime of the HIGH state, and persistence of this state may correlate with long-term memory formation. A critical number of molecules was also required to sustain robust oscillations of CREB expression. If a steady state was near a deterministic Hopf bifurcation point, stochastic resonance could induce oscillations. This comparative analysis of deterministic and stochastic dynamics not only provides insights into the possible dynamics of CREB regulatory motifs, but also demonstrates a framework for understanding other regulatory processes with similar network architecture.

Laser-induced transient grating analysis of dynamics of interaction between sensory rhodopsin II D75N and the HtrII transducer.

The interaction between sensory rhodopsin II (SRII) and its transducer HtrII was studied by the time-resolved laser-induced transient grating method using the D75N mutant of SRII, which exhibits minimal visible light absorption changes during its photocycle, but mediates normal phototaxis responses. Flash-induced transient absorption spectra of transducer-free D75N and D75N joined to 120 amino-acid residues of the N-terminal part of the SRII transducer protein HtrII (DeltaHtrII) showed only one spectrally distinct K-like intermediate in their photocycles, but the transient grating method resolved four intermediates (K(1)-K(4)) distinct in their volumes. D75N bound to HtrII exhibited one additional slower kinetic species, which persists after complete recovery of the initial state as assessed by absorption changes in the UV-visible region. The kinetics indicate a conformationally changed form of the transducer portion (designated Tr*), which persists after the photoreceptor returns to the unphotolyzed state. The largest conformational change in the DeltaHtrII portion was found to cause a DeltaHtrII-dependent increase in volume rising in 8 micros in the K(4) state and a drastic decrease in the diffusion coefficient (D) of K(4) relatively to those of the unphotolyzed state and Tr*. The magnitude of the decrease in D indicates a large structural change, presumably in the solvent-exposed HAMP domain of DeltaHtrII, where rearrangement of interacting molecules in the solvent would substantially change friction between the protein and the solvent.

Synaptic vesicle dynamics in mouse rod bipolar cells.

To better understand synaptic signaling at the mammalian rod bipolar cell terminal and pave the way for applying genetic approaches to the study of visual information processing in the mammalian retina, synaptic vesicle dynamics and intraterminal calcium were monitored in terminals of acutely isolated mouse rod bipolar cells and the number of ribbon-style active zones quantified. We identified a releasable pool, corresponding to a maximum of 7 s. The presence of a smaller, rapidly releasing pool and a small, fast component of refilling was also suggested. Following calcium channel closure, membrane surface area was restored to baseline with a time constant that ranged from 2 to 21 s depending on the magnitude of the preceding Ca2+ transient. In addition, a brief, calcium-dependent delay often preceded the start of onset of membrane recovery. Thus, several aspects of synaptic vesicle dynamics appear to be conserved between rod-dominant bipolar cells of fish and mammalian rod bipolar cells. A major difference is that the number of vesicles available for release is significantly smaller in the mouse rod bipolar cell, both as a function of the total number per neuron and on a per active zone basis.

Dynamics of a minimal model of interlocked positive and negative feedback loops of transcriptional regulation by cAMP-response element binding proteins.

cAMP-response element binding (CREB) proteins are involved in transcriptional regulation in a number of cellular processes (e.g., neural plasticity and circadian rhythms). The CREB family contains activators and repressors that may interact through positive and negative feedback loops. These loops can be generated by auto- and cross-regulation of expression of CREB proteins, via CRE elements in or near their genes. Experiments suggest that such feedback loops may operate in several systems (e.g., Aplysia and rat). To understand the functional implications of such feedback loops, which are interlocked via cross-regulation of transcription, a minimal model with a positive and negative loop was developed and investigated using bifurcation analysis. Bifurcation analysis revealed diverse nonlinear dynamics (e.g., bistability and oscillations). The stability of steady states or oscillations could be changed by time delays in the synthesis of the activator (CREB1) or the repressor (CREB2). Investigation of stochastic fluctuations due to small numbers of molecules of CREB1 and CREB2 revealed a bimodal distribution of CREB molecules in the bistability region. The robustness of the stable HIGH and LOW states of CREB expression to stochastic noise differs, and a critical number of molecules was required to sustain the HIGH state for days or longer. Increasing positive feedback or decreasing negative feedback also increased the lifetime of the HIGH state, and persistence of this state may correlate with long-term memory formation. A critical number of molecules was also required to sustain robust oscillations of CREB expression. If a steady state was near a deterministic Hopf bifurcation point, stochastic resonance could induce oscillations. This comparative analysis of deterministic and stochastic dynamics not only provides insights into the possible dynamics of CREB regulatory motifs, but also demonstrates a framework for understanding other regulatory processes with similar network architecture.

The dynamics of calmodulin signaling revealed using optical methods

The Ca2+-binding protein calmodulin (CaM) is a key transducer of Ca2+ oscillations by virtue of its ability to bind Ca 2+ selectively and then interact specifically with a large number of downstream enzymes and proteins. It remains unclear whether Ca2+ -dependent signaling alone can activate the full range of Ca 2+/CaM regulated processes or whether other regulatory schemes in the cell exist that allow specific targeting of CaM to subsets of Ca 2+/CaM binding sites or regions of the cell. Here we investigate the possibility that alterations of the availability of CaM may serve as a potential cellular mechanism for regulating the activation of CaM-dependent targets. By utilizing sensitive optical techniques with high spatial and temporal resolution, we examine the intracellular dynamics of CaM signaling at a resolution previously unattainable. After optimizing and characterizing both the optical methods and fluorescently labeled probes for intracellular measurements, the diffusion of CaM in the cytoplasm of HEK293 cells was analyzed. It was discovered that the diffusion characteristics of CaM are similar to that of a comparably sized inert molecule. Independent manipulation of experimental parameters, including increases in total concentrations of CaM and intracellular Ca2+ levels, did not change the diffusion of CaM in the cytoplasm. However, changes in diffusion were seen when the concentration of Ca2+/CaM-binding targets was increased in conjunction with elevated Ca2+. This indicates that CaM is not normally limiting for the activation of Ca 2+/CaM-dependent enzymes in HEK293 cells but reveals that the ratio of CaM to CaM-dependent targets is a potential mechanism for changing CaM availability. Next we considered whether cellular compartmentalization may act to regulate concentrations of available Ca2+/CaM in hippocampal neurons. We discovered changes in diffusion parameters of CaM under elevated Ca2+ conditions in the soma, neurite and nucleus which suggest that either the composition of cytoplasm is different in these compartments and/or they are composed of unique families of CaM-binding proteins. Finally, we return to the HEK293 cell and for the first time directly show the intracellular binding of CaM and CaMKII, an important target for CaM critical for neuronal function and plasticity. Furthermore, we analyzed the complex binding stoichiometry of this molecular interaction in the basal, activated and autophosphorylated states of CaMKII and determined the impact of this binding on CaM availability in the cell. Overall these results demonstrate that regulation of CaM availability is a viable cellular mechanism for regulating the output of CaM-dependent processes and that this process is tuned to the specific functional needs of a particular cell type and subcellular compartment. ^

IDENTIFICATION AND ANALYSIS OF A NOVEL ROLE FOR THE TOUSLED-LIKE KINASE IN REGULATING MITOTIC SPINDLE DYNAMICS

Deregulation of kinase activity is one example of how cells become cancerous by evading evolutionary constraints. The Tousled kinase (Tsl) was initially identified in Arabidopsis thaliana as a developmentally important kinase. There are two mammalian orthologues of Tsl and one orthologue in C. elegans, TLK-1, which is essential for embryonic viability and germ cell development. Depletion of TLK-1 leads to embryonic arrest large, distended nuclei, and ultimately embryonic lethality. Prior to terminal arrest, TLK-1-depleted embryos undergo aberrant mitoses characterized by poor metaphase chromosome alignment, delayed mitotic progression, lagging chromosomes, and supernumerary centrosomes. I discovered an unanticipated requirement for TLK-1 in mitotic spindle assembly and positioning. Normally, in the newly-fertilized zygote (P0) the maternal pronucleus migrates toward the paternal pronucleus at the posterior end of the embryo. After pronuclear meeting, the pronuclear-centrosome complex rotates 90° during centration to align on the anteroposterior axis followed by nuclear envelope breakdown (NEBD). However, in TLK-1-depleted P0 embryos, the centrosome-pronuclear complex rotation is significantly delayed with respect to NEBD and chromosome congression, Additionally, centrosome positions over time in tlk-1(RNAi) early embryos revealed a defect in posterior centrosome positioning during spindle-pronuclear centration, and 4D analysis of centrosome positions and movement in newly fertilized embryos showed aberrant centrosome dynamics in TLK-1-depleted embryos. Several mechanisms contribute to spindle rotation, one of which is the anchoring of astral microtubules to the cell cortex. Attachment of these microtubules to the cortices is thought to confer the necessary stability and forces in order to rotate the centrosome-pronuclear complex in a timely fashion. Analysis of a microtubule end-binding protein revealed that TLK-1-depleted embryos exhibit a more stochastic distribution of microtubule growth toward the cell cortices, and the types of microtubule attachments appear to differ from wild-type embryos. Additionally, fewer astral microtubules are in the vicinity of the cell cortex, thus suggesting that the delayed spindle rotation could be in part due to a lack of appropriate microtubule attachments to the cell cortex. Together with recently published biochemical data revealing the Tousled-like kinases associate with components of the dynein microtubule motor complex in humans, these data suggest that Tousled-like kinases play an important role in mitotic spindle assembly and positioning.