**Contemporary Mathematics**

Volume: 746;
2020;
246 pp;
Softcover

MSC: Primary 57; 05; 92; 82; 65;

**Print ISBN: 978-1-4704-4840-0
Product Code: CONM/746**

List Price: $120.00

AMS Member Price: $96.00

MAA Member Price: $108.00

**Electronic ISBN: 978-1-4704-5456-2
Product Code: CONM/746.E**

List Price: $120.00

AMS Member Price: $96.00

MAA Member Price: $108.00

# Topology and Geometry of Biopolymers

Share this page *Edited by *
*Erica Flapan; Helen Wong*

This book contains the proceedings of the AMS
Special Session on Topology of Biopolymers, held from April
21–22, 2018, at Northeastern University, Boston, MA.

The papers cover recent results on the topology and geometry of DNA
and protein knotting using techniques from knot theory, spatial graph
theory, differential geometry, molecular simulations, and laboratory
experimentation. They include current work on the following topics:
the density and supercoiling of DNA minicircles; the dependence of DNA
geometry on its amino acid sequence; random models of DNA knotting;
topological models of DNA replication and recombination; theories of
how and why proteins knot; topological and geometric approaches to
identifying entanglements in proteins; and topological and geometric
techniques to predict protein folding rates.

All of the articles are written as surveys intended for a broad
interdisciplinary audience with a minimum of prerequisites. In
addition to being a useful reference for experts, this book also
provides an excellent introduction to the fast-moving field of
topology and geometry of biopolymers.

#### Readership

Graduate students and research mathematicians interested in applications of topology and DNA proteins.

# Table of Contents

## Topology and Geometry of Biopolymers

- Cover Cover11
- Title page iii4
- Contents v6
- Preface vii8
- Part I . The Topology and Geometry of DNA 112
- Beyond the static DNA model of Watson and Crick 314
- Characterizing the topology of kinetoplast DNA using random knotting 1728
- 1. Introduction 1728
- 2. Review of previous works 1829
- 3. Mathematical models for testing the confinement hypothesis 2132
- 4. Results: The linking probability of two minicircles 2637
- 5. Results: The mean valence of a minicircle network 2940
- 6. Results: Percolation and saturation phenomena in a minicircle network 3243
- 7. Applications to kDNA 3647
- 8. Discussion 3748
- Acknowledgments 3849
- References 3849

- Did sequence dependent geometry influence the evolutionof the genetic code? 4152
- 1. Measuring the efficiency of duplexed codes 4152
- 2. A natural example of duplexed codes associated to DNA 4455
- 3. How well duplexed is the real genetic code as compareswith alternatives? 4960
- 4. New results: Mutual information via a Monte-Carlostyle discretization 5263
- 5. Conclusions 5566
- References 5566

- Topological sum rules in the knotting probabilities of DNA 5768
- 1. Introduction 5768
- 2. Numerical methods 6071
- 3. Review on the formula of the knotting probability and the sum rules 6374
- 4. Numerical support for the sum rules 6576
- 5. Trefoil knot dominance in thick ring polymer chains 6879
- 6. Fitting parameters being close to those of the asymptotic expansion 7384
- 7. Gradual monotonic flow of the knot exponent toward that of the large excluded volume 7586
- 8. Discussion 7889
- Appendix A. Estimates of the parameters in asymptotic expansion (1.5) 7889
- Acknowledgments 8192
- References 8192

- Knotting of replication intermediates is narrowly restricted 8596
- Recent advances on the non-coherent band surgery model for site-specific recombination 101112
- 1. Introduction 101112
- 2. DNA recombination 103114
- 3. The band-surgery model for site-specific recombination 108119
- 4. Band surgery obstructions via Heegaard Floer homology 114125
- 5. The frequency of chirally cosmetic bandings, as assessed by numerical simulations 118129
- Acknowledgments 120131
- References 120131

- Part II . The Topology and Geometry of Proteins 127138
- Why are there knots in proteins? 129140
- Knotted proteins: Tie etiquette in structural biology 155166
- 1. Introduction 155166
- 2. Structure-based models 158169
- 3. Knotting physics and why knotted proteins are rare 160171
- 4. First insights into the folding of knotted proteins via molecular simulation 163174
- 5. Insights into the knotting mechanism from molecular simulations 164175
- 6. Knot type and knotting mechanism 167178
- 7. Folding properties of knotted proteins 168179
- 8. Towards a mechanistic understanding of knotting in vivo 170181
- 9. Co-translational folding of knotted proteins 173184
- 10. Functional advantages of knots in proteins 174185
- 11. Conclusions and future prospects 175186
- Acknowledgments 176187
- References 176187

- Knotoids and protein structure 185196
- 1. Introduction 185196
- 2. Knotoids 186197
- 3. Knotoid invariants 188199
- 4. Embedded open curves in 3-space and knotoids 192203
- 5. Application to protein studies 194205
- 6. Knotoids and intra-chain protein bonds 196207
- 7. Knoto-ID: the first software for the analysis of open ended curves using knotoids. 197208
- Acknowledgements 198209
- References 198209

- Topological linking and entanglement in proteins 201212
- 1. Introduction 201212
- 2. Linking entanglement in families of macromolecules 203214
- 3. Lassos and LassoProt: An analysis of linking entanglementin the presence of a single bridge 214225
- 4. LinkProt: An analysis of linking entanglementin the presence of multiple bridges 215226
- 5. PyLasso & PyLink: PyMol tools to study linking entanglement due to the presence of one or more bridges in proteins 216227
- 6. Conclusion 220231
- References 221232

- A topological study of protein folding kinetics 223234
- Back Cover Back Cover1248