分形polymer链和fBm轨迹中的多体接触
Many-body contacts in fractal polymer chains and fBm trajectories
摘要：
本文计算了具有任意分形维数$d_f$的分数布朗运动的轨迹在给定时刻$t_1, ..., t_n$到达同一点$n \ge 3$次的概率，并得到了这些概率的位移-位移协方差矩阵的行列式表达式.除了具有$d_f = 2$的标准布朗轨迹外，所得到的多体接触概率不能被因式分解为单圈贡献的乘积。在我们最近提出的自相互作用聚合物链的高斯网络模型中，我们在这里计算的概率可以解释为具有相同分维$d_f$的分形聚合物构象中多体接触的概率。这种高斯方法意味着从分数布朗运动轨迹到聚合物构象的映射，可以用作拓扑稳定构象中聚合物链的半定量模型，例如，在非连接环的熔化物中或在染色质纤维中，染色质纤维是包含遗传信息的物质介质。因此，这里提出的模型可以用作解释基因组中多体接触数据的基准，我们预计很快就会在，例如，Hi-C实验
英文摘要：
We calculate the probabilities that a trajectory of a fractional Brownian motion with arbitrary fractal dimension $d_f$ visits the same spot $n \ge 3$ times, at given moments $t_1, ..., t_n$, and obtain a determinant expression for these probabilities in terms of a displacement-displacement covariance matrix. Except for the standard Brownian trajectories with $d_f = 2$, the resulting many-body contact probabilities cannot be factorized into a product of single loop contributions. Within a Gaussian network model of a self-interacting polymer chain, which we suggested recently, the probabilities we calculate here can be interpreted as probabilities of multi-body contacts in a fractal polymer conformation with the same fractal dimension $d_f$. This Gaussian approach, which implies a mapping from fractional Brownian motion trajectories to polymer conformations, can be used as a semiquantitative model of polymer chains in topologically-stabilized conformations, e.g., in melts of unconcatenated rings or in the chromatin fiber, which is the material medium containing genetic information. The model presented here can be used, therefore, as a benchmark for interpretation of the data of many-body contacts in genomes, which we expect to be available soon in, e.g., Hi-C experiments.
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