Learning Dynamic Boltzmann Distributions as Reduced Models of Spatial Chemical Kinetics

Finding reduced models of spatially-distributed chemical reaction networks requires an estimation of which effective dynamics are relevant. We propose a machine learning approach to this coarse graining problem, where a maximum entropy approximation is constructed that evolves slowly in time. The dynamical model governing the approximation is expressed as a functional, allowing a general treatment of spatial interactions. In contrast to typical machine learning approaches which estimate the interaction parameters of a graphical model, we derive Boltzmann-machine like learning algorithms to estimate directly the functionals dictating the time evolution of these parameters. By incorporating analytic solutions from simple reaction motifs, an efficient simulation method is demonstrated for systems ranging from toy problems to basic biologically relevant networks. The broadly applicable nature of our approach to learning spatial dynamics suggests promising applications to multiscale methods for spatial networks, as well as to further problems in machine learning

Geometric principles of second messenger dynamics in dendritic spines.

Dendritic spines are small, bulbous protrusions along dendrites in neurons and play a critical role in synaptic transmission. Dendritic spines come in a variety of shapes that depend on their developmental state. Additionally, roughly 14-19% of mature spines have a specialized endoplasmic reticulum called the spine apparatus. How does the shape of a postsynaptic spine and its internal organization affect the spatio-temporal dynamics of short timescale signaling? Answers to this question are central to our understanding the initiation of synaptic transmission, learning, and memory formation. In this work, we investigated the effect of spine and spine apparatus size and shape on the spatio-temporal dynamics of second messengers using mathematical modeling using reaction-diffusion equations in idealized geometries (ellipsoids, spheres, and mushroom-shaped). Our analyses and simulations showed that in the short timescale, spine size and shape coupled with the spine apparatus geometries govern the spatiotemporal dynamics of second messengers. We show that the curvature of the geometries gives rise to pseudo-harmonic functions, which predict the locations of maximum and minimum concentrations along the spine head. Furthermore, we showed that the lifetime of the concentration gradient can be fine-tuned by localization of fluxes on the spine head and varying the relative curvatures and distances between the spine apparatus and the spine head. Thus, we have identified several key geometric determinants of how the spine head and spine apparatus may regulate the short timescale chemical dynamics of small molecules that control synaptic plasticity

Multi-Stage Modeling of the Kinetics of Activation of CaMKII

Ca 2+ /calmodulin-dependent protein kinase 2 (CaMKII) plays an important role in induction of long-term potentiation and formation of memory. It is abundant in dendritic spines, and is activated when Ca 2+ flows into the postsynaptic cytosol through open NMDA-type glutamate receptors. Its function is fine-tuned through interaction with other proteins as well as through subunit interactions and regulatory autophosphorylation. We have undertaken a multi-stage project to study the critical kinetics of activation of CaMKII in the spine by combining modeling and experimental studies. We are using computational modeling and simulations on various platforms, coupled with biochemical experiments in vitro, and eventually in vivo, to understand CaMKII regulation. The project includes the following steps: 1. Determining the parameters governing activation of a monomeric subunit. The CaMKII holoenzyme is a large dodecamer of similar, homologous subunits held together by interactions between the association domains located at the carboxyl end of each subunit. Individual, monomeric subunits can be expressed recombinantly by removing the association domain. Computer simulations of activation of monomeric CaMKII by Ca 2+ /calmodulin at both saturating and non-saturating concentrations in a test tube have helped to identify the binding parameters that are most crucial for modeling of regulation of CaMKII and thus have indicated the most useful biochemical assays to measure those parameters (Pepke et al., 2010). We are using these measurements to fine-tune our model of activation of individual catalytic subunits. 2. Building a model of the holoenzyme. Because a CaMKII holoenzyme contains 12 similar subunits, each of which can exist in several states, the holoenzyme can have a large number of state combinations. Thus, modeling the entire holoenzyme requires a computational framework that avoids the ensuing combinatorial complexity. The stochastic simulator MCell provides an elegant, rule-based way of modeling state changes in the CaMKII holoenzyme. 3. Modeling cooperativity that arises from the dodecameric structure of CaMKII. Autophosphorylation at threonine-286, which activates CaMKII subunits, is an inter-subunit event. Thus, it is greatly facilitated by the close proximity of subunits in the holoenzyme. In addition, the subunits within the holoenzyme are arranged as dimers which appears to result in cooperativity in the binding of Ca 2+ /CaM to individual subunits of the dimer (Chao et al., 2010). An accurate model of activation of subunits in the holoenzyme and their autophosphorylation will allow us to explore the effects of cooperativity on CaMKII activation on various time scales. 4. Modeling CaMKII within the context of a postsynaptic spine CaMKII interacts with a variety of other proteins, both in the postsynaptic density (PSD), close to major sources of Ca 2+ influx, and in other parts of the spine. In the fourth stage of this project we plan to implement kinetic models of activation of CaMKII in the context of an MCell model of Ca 2+ influx into a spine upon activation of NMDA-type glutamate receptors (Keller et al., 2008; Keller et al., 2011, submitted). We will explore the effects of different localization and numbers of CaMKII holoenzymes in the spine on CaMKII activation. References: Pepke, S., Kinzer-Ursem, T., Mihalas, S., and Kennedy, M.B. (2010). A dynamic model of interactions of Ca 2+ , calmodulin, and catalytic subunits of Ca 2+ /calmodulin-dependent protein kinase II. PLoS Comput Biol 6, e1000675. Chao, L.H., Pellicena, P., Deindl, S., Barclay, L.A., Schulman, H., and Kuriyan, J. (2010). Intersubunit capture of regulatory segments is a component of cooperative CaMKII activation. Nat Struct Mol Biol 17, 264-272. Keller, D.X., Franks, K.M., Bartol, T.M., Jr., and Sejnowski, T.J. (2008). Calmodulin activation by calcium transients in the postsynaptic density of dendritic spines. PLoS ONE 3, e2045. Keller, D.X., Bartol, T.M., Kinney, J.P, Kennedy, M.B., Bajaj, C., Harris, K.M., and Sejnowski, T.J. Regulation of synaptic calcium transients in reconstructed dendritic spines of hippocampal CA1 pyramidal neurons, submitted

Nanoconnectomic upper bound on the variability of synaptic plasticity

Information in a computer is quantified by the number of bits that can be stored and recovered. An important question about the brain is how much information can be stored at a synapse through synaptic plasticity, which depends on the history of probabilistic synaptic activity. The strong correlation between size and efficacy of a synapse allowed us to estimate the variability of synaptic plasticity. In an EM reconstruction of hippocampal neuropil we found single axons making two or more synaptic contacts onto the same dendrites, having shared histories of presynaptic and postsynaptic activity. The spine heads and neck diameters, but not neck lengths, of these pairs were nearly identical in size. We found that there is a minimum of 26 distinguishable synaptic strengths, corresponding to storing 4.7 bits of information at each synapse. Because of stochastic variability of synaptic activation the observed precision requires averaging activity over several minutes