We present a derivation of a coarse-grained description, in the form of a generalized Langevin equation, from the Langevin dynamics model that describes the dynamics of bio-molecules. The focus is placed on the form of the memory kernel function, the colored noise, and the second fluctuation-dissipation theorem that connects them. Also presented is a hierarchy of approximations for the memory and random noise terms, using rational approximations in the Laplace domain. These approximations offer increasing accuracy. More importantly, they eliminate the need to evaluate the integral associated with the memory term at each time step. Direct sampling of the colored noise can also be avoided within this framework. Therefore, the numerical implementation of the generalized Langevin equation is much more efficient.

This paper considers the reduction of the Langevin equation arising from bio-molecular models. To facilitate the construction and implementation of the reduced models, the problem is formulated as a reduced-order modeling problem. The reduced models can then be directly obtained from a Galerkin projection to appropriately defined Krylov subspaces. The equivalence to a moment-matching procedure, previously implemented in , 2), is proved. A particular emphasis is placed on the reduction of the stochastic noise, which is absent in many order-reduction problems. In particular, for order less than six we can show the reduced model obtained from the subspace projection automatically satisfies the fluctuation-dissipation theorem. Details for the implementations, including a bi-orthogonalization procedure and the minimization of the number of matrix multiplications, will be discussed as well.There are multiple benefits from such an approach. For example, reduced models can capture directly the dynamics of certain quantities of interest. Secondly, with the reduction of the dimension, the computational cost can be reduced dramatically. In addition, the quantities of interest often correspond to slow variables. By eliminating fast variables, the time step can also be increased considerably. This allows one to access longer time scales ?.There has been tremendous recent progress in the development of coarse-grained models , 7, 9, 1, 3, 7,?, 5); , 0). Most effort, however, is thermodynamics based. Namely, one aims to construct the free energy associated with the reduced variables, which then yields the driving force for the reduced dynamics, known as the potential of mean forces (PMF) , 3, 4). As pointed out in , 3, 4), the damping mechanics, which also plays an important role in the reduced dynamics, is not part of the construction.In this work, we are interested in an equation-based derivation, where the reduced model can be derived directly from the Langevin dynamics. Deriving reduced models from a stochastic dynamical system has been a subject of extensive studies, the most well known of which is the homogenization approach , 9). Another important approach is to employ a coordinate transformation using normal forms to separate out the degrees of freedom that are less relevant , 3). More recently, Legoll and Lelievre proposed to use conditional expectations to derived reduced models , 5). Overall, these methods require either significant scale separation assumption, or simple functions forms in the stochastic differential equations, which for bi-molecular models, does not apply. For example, the force-field for biomolecular models typically involves complicated function forms.Meanwhile, in the field of molecular modeling there are also many methods that were proposed to coarse-grain a molecular dynamics model. Most of these methods are derived from a Hamiltonian system of ODEs ,9,1,7,8,4); ; ; , either motivated by or directly obtained, from the Mori-Zwanzig projection formalism , 5, 9). Strictly speaking, such a procedure...

We present the reduction of generalized Langevin equations to a coordinateonly stochastic model, which in its exact form, involves a forcing term with memory and a general Gaussian noise. It will be shown that a similar fluctuation-dissipation theorem still holds at this level. We study the approximation by the typical Brownian dynamics as a first approximation. Our numerical test indicates how the intrinsic frequency of the kernel function influences the accuracy of this approximation. In the case when such an approximate is inadequate, further approximations can be derived by embedding the nonlocal model into an extended dynamics without memory. By imposing noises in the auxiliary variables, we show how the second fluctuation-dissipation theorem is still exactly satisfied.

In this study, the DNA logic computing model is established based on the methods of DNA self-assembly and strand branch migration. By adding the signal strands, the preprogrammed signals are released with the disintegrating of initial assembly structures. Then, the computing results are able to be detected by gel electrophoresis. The whole process is controlled automatically and parallely, even triggered by the mixture of input signals. In addition, the conception of single polar and bipolar is introduced into system designing, which leads to synchronization and modularization. Recognizing the specific signal DNA strands, the computing model gives all correct results by gel experiment. In recent years, with the approaching of the Moore's Law, the pressure of traditional electronic computers increased greatly for handling mass information. The interests of researchers have been attracted to the area of novel computing. Taking advantage of some new methods of quantum and molecular computing, researchers attempt to use nano-materials and technologies to implement super large scale information processing. In particular, DNA computing has become a research focus in molecular computing, combined with information science, biology and nanotechnology [1][2][3][4][5]. Because the DNA molecules have lots of natural advantages in huge parallelism and microscopic, the mass parallel information processing could be achieved by using DNA computing. Thus, DNA computing may become a flourishing route in future computing. In fact, since the end of last century, it has made a great progress, both in theoretical models and experimental operations of DNA computing. In addition, it has developed greatly in the interdisciplinary fields of information processing, molecular encoding, nanomachines and so on [6][7][8][9][10][11]. In the development of DNA computing, a variety of molecular operations have been utilized such as polymerase chain reaction (PCR), fluorescence techniques, strand branch migration and self assembly techniques. In these methods, DNA strand branch migration with fluorescent labeling develops rapidly for constructing various molecular computing models [12][13][14][15]. Importantly, Professor Winfree used DNA strand branch migration to implement simple squareroot computing and neural networks computing in 2011. These works were reported in Science and Nature, and attracted lots of attentions from researchers in the field of information computing [4,5]. DNA strand branch migration is able to be combined with not only fluorescent detecting and DNA self-assembly, but also nano-particles, quantum dots and proteins. Moreover, it has promoted the development of research fields as parallel computing, cryptography and nanoelectronics.

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