A detailed algorithmic explanation is required for how a network of chemical reactions can generate the sophisticated behavior displayed by living cells. Though several previous works have shown that reaction networks are computationally universal and can in principle implement any algorithm, there is scope for constructions that map well onto biological reality, make efficient use of the computational potential of the native dynamics of reaction networks, and make contact with statistical mechanics. We describe a new reaction network scheme for solving a large class of statistical problems. Specifically we show how reaction networks can implement information projection, and consequently a generalized Expectation-Maximization algorithm, to solve maximum likelihood estimation problems in partially-observed exponential families on categorical data. Our scheme can be thought of as an algorithmic interpretation of E. T. Jaynes’s vision of statistical mechanics as statistical inference.
In a brilliant paper, @cite_29 show how to implement a Kalman filter with reaction networks, and implement this scheme in vivo in E. Coli. Their approach is to write down a dynamical system describing the filter, then change variables if required so the dynamical system falls within the class of systems that are implementable with CRNs, and finally obtain a DNA strand displacement implementation for the dynamical system. Their work shows that filters, being self-correcting, are robust, and can tolerate some amount of model mismatch. Further, such systems when implemented in vitro and in vivo work as advertised. This is very encouraging for the empirical prospects of our schemes.
@cite_29
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The invention of the Kalman filter is a crowning achievement of filtering theory—one that has revolutionized technology in countless ways. By dealing effectively with noise, the Kalman filter has enabled various applications in positioning, navigation, control, and telecommunications. In the emerging field of synthetic biology, noise and context dependency are among the key challenges facing the successful implementation of reliable, complex, and scalable synthetic circuits. Although substantial further advancement in the field may very well rely on effectively addressing these issues, a principled protocol to deal with noise—as provided by the Kalman filter—remains completely missing. Here we develop an optimal filtering theory that is suitable for noisy biochemical networks. We show how the resulting filters can be implemented at the molecular level and provide various simulations related to estimation, system identification, and noise cancellation problems. We demonstrate our approach in vitro using DNA strand displacement cascades as well as in vivo using flow cytometry measurements of a light-inducible circuit in Escherichia coli. We propose an abstraction method for medium-scale biomolecular networks, based on hybrid dynamical systems with continuous multi-affine dynamics. This abstraction method follows naturally from the notion of approximating nonlinear rate laws with continuous piecewise linear functions and can be easily automated. An efficient reachability algorithm is possible for the resulting class of hybrid systems. An efficient reachability algorithm is possible for the resulting class of hybrid systems. An approximation for an ordinary differential equation model of the lac operon is constructed, and it is shown that the abstraction passes the same experimental tests as were used to validate the original model. The well studied biological system exhibits bistability and switching behaviour, arising from positive feedback in the expression mechanism of the lac operon. The switching property of the lac system is an example of the major qualitative features that are the building blocks of higher level, more coarse-grained descriptions. The present approach is useful in helping to correctly identify such properties and in connecting them to the underlying molecular dynamical details. Reachability analysis together with the knowledge of the steady-state structure are used to identify ranges of parameter values for which the system maintains the bistable switching property. Filters in convolutional networks are typically parameterized in a pixel basis, that does not take prior knowledge about the visual world into account. We investigate the generalized notion of frames designed with image properties in mind, as alternatives to this parametrization. We show that frame-based ResNets and Densenets can improve performance on Cifar-10+ consistently, while having additional pleasant properties like steerability. By exploiting these transformation properties explicitly, we arrive at dynamic steerable blocks. They are an extension of residual blocks, that are able to seamlessly transform filters under pre-defined transformations, conditioned on the input at training and inference time. Dynamic steerable blocks learn the degree of invariance from data and locally adapt filters, allowing them to apply a different geometrical variant of the same filter to each location of the feature map. When evaluated on the Berkeley Segmentation contour detection dataset, our approach outperforms all competing approaches that do not utilize pre-training. Our results highlight the benefits of image-based regularization to deep networks. Particle filters (PFs) are powerful sampling-based inference learning algorithms for dynamic Bayesian networks (DBNs). They allow us to treat, in a principled way, any type of probability distribution, nonlinearity and non-stationarity. They have appeared in several fields under such names as "condensation", "sequential Monte Carlo" and "survival of the fittest". In this paper, we show how we can exploit the structure of the DBN to increase the efficiency of particle filtering, using a technique known as Rao-Blackwellisation. Essentially, this samples some of the variables, and marginalizes out the rest exactly, using the Kalman filter, HMM filter, junction tree algorithm, or any other finite dimensional optimal filter. We show that Rao-Blackwellised particle filters (RBPFs) lead to more accurate estimates than standard PFs. We demonstrate RBPFs on two problems, namely non-stationary online regression with radial basis function networks and robot localization and map building. We also discuss other potential application areas and provide references to some finite dimensional optimal filters.