Systematic approximation methods for stochastic biochemical kinetics

Experimental studies have shown that the protein abundance in living cells varies from few tens to several thousands molecules per species. Molecular fluctuations roughly scale as the inverse square root of the number of molecules due to the random timing of reactions. It is hence expected that intrinsic noise plays an important role in the dynamics of biochemical networks. The Chemical Master Equation is the accepted description of these systems under well-mixed conditions. Because analytical solutions to this equation are available only for simple systems, one often has to resort to approximation methods. A popular technique is an expansion in the inverse volume to which the reactants are confined, called van Kampen's system size expansion. Its leading order terms are given by the phenomenological rate equations and the linear noise approximation that quantify the mean concentrations and the Gaussian fluctuations about them, respectively. While these approximations are valid in the limit of large molecule numbers, it is known that physiological conditions often imply low molecule numbers. We here develop systematic approximation methods based on higher terms in the system size expansion for general biochemical networks. We present an asymptotic series for the moments of the Chemical Master Equation that can be computed to arbitrary precision in the system size expansion. We then derive an analytical approximation of the corresponding time-dependent probability distribution. Finally, we devise a diagrammatic technique based on the path-integral method that allows to compute time-correlation functions. We show through the use of biological examples that the first few terms of the expansion yield accurate approximations even for low number of molecules. The theory is hence expected to closely resemble the outcomes of single cell experiments

Anomalous transport in the crowded world of biological cells

A ubiquitous observation in cell biology is that diffusion of macromolecules and organelles is anomalous, and a description simply based on the conventional diffusion equation with diffusion constants measured in dilute solution fails. This is commonly attributed to macromolecular crowding in the interior of cells and in cellular membranes, summarising their densely packed and heterogeneous structures. The most familiar phenomenon is a power-law increase of the MSD, but there are other manifestations like strongly reduced and time-dependent diffusion coefficients, persistent correlations, non-gaussian distributions of the displacements, heterogeneous diffusion, and immobile particles. After a general introduction to the statistical description of slow, anomalous transport, we summarise some widely used theoretical models: gaussian models like FBM and Langevin equations for visco-elastic media, the CTRW model, and the Lorentz model describing obstructed transport in a heterogeneous environment. Emphasis is put on the spatio-temporal properties of the transport in terms of 2-point correlation functions, dynamic scaling behaviour, and how the models are distinguished by their propagators even for identical MSDs. Then, we review the theory underlying common experimental techniques in the presence of anomalous transport: single-particle tracking, FCS, and FRAP. We report on the large body of recent experimental evidence for anomalous transport in crowded biological media: in cyto- and nucleoplasm as well as in cellular membranes, complemented by in vitro experiments where model systems mimic physiological crowding conditions. Finally, computer simulations play an important role in testing the theoretical models and corroborating the experimental findings. The review is completed by a synthesis of the theoretical and experimental progress identifying open questions for future investigation.Comment: review article, to appear in Rep. Prog. Phy

Modeling biomolecules: interactions, forces and free energies

La biología ha sido tradicionalmente una ciencia cualitativa. El principal problema que presenta es que trata con sistemas muy complejos, mucho más que las moléculas de las que se ocupa la química, o que muchos sistemas físicos. Sin embargo, en los últimos años, hemos sido testigos de un desarrollo enorme hacia planteamientos cuantitativos para resolver problemas biológicos, impulsado principalmente por el desarrollo de diversas técnicas avanzadas en biofísica, o por la emergencia de las herramientas computacionales. En particular, en biofísica computacional, dado un determinado problema a estudiar, la estrategia es proponer un modelo que describa el comportamiento de nuestro sistema y realizar simulaciones numéricas sobre este modelo. Este planteamiento presenta una dificultad principal que es la elección de la escala a la cual realizamos nuestro modelo. Es necesario llegar a un compromiso entre el nivel de detalle y la capacidad computacional de que disponemos. Así, modelos muy detallados son capaces de proporcionar información de gran resolución, sin embargo sólo para sistemas moleculares de tamaño limitado, con propiedades que se manifiesten a escalas temporales cortas. Si necesitamos tratar con sistemas de mayor tamaño, o nos interesan propiedades que se manifiestan en escalas temporales mayores, es necesario identificar cuáles son los grados de libertad relevantes para nuestro sistema y despreciar el resto. Aparte de este problema, el siguiente reto que se nos plantea es transformar todos los datos numéricos producidos en información relevante que pueda responder de manera objetiva a las preguntas que nos planteamos. Para ello, debemos disponer de métodos de análisis lo bastante robustos como para transformar la información en bruto producida en nuestras simulaciones, en conocimiento directo de una manera no sesgada. La presente Tesis Doctoral se enmarca en este ámbito, ya que estudiaremos tres problemas biológicos diferentes haciendo énfasis en la fase de modelización de nuestro sistema, así como en el empleo de técnicas de análisis avanzadas para comprenderlo. En la primera parte, nos centramos en el análisis de la dinámica de proteínas, enfatizando las distintas descripciones que pueden usarse para comprender su paisaje de energía libre. Para ello escogemos un sistema relativamente simple, una proteína modelo coarse-grained a la cual aplicamos una fuerza constante para promover su desplegamiento. Realizaremos simulaciones numéricas en este sistema y nos plantearemos cuál es la mejor manera de obtener una descripción fiel de su espacio configuracional así como de su mecanismo de desplegamiento. Para ello emplearemos dos métodos distintos. Primero, proyectaremos su paisaje de energía libre –de gran dimensión- sobre distintos parámetros de orden, obteniendo representaciones unidimensionales. Éstas proporcionarán una visión globalmente correcta del sistema, sin embargo fallarán en la descripción adecuada de su mecanismo de desnaturalización. Por otra parte, emplearemos modelos de Markov para representar el paisaje de energía libre. Estos revelarán un espacio configuracional más complejo que el previsto anteriormente, con varios intermediarios que tendrán un papel relevante, especialmente para comprender el mecanismo de desplegamiento. En la segunda parte de la Tesis Doctoral, mostramos el estudio de un modelo de DNA al nivel del par de bases, el modelo de Peyrard-Bishop-Dauxois. En particular, extenderemos este modelo para introducir la interacción proteína-DNA. Proponiendo un método de análisis adecuado basado en modelos de Markov, podremos emplear este modelo para analizar secuencias de promotores, relacionando los estados que encontramos en la dinámica del sistema con sitios de unión proteína-DNA. Este modelo lo emplearemos para el análisis de nueve secuencias de promotores de una cianobacteria en particular. Nos centraremos en la identificación del sitio de inicio de la transcripción (TSS), región donde se une la RNA polimerasa para iniciar este proceso. En cada uno de los promotores, gracias al modelo somos capaces de identificar esta región como un estado de relevancia en la dinámica, con tendencia a que la partícula se una, formando una burbuja. Asimismo, gracias al método de análisis, cuantificamos estos estados, proporcionando magnitudes estadísticas que podemos relacionar con el conocimiento biológica acerca de estos promotores. La tercera parte está dedicada a los experimentos de molécula individual. Presentamos una colaboración experimental en la cual analizamos experimentos de disociación mecánica de dos complejos proteína:proteína. Nuestro objetivo es proporcionar una visión adecuada del paisaje de energía libre que gobierna este proceso. Para ello proponemos un método que permite recuperar la barrera de energía libre así como la energía libre de disociación para complejos biológicos. En particular, emplearemos este método para analizar experimentos de espetroscopía de fuerza, permitiendo obtener estas magnitudes y discutirlas en el contexto de la biología del sistema. Asimismo, proponemos un modelo físico para este tipo de experimentos, sobre el cual realizamos simulaciones numéricas que analizamos con el mismo método, con objeto de validarlo y respaldar su empleo

Non-equilibrium statistical mechanics: From a paradigmatic model to biological transport

Unlike equilibrium statistical mechanics, with its well-established foundations, a similar widely-accepted framework for non-equilibrium statistical mechanics (NESM) remains elusive. Here, we review some of the many recent activities on NESM, focusing on some of the fundamental issues and general aspects. Using the language of stochastic Markov processes, we emphasize general properties of the evolution of configurational probabilities, as described by master equations. Of particular interest are systems in which the dynamics violate detailed balance, since such systems serve to model a wide variety of phenomena in nature. We next review two distinct approaches for investigating such problems. One approach focuses on models sufficiently simple to allow us to find exact, analytic, non-trivial results. We provide detailed mathematical analyses of a one-dimensional continuous-time lattice gas, the totally asymmetric exclusion process (TASEP). It is regarded as a paradigmatic model for NESM, much like the role the Ising model played for equilibrium statistical mechanics. It is also the starting point for the second approach, which attempts to include more realistic ingredients in order to be more applicable to systems in nature. Restricting ourselves to the area of biophysics and cellular biology, we review a number of models that are relevant for transport phenomena. Successes and limitations of these simple models are also highlighted.Comment: 72 pages, 18 figures, Accepted to: Reports on Progress in Physic

심층학습을 이용한 액체계의 성질 예측

학위논문(박사)--서울대학교 대학원 :자연과학대학 화학부,2020. 2. 정연준.최근 기계학습 기술의 급격한 발전과 이의 화학 분야에 대한 적용은 다양한 화학적 성질에 대한 구조-성질 정량 관계를 기반으로 한 예측 모형의 개발을 가속하고 있다. 용매화 자유 에너지는 그러한 기계학습의 적용 예중 하나이며 다양한 용매 내의 화학반응에서 중요한 역할을 하는 근본적 성질 중 하나이다. 본 연구에서 우리는 목표로 하는 용매화 자유 에너지를 원자간의 상호작용으로부터 구할 수 있는 새로운 심층학습 기반 용매화 모형을 소개한다. 제안된 심층학습 모형의 계산 과정은 용매와 용질 분자에 대한 부호화 함수가 각 원자와 분자들의 구조적 성질에 대한 벡터 표현을 추출하며, 이를 토대로 원자간 상호작용을 복잡한 퍼셉트론 신경망 대신 벡터간의 간단한 내적으로 구할 수 있다. 952가지의 유기용질과 147가지의 유기용매를 포함하는 6,493가지의 실험치를 토대로 기계학습 모형의 교차 검증 시험을 실시한 결과, 평균 절대 오차 기준 0.2 kcal/mol 수준으로 매우 높은 정확도를 가진다. 스캐폴드-기반 교차 검증의 결과 역시 0.6 kcal/mol 수준으로, 외삽으로 분류할 수 있는 비교적 새로운 분자 구조에 대한 예측에 대해서도 우수한 정확도를 보인다. 또한, 제안된 특정 기계학습 모형은 그 구조 상 특정 용매에 특화되지 않았기 때문에 높은 양도성을 가지며 학습에 이용할 데이터의 수를 늘이는 데 용이하다. 원자간 상호작용에 대한 분석을 통해 제안된 심층학습 모형 용매화 자유 에너지에 대한 그룹-기여도를 잘 재현할 수 있음을 알 수 있으며, 기계학습을 통해 단순히 목표로 하는 성질만을 예측하는 것을 넘어 더욱 상세한 물리화학적 이해를 하는 것이 가능할 것이라 기대할 수 있다.Recent advances in machine learning technologies and their chemical applications lead to the developments of diverse structure-property relationship based prediction models for various chemical properties; the free energy of solvation is one of them and plays a dominant role as a fundamental measure of solvation chemistry. Here, we introduce a novel machine learning-based solvation model, which calculates the target solvation free energy from pairwise atomistic interactions. The novelty of our proposed solvation model involves rather simple architecture: two encoding function extracts vector representations of the atomic and the molecular features from the given chemical structure, while the inner product between two atomistic features calculates their interactions, instead of black-boxed perceptron networks. The cross-validation result on 6,493 experimental measurements for 952 organic solutes and 147 organic solvents achieves an outstanding performance, which is 0.2 kcal/mol in MUE. The scaffold-based split method exhibits 0.6 kcal/mol, which shows that the proposed model guarantees reasonable accuracy even for extrapolated cases. Moreover, the proposed model shows an excellent transferability for enlarging training data due to its solvent-non-specific nature. Analysis of the atomistic interaction map shows there is a great potential that our proposed model reproduces group contributions on the solvation energy, which makes us believe that the proposed model not only provides the predicted target property, but also gives us more detailed physicochemical insights.1. Introduction 1 2. Delfos: Deep Learning Model for Prediction of Solvation Free Energies in Generic Organic Solvents 7 2.1. Methods 7 2.1.1. Embedding of Chemical Contexts 7 2.1.2. Encoder-Predictor Network 9 2.2. Results and Discussions 13 2.2.1. Computational Setup and Results 13 2.2.2. Transferability of the Model for New Compounds 17 2.2.3. Visualization of Attention Mechanism 26 3. Group Contribution Method for the Solvation Energy Estimation with Vector Representations of Atom 29 3.1. Model Description 29 3.1.1. Word Embedding 29 3.1.2. Network Architecture 33 3.2. Results and Discussions 39 3.2.1. Computational Details 39 3.2.2. Prediction Accuracy 42 3.2.3. Model Transferability 44 3.2.4. Group Contributions of Solvation Energy 49 4. Empirical Structure-Property Relationship Model for Liquid Transport Properties 55 5. Concluding Remarks 61 A. Analyzing Kinetic Trapping as a First-Order Dynamical Phase Transition in the Ensemble of Stochastic Trajectories 65 A1. Introduction 65 A2. Theory 68 A3. Lattice Gas Model 70 A4. Mathematical Model 73 A5. Dynamical Phase Transitions 75 A6. Conclusion 82 B. Reaction-Path Thermodynamics of the Michaelis-Menten Kinetics 85 B1. Introduction 85 B2. Reaction Path Thermodynamics 88 B3. Fixed Observation Time 94 B4. Conclusions 101Docto

Theoretical investigation of the quantum-confined Stark effect

The two main objectives of this dissertation are the systematic development of explicitly correlated electron-hole wave function based methods and the application of these methods to chemical systems with an emphasis on nanoparticles. The understanding of the basic physics of excited electronic states is an important consideration when developing new methods and applications. In this dissertation, excited electronic states were studied using the electron-hole quasiparticle representation. Theoretical treatment of electronic excitation in large quantum dots and nanoparticles is challenging because of the large number of electrons in the system. The quasiparticle representation provides an alternative representation that can partially alleviate the computational bottleneck associated with investigating these systems. However, in this representation, the effects of electron-hole correlation must be understood in order to accurately describe the system\u27s optical and electronic properties. The electron-hole wave function consists of two separate mathematical components which are the explicitly correlated part of the wave function and the reference wave function which is operated on by the explicitly correlated operator. This dissertation presents theoretical development of both of these components. In the first part, a systematic formulation for deriving the explicitly correlated form of the electron-hole wave function was performed. Towards that goal, the electron-hole correlation length was defined using the electron-hole cumulant. The construction of explicitly correlated wave function was improved by the introduction of the electron-hole correlation length which was determined using the electron-hole cumulant. The electron-hole correlation length allowed the determination of parameters in the explicitly correlated operator without the performance of energy minimizations. In the second part, the electron-hole reference wave function was improved by combining full configuration electron-hole wave function with the explicitly correlated operator. The developed methods were used to investigate the quantum-confined Stark effect (QCSE) and the effect of pH on the optical properties of quantum dots. The effect of applied electric fields on nanoparticles is known as the quantum-confined Stark effect. In this dissertation, the effect of both homogeneous and inhomogeneous electric fields on the optical and electronic properties of quantum dots was investigated. The effect of electric fields on the optical and electronic properties of a GaAs quantum dot was determined by combining the variational polaron transformation with the explicitly correlated electron-hole wave function. The presence of charged ligands also influenced the optical properties of quantum dot and this effect is known as the ligand-induced quantum-confined Stark effect. In this dissertation, the effect of pH on the optical properties of functionalized quantum dots were investigated by first calculating the charged states of the surface ligands at a given pH and then performing electron-hole explicitly correlated wave function based calculations in the electrostatic field generated by the charged ligands. Theoretical methods developed in this dissertation have impacted the field of computational nanoscience by reducing the computational bottleneck to investigate nanoparticles and by providing novel avenues for improving accuracy of existing methods

Molecular Dynamics Simulation

Condensed matter systems, ranging from simple fluids and solids to complex multicomponent materials and even biological matter, are governed by well understood laws of physics, within the formal theoretical framework of quantum theory and statistical mechanics. On the relevant scales of length and time, the appropriate ‘first-principles’ description needs only the Schroedinger equation together with Gibbs averaging over the relevant statistical ensemble. However, this program cannot be carried out straightforwardly—dealing with electron correlations is still a challenge for the methods of quantum chemistry. Similarly, standard statistical mechanics makes precise explicit statements only on the properties of systems for which the many-body problem can be effectively reduced to one of independent particles or quasi-particles. [...

Electron counting statistics of open quantum systems

Electron transport through organic molecules is a process fundamental to life and plays a central role in the emerging field of molecular electronics. This thesis presents an investigation of electron transport through molecular systems from the perspective of full counting statistics. An extension of a Markovian counting statistics framework to a non-perturbative setting is presented which allows for an exact treatment of the phonon bath. This framework is applied to a theoretical photocell device inspired by the photosystem II reaction centre. It is demonstrated that the asymmetric coupling of excitation and charge transfer states to a structured spectral density rather than a smooth low energy background has the effect of reducing the output current along with an associated reduction in the current fluctuations. The insights gained from this are discussed in terms of design principles for pigmentprotein complexes used in nano-electronic devices and their relevance for biological function in vivo. Finally, the asymmetric coupling of excitation and charge transfer states to their vibrational environment is investigated more closely through the dynamics of a dimer model and the effect of the output current statistics of a prototype photocell