Persistence homology of entangled rings

Abstract

Topological constraints (TCs) between polymers determine the behavior of complex fluids such as creams, oils, and plastics. Most of the polymer solutions used in everyday life employ linear chains; their behavior is accurately captured by the reptation and tube theories which connect microscopic TCs to macroscopic viscoelasticity. On the other hand, polymers with nontrivial topology, such as rings, hold great promise for new technology but pose a challenging problem as they do not obey standard theories; additionally, topological invariance, i.e., the fact that rings must remain unknotted and unlinked if prepared so, precludes any serious analytical treatment. Here we propose an unambiguous, parameter-free algorithm to characterize TCs in polymeric solutions and show its power in characterizing TCs of entangled rings. We analyze large-scale molecular dynamics simulations via persistent homology, a key mathematical tool to extract robust topological information from large datasets. This method allows us to identify ring-specific TCs which we call "homological threadings" (H-threadings) and to connect them to polymer behavior. It also allows us to identify, in a physically appealing and unambiguous way, scale-dependent loops which have eluded precise quantification so far. We discover that while threaded neighbors slowly grow with ring length, the ensuing TCs are extensive also in the asymptotic limit. Our proposed method is not restricted to ring polymers and can find broader applications for the study of TCs in generic polymeric materials.

Efficient compressed database of equilibrated configurations of ring-linear polymer blends for MD simulations

“…For such analyses, it is necessary to develop mathematical methods that can accurately evaluate how the linear chain penetrates the ring polymer; this has long been an important problem in the mathematics of topology 5 – 14 . The relevance of this task is not limited to ring-linear polymer blends 13 , 14 ; research on knots in proteins 15 – 18 , threading of ring polymers 19 – 22 , and cross-linked networks 23 , 24 is greatly concerned with the linkage between loops and chains owing to its impact on the material properties. Therefore, public availability of MD coordinate data is expected to promote the development of analysis methods by applied mathematicians.…”

Persistent Homology Analysis for Materials Research and Persistent Homology Software: HomCloud

“…As a study that focuses on the structure itself, a method to quantify the threading of ring polymers has been proposed. 40)…”

“…Of course, other data than materials science is also available. HomCloud has been already used in various scientific researches, including materials science, 17,34,35,37,38,40,[61][62][63][64][65][66][67][68] geology, 20,69) structural biology, 70) and medical image analysis. 71,72) HomCloud has useful functionalities such as visualization, inverse analysis, machine learning.…”

Quantifying the mobility of chromatin during embryogenesis: Nuclear size matters

“…Rather, the chromosome conformational capture experiments invoke the abundance of large loop structures on the order of Mbp [23], which suggests the similarity with the dense solution of non-concatenated ring polymers [24,25]. In the latter, rings are not entangled in a classical sense, but their conformation and dynamics are dictated by the immense TCs [12,[26][27][28][29][30][31].…”

“…where a is a characteristic length scale relevant to the process under consideration, τ e is the corresponding time scale, and α is a nontrivial exponent characterizing the large-scale loci dynamics [4][5][6][7][8][9][10][11]. Over long length and time scales, the conformation and dynamics of chromatin will be dominated by TC [12,[24][25][26][27][28][29][30][31], which set our basic length scale to be a so-called tube diameter [21,22]. In a larger length scale (r > a), the chromatin will exhibit a non-trivial conformation, which also affects the motion of chromatin loci at long time scale (t > τ e ).…”