Transfer Matrix Algorithm for Computing the Geometric Quantities of a Square Lattice Polymer

Julian Lee 한국물리학회 2018 THE JOURNAL OF THE KOREAN PHYSICAL SOCIETY Vol.73 No.12

I develop a transfer matrix algorithm for computing the geometric quantities of a square lattice polymer with nearest-neighbor interactions. The radius of gyration, the end-to-end distance, and the monomer-to-end distance were computed as functions of the temperature. The computation time scales as . 1:8N with a chain length N, in contrast to the explicit enumeration where the scaling is 2:7N. Various techniques for reducing memory requirements are implemented.

Steady state of an auto-regulatory gene network with partial binding and transcription leakage

Lee Julian 한국물리학회 2021 THE JOURNAL OF THE KOREAN PHYSICAL SOCIETY Vol.78 No.5

A positive feedback loop, where the gene product directly activates its own production, is the simplest form of a gene regulatory network that exhibits a bistable behavior. An analytic solution for the steady state can be obtained only under various simplifying assumptions, such as simultaneous binding of the activator molecules to the gene or complete inactivation of the gene in the absence of protein binding. Here, I consider a feedback loop without such unrealistic simplification and analyze the fixed point structure of the resulting rate equation. The current model is the most general deterministic model that has been solved analytically. I find that the bifurcation diagram is qualitatively different from that obtained in the absence of transcription leakage.

Protein Structure Prediction Based on Fragment Assembly and $\beta$-Strand Pairing Energy Function

Julian Lee,Jooyoung Lee,Seung-Yeon Kim 한국물리학회 2005 THE JOURNAL OF THE KOREAN PHYSICAL SOCIETY Vol.46 No.3

We develop an improved version of PROFESY, a novel method for ab-initio prediction of protein tertiary structures based on fragment assembly and global optimization. In contrast to the primitive version presented earlier, where the hydrogen bond was dened only in terms of inter-atom distance, the angle dependence is now correctly incorporated. This new feature allows us to obtain low-energy conformations with a reasonable number of beta strands, in contrast to the earlier version in which the fraction of alpha helices was excessively large on average. In order to enhance the performance of the prediction method, we optimize the linear parameters of the energy function so that nativelike conformations become energetically more favorable than non-native ones for proteins with known structures. We test the feasibility of the parameter optimization procedure by applying it to a training set consisting of two proteins of structural class + : 1FSD and 1PQS. We use the resulting parameter set for the jackknife test on several proteins from various structural classes. The results are quite promising. In particular, for protein 2GB1, the predictions improve dramatically with the optimized parameter set compared to the original parameters, despite the fact that 2GB1 was not included in the training set. This result suggests that parameters trained for a relatively small number of proteins are transferrable to other proteins to some extent.

Bifurcations of a Positive Feedback Loop with Partial Binding

Julian Lee 한국물리학회 2020 THE JOURNAL OF THE KOREAN PHYSICAL SOCIETY Vol.76 No.8

The positive feedback loop with cooperative binding of the auto-regulatory transcription factor molecules, is the simplest form of a genetic regulatory network (GRN) that can exhibit bistable behavior. The steady states of such systems have been analyzed, but only under the simplifying assumption that the protein molecules bind to the gene simultaneously. In this work, I consider a positive feedback loop where two molecules of the gene product bind to the gene, but one molecule at a time, to activate its own production. I find an analytic expression for the stable fixed point of the corresponding rate equation, and obtain the full phase diagram. Both the model with one-molecule binding and the model with the simultaneous binding of two molecules are shown to be the special limits of the current model. Moreover, the current model exhibits novel bifurcation behavior unseen in previous models.

Exact partition function zeros of two-dimensional lattice polymers

Julian Lee 한국물리학회 2004 THE JOURNAL OF THE KOREAN PHYSICAL SOCIETY Vol.44 No.32

We study the zeros of the exact partition function of lattice polymers on two-dimensional square lattices up to the chain length 28. We observe that the leading zeros tend to approach the real axis as the chain length increases. The results suggest that the locus of zeros may intersect the real axis in the limit of innite chain length, which is the necessary condition for the existence of the collapse transition.

Conformational Space Annealing and a Lattice Model Protein

Julian Lee 한국물리학회 2004 THE JOURNAL OF THE KOREAN PHYSICAL SOCIETY Vol.45 No.3

The conformational space annealing (CSA) method is a powerful global optimization method for sampling low-lying local minimum energy conformations of a physical system. In this work, I apply CSA to the study of a two-dimensional HP lattice model of a protein, where a conformation is dened as a self-avoiding chain on a lattice. I study the 36-residue chain with a particular sequence HPH2P2H2P3H2PHP3H2P2H2P4HPH2PHP2 presented by Li et al., for which by exhaustive enumeration of compact conformations, only one such conformation with the lowest energy was shown to exist. The CSA algorithm nds conformations with energies lower than those found by Li et al. for 100 independent runs, demonstrating that the global minimum energy conformation is not necessarily the most compact structure.

Global Optimization of a Three-Dimensional HP Protein

Julian Lee 한국물리학회 2008 THE JOURNAL OF THE KOREAN PHYSICAL SOCIETY Vol.52 No.2

The conformational space annealing (CSA) method is an efficient global optimization algorithm for sampling low-lying local-minimum-energy conformations of a physical system. In this work, the CSA method is applied to the three-dimensional HP lattice model of a protein, where a conformation is defined as a self-avoiding chain on the three-dimensional cubic lattice. The 27-residue chain with a particular sequence H₂PHP₂HPHPHPH₂P₃HP₂H₃P₂HP is considered, for which compact conformations filling a 3 × 3 × 3 cube have been exhaustively enumerated in an earlier work, and only one compact conformation with the lowest energy has been shown to exist. The CSA algorithm finds non-compact conformations with energies lower than any compact conformation, for several independent runs, proving that the global minimum energy conformation is not a compact structure. The conformational space annealing (CSA) method is an efficient global optimization algorithm for sampling low-lying local-minimum-energy conformations of a physical system. In this work, the CSA method is applied to the three-dimensional HP lattice model of a protein, where a conformation is defined as a self-avoiding chain on the three-dimensional cubic lattice. The 27-residue chain with a particular sequence H₂PHP₂HPHPHPH₂P₃HP₂H₃P₂HP is considered, for which compact conformations filling a 3 × 3 × 3 cube have been exhaustively enumerated in an earlier work, and only one compact conformation with the lowest energy has been shown to exist. The CSA algorithm finds non-compact conformations with energies lower than any compact conformation, for several independent runs, proving that the global minimum energy conformation is not a compact structure.

Optimization of Potential-Energy Parameters for Folding of Several Proteins

Julian Lee,Jooyoung Lee,Seung-Yeon Kim 한국물리학회 2004 THE JOURNAL OF THE KOREAN PHYSICAL SOCIETY Vol.44 No.32

We introduce a novel approach to the study of the folding of proteins whose native structures are already known. We use an o-lattice atomistic potential energy. The parameters of the potential energy are simultaneously optimized for several proteins. The low-lying local-energy minima for these proteins are found by conformational space annealing. The parameters are modied in such a way that the native-like conformations are energetically more favored than the others. After the parameter optimization, one set of the parameters is obtained for the proteins. We then investigate Monte Carlo dynamics of these proteins by using this optimized potential energy. Our work is distinguished from earlier work in the literature, where folding was achieved by using simplied models such as lattice models. We apply our method to four proteins: betanova, 1fsd, 1vii, and 1bdd, and observe that at appropriate temperatures they fold into their native structure, starting from various non-native states. In all cases, rapid collapse is followed by a subsequent folding process, that takes place on a longer timescale. We also observe that for all proteins at low temperatures, the probability distributions of various quantities such as RMSD depend on initial conformations, showing their glassy behavior. At higher temperatures, this non-ergodic glassy behavior disappears. The results provide new insights into the folding mechanism, which is controlled not only by thermodynamic factors but also by kinetic factors. The way a protein folds into its native structure is also determined by the convergence point of early folding trajectories, which cannot be obtained from the free-energy surface.

Analytic Form of the Quasi-stationary Distribution of a Simple Birth-Death Process

Lee Julian 한국물리학회 2020 THE JOURNAL OF THE KOREAN PHYSICAL SOCIETY Vol.77 No.6

I consider a simple birth-death model with an absorbing state, where the stable fixed point of the corresponding deterministic mean-field dynamics turns into a transient peak of the probability distribution due to the presence of a tiny fluctuation. The model satisfies the detailed-balance condition, enabling one not only to obtain the analytic form of a quasi-stationary distribution, but also to obtain the analytic form of the escape time under the assumption of quasi-stationarity. I argue that the quasi-steady distribution with exponentially decaying normalization is an excellent approximation of the dynamics at late times, especially for small fluctuations. The analytic expressions for the quasi-stationary distribution and the escape time are expected to be more accurate, hence more useful, for systems with larger sizes.

Phase Diagram of the Wako-Saitˆo-Mu˜noz-Eaton Hairpin Model Obtained from the Partition-function Zeros

Julian Lee 한국물리학회 2014 THE JOURNAL OF THE KOREAN PHYSICAL SOCIETY Vol.65 No.5

I study the partition-function zeros of the Wako-Saitˆo-Mu˜noz-Eaton (WSME) hairpin modelin the complex temperature plane when the number of native contacts is 10. For various values ofthe entropy cost of disordering a bond, the zeros show a clear locus corresponding to the foldingtransition. The result suggests a new definition of the transition temperature, the temperatureobtained by extrapolating the locus to the positive real axis. By computing the transition temperaturefor various values of the entropy cost, we obtain the phase diagram of the WSME hairpinmodel. I compare the present definition of the transition temperature with the transition midpointtemperature for several values of the entropy parameter and show that they are in a reasonablygood agreement.