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State-of-the-art biochemical systems for medical applications and chemical computing are application-specific and cannot be re-programmed or trained once fabricated. The implementation of adaptive biochemical systems that would offer flexibility through programmability and autonomous adaptation faces major challenges because of the large number of required chemical species as well as the timing-sensitive feedback loops required for learning. Currently, biochemistry lacks a systems vision on how the user-level programming interface and abstraction with a subsequent translation to chemistry should look like. By developing adaptation in chemistry, we could replace multiple hard-wired systems with a single programmable template that can be (re)trained to match a desired input-output profile benefiting smart drug delivery, pattern recognition, and chemical computing. I aimed to address these challenges by proposing several approaches to learning and adaptation in Chemical Reaction Networks (CRNs), a type of simulated chemistry, where species are unstructured, i.e., they are identified by symbols rather than molecular structure, and their dynamics or concentration evolution are driven by reactions and reaction rates that follow mass-action and Michaelis-Menten kinetics. Several CRN and experimental DNA-based models of neural networks exist. However, these models successfully implement only the forward-pass, i.e., the input-weight integration part of a perceptron model. Learning is delegated to a non-chemical system that computes the weights before converting them to molecular concentrations. Autonomous learning, i.e., learning implemented fully inside chemistry has been absent from both theoretical and experimental research. The research in this thesis offers the first constructive evidence that learning in CRNs is, in fact, possible. I have introduced the original concept of a chemical binary perceptron that can learn all 14 linearly-separable logic functions and is robust to the perturbation of rate constants. That shows learning is universal and substrate-free. To simplify the model I later proposed and applied the "asymmetric" chemical arithmetic providing a compact solution for representing negative numbers in chemistry. To tackle more difficult tasks and to serve more complicated biochemical applications, I introduced several key modular building blocks, each addressing certain aspects of chemical information processing and learning. These parts organically combined into gradually more complex systems. First, instead of simple static Boolean functions, I tackled analog time-series learning and signal processing by modeling an analog chemical perceptron. To store past input concentrations as a sliding window I implemented a chemical delay line, which feeds the values to the underlying chemical perceptron. That allows the system to learn, e.g., the linear moving-average and to some degree predict a highly nonlinear NARMA benchmark series. Another important contribution to the area of chemical learning, which I have helped to shape, is the composability of perceptrons into larger multi-compartment networks. Each compartment hosts a single chemical perceptron and compartments communicate with each other through a channel-mediated exchange of molecular species. Besides the feedforward pass, I implemented the chemical error backpropagation analogous to that of feedforward neural networks. Also, after applying mass-action kinetics for the catalytic reactions, I succeeded to systematically analyze the ODEs of my models and derive the closed exact and approximative formulas for both the input-weight integration and the weight update with a learning rate annealing. I proved mathematically that the formulas of certain chemical perceptrons equal the formal linear and sigmoid neurons, essentially bridging neural networks and adaptive CRNs. For all my models the basic methodology was to first design species and reactions, and then set the rate constants either "empirically" by hand, automatically by a standard genetic algorithm (GA), or analytically if possible. I performed all simulations in my COEL framework, which is the first cloud-based chemistry modeling tool, accessible at http://coel-sim.org. I minimized the amount of required molecular species and reactions to make wet chemical implementation possible. I applied an automatized mapping technique, Soloveichik's CRN-to-DNA-strand-displacement transformation, to the chemical linear perceptron and the manual signalling delay line and obtained their full DNA-strand specified implementations. As an alternative DNA-based substrate, I mapped these two models also to deoxyribozyme-mediated cleavage reactions reducing the size of the displacement variant to a third. Both DNA-based incarnations could directly serve as blue-prints for wet biochemicals. Besides an actual synthesis of my models and conducting an experiment in a biochemical laboratory, the most promising future work is to employ so-called reservoir computing (RC), which is a novel machine learning method based on recurrent neural networks. The RC approach is relevant because for time-series prediction it is clearly superior to classical recurrent networks. It can also be implemented in various ways, such as electrical circuits, physical systems, such as a colony of Escherichia Coli, and water. RC's loose structural assumptions therefore suggest that it could be expressed in a chemical form as well. This could further enhance the expressivity and capabilities of chemically-embedded learning. My chemical learning systems may have applications in the area of medical diagnosis and smart medication, e.g., concentration signal processing and monitoring, and the detection of harmful species, such as chemicals produced by cancer cells in a host (cancer miRNAs) or the detection of a severe event, defined as a linear or nonlinear temporal concentration pattern. My approach could replace hard-coded solutions and would allow to specify, train, and reuse chemical systems without redesigning them. With time-series integration, biochemical computers could keep a record of changing biological systems and act as diagnostic aids and tools in preventative and highly personalized medicine.

Chemical reactions involving small number of reacting molecules are noisy processes. They are simulated using stochastic simulation algorithms like the Gillespie SSA, which are valid when the reaction environment is well-mixed. This is not the case in reactions occuring on biological media like cell membranes, where alternative simulation methods have to be used to account for the crowded nature of the reacting environment. Ion channels, which are membrane proteins controlling the flow of ions into and out of the cell, offer excellent single molecule conditions to test stochastic simulation schemes in crowded biological media. Single molecule reactions are of great importance in determining the functions of biological molecules. Access to their experimental data have increased the scope of com-putational modeling of biological processes. Recently, single molecule experiments have revealed the non-Markovian nature of chemical reactions, due to a phenomenon called `dynamic disorder', which makes the rate constants a deterministic function of time or a random process. This happens when there are additional slow scale conformational transitions, giving the molecule a memory of its previous states. In a previous work, the hTREK1 two pore domain potassium channel was revealed to have long term memory in its kinetics, prompting alternate non-Markovian schemes to analyze its gating. Traditionally, ion channel gating is modeled as Markovian transitions between fixed states. In this work, we have used single channel data from hTREK1 ion channel and have provided a simple diffusion model for its gating. The main assumption of this model is that the ion channel diffuses through a continuum of states on its potential energy landscape, which is derived from the steady state probability distribution of ionic current recorded from patch clamp experiments. A stochastic differential equation (SDE) driven by Gaussian white noise is proposed to model this motion in an asymmetric double well potential. The method is computationally very simple and efficient and reproduces the amplitude histogram very well. For the case when ligands are added, leading to incorporation of long term memory in the kinetics, the SDE is modified to run on coloured noise. This has been done by introducing an auxiliary variable into the equation. It has been shown that increasing the noise correlation with ligand concentration improves the fits to the experimental data. This has been validated for several datasets. These methods are more advantageous for simulation than the Markovian models as they are true to the physical picture of gating and also computationally very efficient. Reproducing the whole raw data trace takes no more than a few seconds with our scheme, with the only input being the amplitude histogram and four parameters. Finally a quantitative model based on a modified version of the Chemical Langevin equation is given, which works on random rate parameters. This model is computationally simple to implement and reproduces the catalytic activity of the channel as a function of time. From the computational analysis undertaken in this work, we can infer that ion channel activity can be modeled using the framework of non-Markovian processes, lending credence to the recent understanding that single molecule reactions are basically processes with long-term memory. Since the ion channel is basically a protein, we can also hypothesize that the some of the properties that make proteins so vital to living organ-isms could be attributed to long-term memory in their folding kinetics, giving them the ability to sample specific regions of their conformation space, which are of interest to biological functions.