Dynamics of the biomolecules are complex, which enables the complex func-tionality of the biomolecular systems. One of the best ways to understand suchcomplex systems is, simply, seeing how it works. Molecular simulation enables usto make a _sandbox_ of the proteins inside computers, with which we researcherscan observe its intrinsic motions or its response to changes in the system; we cansee how it works. Combined with other experimental techniques, it nowadaysbecame a mature and an essential tool for protein science.

One of the problems with the molecular simulation is how we understandthe results. Proteins are high dimensional systems, and the simulations of them generate even tera-bytes sized data within a week. The ways to extract datafrom the enormous data space is necessary, and the result of extraction has to be human-understandable. Herein, I propose two approaches to cope with the problem. One is a method to estimate the stability of the protein's substates, without requiring long sequential run. The other is a method to determine the best axes to represent protein's dynamics, or reaction coordinates of the proteins, automatically from the simulation.

1 Combining Multiple Non-equilibrium Dynam-ics Simulation Based on Markov Model

In the molecular simulation, the snapshots of the system are taken with interval,which consists of trajectory. Trajectory is considered as the representative set for the molecule. However, even with today's hardware and simulation algo-rithm, there are still a large gap between the time-frame of simulation and that of protein's functional motion, such as enzyme activity or receptor binding. Pos-sible workaround is to improve sampling with modi_ed dynamics. These meth-ods, usually called generalized ensemble, sample snapshots from simulations with modi_ed dynamics; after obtaining data, each snapshot of the trajectory is weighed according to the magnitude of modi_cation. The pair of trajectory and weigh represents the protein's character. One of the drawbacks of these methods is that methods rely on the equivalence of statistical ensemble average and time average. Because of this, for the large systems these methods require long equilibration time before researchers can start sampling snapshots. Thus generalized ensemble is at this moment impractical for the biomolecules.

In this research I pursued the new analysis method which calculates the statistical ensemble of the system, but which does not require total equilibration of the system. Starting from the assumption of Markovity, the dynamics of the system can be represented by a transition matrix M. The equilibrium probability density _ can be obtaind by the left diagonalization of the M, since_ is the left eigenvector of M corresponding to eigenvalue 1.

Simple form to compute equilibrium probability distribution was therefore ob-tained. Extending this form to combine multiple simulations, I developed a method called Multiple Markovian transition Matrix Method (MMMM), based on the error analysis of the eigenproblems. MMMM was compared with other state-of-art ensemble value determination methods, and it showed better result with the case that the equilibration cannot be expected. Also, the method was tested with peptide energy landscape as an application, giving proper transition state structure. With these results, I showed the method is applicable to even a practical case.

2 Decomposing Protein into Components with Correlations

In section 1, reaction coordinates were considered to be given, but _nding proper reaction coordinates is a non-trivial task. Also, because of the curse of dimen-sionality, above problem does not scale well. The number of states blows-up with the increase of dimension. One of the reasons is that proteins consist of strongly coupled components. If there are many uncoupled components in a protein, each component can be analyzed in divide-and-conquer fashion. In reality, it is non-trivial task to _nd uncoupled components.

In order to identify the correlated modes and to decompose proteins into uncoupled components, I borrowed an idea from the _eld of the signal process-ing. In this thesis Independent subspace analysis (ISA) method is introduced to the concept of the biomolecular collective motion. A linear projection of the coordinates x is considered:〓With ISA, projected vector s and its rotation matrix A can be determined so as to minimize the number of strongly correlated modes. Procedure based on subspace joint approximate diagonalization of eigenmatrices (SJADE) algo-rithm is employed to perform ISA. ISA/SJADE dissects multiple dimensions into irreducible _blocks_, in which projection to each dimension have strong cor-relation within the same block, while there are no or very small correlation over di_erent blocks. With ISA/SJADE, the result of 100-ns MD simulation of T4 lysozyme was analyzed for testing purpose. Result showed the modes determined from ISA explain long-ranged correlation of modes, and it success-fully found the modes which are strongly correlated to functionally important residues reported from mutation experiments. From these results, ISA is shown to be powerful technique for analyzing protein dynamics.

Figure 1: Energy landscape of 5-residue peptide Met-enkephalin, obtained with MMMM. Representative structures for each characteristic point is also shown on the right side.

Figure 2: Arrow representation of SJADE mode 5, and the residues which strongly correlated to mode 5. Presented residues are con_rmed to be strongly correlated to the motion of the mode 5.

ペプチドおよび蛋白質の運動を推定し分析することは、膨大な数の原子から構成される3次元構造の持つ複雑性により、困難な問題として残されている。本論文では2つの問題を吟味し、複雑な系を単純化して記述する縮約表現を提案し、分析精度の向上と推定速度の効率化を図ろうとしている。

第一の問題はペプチドの自由エネルギーの推定である。ペプチドは構造変化を起こすと自由エネルギーが多様に変化する。各構造における自由エネルギーを高速に推定できれば、自由エネルギーの地図を描ける。すると安定した構造を把握でき、構造変化のダイナミクスを理解することが容易になる。従来から普及している方法にWHAMがある。WHAMでは複数のシミュレーション結果を組合せることにより平衡確率分布を計算している。しかし個々のシミュレーションの平衡を仮定する制約のため、実験データを精度よく推定できない場合が多い。本論文では、構造を状態、構造間の遷移を確率として表現するマルコフモデルにより平衡確率分布を推定している。詳しくは、個々のシミュレーションから構造間の遷移回数を計数し、確率分布を更新する式を導き、その極限として平衡確率分布を推定している。個々のシミュレーションの平衡性を仮定しない点で前進している。しかし短時間のシミュレーションでは、遷移回数は少ない傾向にあり確率分布が粗くなる。長時間シミュレーションすれば確率分布を高解像度化できるが、効率的でない。この問題を回避するため、短時間のシミュレーションを多数並列に実行して連結し、確率分布推定の誤差を抑えるboosting法を提案している。WHAM法に比べ高速かつ高精度であることを実データにより確認している。

第二の問題は、蛋白質の機能的部分構造の同定である。蛋白質構造の時系列的変化を、原子毎の座標変化として詳細に収集したとする。このとき同期して構造変化を起こす部分構造を同定できれば、機能にかかわる重要な部分や、遠く離れていても同期して変化する部分構造間の相互作用を理解できる可能性がある。そこで、原子や原子の集まりの座標を、互いに独立な部分空間群へと分類し、各部分空間が蛋白質の部分構造を表現するという立場を展開している。しかし原子数が膨大になると、部分構造へ分解する組合せ数も膨らむため、計算が容易でない。本論文では2006年に提案された比較的効率的なSJADE法を利用してこの問題ヘアプローチしている。SJADE法は、相関をキュムラントとして定義し、同時ブロック対角化を実行し、独立な部分空間を導く。本手法の有効性を示すために、T4lysozyme(T4phageの加水分解酵素)のMDシミュレーションデータに適用し、同期して変化する部分構造の特徴づけに成功している。

なお、本論文は北尾彰朗との共同研究であるが、論文提出者が主体となって分析及び検証を行ったもので、論文提出者の寄与が十分であると判断する。したがって、博士(科学)の学位を授与できると認める。