Masaaki Yoshida’s research while affiliated with Saitama University and other places

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Publications (4)


Figure 2. Quantification of the time-course experiments 9. The gel presented in Fig. 1B in Grainge et al. 9 showed a time course of unlinking by XerCD-dif-FtsK50C at 25 °C of newly replicated plasmids containing dif sites. Line scans of the gel were previously published 10. In this figure each topological class is shown as a separate series of points with linear interpolation. The caption assumes the bands observed correspond to the topologies expected from a substrate composed of replication links, i.e. 2m-crossing links (e.g. 2m-cats), and some of the corresponding knotted intermediates (open circle or 0 1 , 3 1 , 5 1 ). " Unlink " corresponds to the two unlinked components in monomeric state (topology type 0 1 2 
Figure 4. The substrate at the top left corner is the link 6 1 2 with two reconnection sites in parallel orientation. The pathways are represented as an oriented graph where the nodes are the knot or link types, and two nodes are connected by an edge if one can go from one to the other via a reconnection event. The substrate and product of each reconnection are indicated by the orientation of the edges. The diagrams above each edge illustrate an example of the corresponding reconnection event by showing the band where the band surgery will be performed. The weights on the edges correspond to transition probabilities obtained numerically. Details of the simulations are in the Numerical Methods section below, and in the Supplementary Methods and Supplementary Data.
(A) Under the assumption that each reconnection step strictly reduces the number of crossings of the substrate, in Shimokawa et al.¹⁰ we showed that there is a unique unlinking pathway starting at a 2m-crossing replication link. In E. coli a replication link is a 2m-cat with parallel dif sites⁶, and this pathway predicts the first product to be a (2m−1)1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{(2}m-\mathrm{1)}}_{1}$$\end{document} knot with two dif sites in direct repeats. Two sites along a knotted chain are in direct repeats if they induce the same orientation into the knot. Replication links are 2m-crossing right-handed torus links with parallel sites (mathematical notation: (2m)12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{(2}m)}_{1}^{2}$$\end{document}). The pathway in the figure illustrates, for m = 6, the only unlinking pathway starting at the parallel 2m-cat under the assumption that each reconnection step strictly reduces the minimal crossing number. All the intermediate topologies are torus links (2m)12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{(2}m)}_{1}^{2}$$\end{document} or torus knots (2m−1)1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{(2}m-\mathrm{1)}}_{1}$$\end{document} with two reconnection sites in direct repeats as in the figure. (B) One reconnection step: here the cleavage regions of the reconnection sites on a 612\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${6}_{1}^{2}$$\end{document} link are brought together to form a synapse (shown as a ball enclosing two strings). The synapse is modeled mathematically as a 2-string tangle. In the case of XerCD site-specific recombination, the strings in the tangle contain the core regions of the dif sites (indicated by two arrows in a tangle P representing two very short segments of double-stranded DNA which physically behave as two almost straight strings) and any bound DNA which does not change during recombination (gray shaded region). Any interesting geometrical or topological complexity of the substrate is captured mathematically as an outside tangle O that remains constant during reconnection. Before strand cleavage, the substrate is modeled by the tangle equation N(O+P)=612\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N(O+P)={6}_{1}^{2}$$\end{document}. The local reconnection is modeled by tangle surgery where P is replaced with R, yielding a product represented as N(O+R)=K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N(O+R)=K$$\end{document}, where K is a knot with two directly repeated sites. (C) Local reconnection is a simple event which can be modeled as a band surgery, where P = (0) is replaced with a tangle R = (w, 0) enclosing a vertical row of w twists, for some integer w. The rational tangle notation (or Conway notation) for such vertical tangle is R = (w, 0). In the case when w=±1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w=\pm 1$$\end{document} the notation simplifies to R=(±1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R=(\pm \mathrm{1)}$$\end{document}. In the simplest cases, P = (0) with sites in parallel alignment goes to R=(±1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R=(\pm \mathrm{1)}$$\end{document}, and P = (0) with sites in anti-parallel alignment goes to R=(0,0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R=\mathrm{(0,}\,\mathrm{0)}$$\end{document} as illustrated in the figure.
Quantification of the time-course experiments⁹. The gel presented in Fig. 1B in Grainge et al.⁹ showed a time course of unlinking by XerCD-dif-FtsK50C at 25 °C of newly replicated plasmids containing dif sites. Line scans of the gel were previously published¹⁰. In this figure each topological class is shown as a separate series of points with linear interpolation. The caption assumes the bands observed correspond to the topologies expected from a substrate composed of replication links, i.e. 2m-crossing links (e.g. 2m-cats), and some of the corresponding knotted intermediates (open circle or 01, 31, 51). “Unlink” corresponds to the two unlinked components in monomeric state (topology type 012\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${0}_{1}^{2}$$\end{document}), and “Unknot” corresponds to the dimeric unknot (01). The quantification clearly illustrates the reduction of replication links by XerCD-FtsK site-specific recombination at dif sites. The complexity of the data is also evident, with the relative proportions of all the different topologies fluctuating from one step to the next, thus obscuring the signal.
(A) Illustration of some of the knots relevant to the present study and their nomenclature. The chirality is consistent with that in Brasher et al.²¹. The green arrows along the unknot 01 represent the two reconnection sites. The sites shown are equidistant and in direct repeats. A complete table of prime knots with up to 10 crossings and information on how they compare to those in Rolfsen²⁰ can be obtained from the authors upon request. (B) Nomenclature for two component links relevant to the present study. The green arrows represent the reconnection sites, which confer an orientation to each link component. The nomenclature is described in the Supplementary Methods and in Supplementary Fig. S5. For 2-component links with 9 or more crossings we revert to the default Knotplot naming convention. (C) The four possible combinations of chirality and orientation for the 4-crossing torus link. A comparison between the nomenclature used in this paper and that in Rolfsen²⁰ and in works by Kanenobu28,29 is included in Supplementary Fig. S5. Arrows indicate the relative orientations of the sites.

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Pathways of DNA unlinking: A story of stepwise simplification
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September 2017

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484 Reads

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36 Citations

Robert Stolz

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Masaaki Yoshida

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Reuben Brasher

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[...]

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In Escherichia coli DNA replication yields interlinked chromosomes. Controlling topological changes associated with replication and returning the newly replicated chromosomes to an unlinked monomeric state is essential to cell survival. In the absence of the topoisomerase topoIV, the site-specific recombination complex XerCD- dif-FtsK can remove replication links by local reconnection. We previously showed mathematically that there is a unique minimal pathway of unlinking replication links by reconnection while stepwise reducing the topological complexity. However, the possibility that reconnection preserves or increases topological complexity is biologically plausible. In this case, are there other unlinking pathways? Which is the most probable? We consider these questions in an analytical and numerical study of minimal unlinking pathways. We use a Markov Chain Monte Carlo algorithm with Multiple Markov Chain sampling to model local reconnection on 491 different substrate topologies, 166 knots and 325 links, and distinguish between pathways connecting a total of 881 different topologies. We conclude that the minimal pathway of unlinking replication links that was found under more stringent assumptions is the most probable. We also present exact results on unlinking a 6-crossing replication link. These results point to a general process of topology simplification by local reconnection, with applications going beyond DNA.

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Figure 2: Quantification of the time-course experiments. 10 The gel presented in Fig. 1B in Grainge et al. 10 showed a time course of unlinking by XerCD-dif-FtsK50C at 25 o C of newly replicated plasmids containing dif sites. Line scans of the gel were previously published. 7 In this figure each topological class is shown as a separate series of points with linear interpolation. The caption assumes the bands observed correspond to the topologies expected from a substrate composed of replication links, i.e. 2m-crossing links (e.g. 2m-cats), and some of the corresponding knotted intermediates (open circle or 0 1 ,3 1 , 5 1 ). "Unlink" corresponds to the two unlinked components in monomeric state (topology type 0 2 1 ), and "Unknot" corresponds to the dimeric unknot (0 1 ). The quantification clearly illustrates the reduction of replication links by XerCD-FtsK site-specific recombination at dif sites. The complexity of the data is also evident, with the relative proportions of all the different topologies fluctuating from one step to the next, thus obscuring the signal.
Figure 4: The substrate at the top left corner is the link 6 2 1 with two reconnection sites in parallel orientation. The pathways are represented as an oriented graph where the nodes are the knot or link types, and two nodes are connected by an edge if one can go from one to the other via a reconnection event. The substrate and product of each reconnection are indicated by the orientation of the edges. The diagrams above each edge illustrate an example of the corresponding reconnection event by showing the band where the band surgery will be performed. The weights on the edges correspond to transition probabilities obtained numerically. Details of the simulations are in the Numerical Methods section below, and in the Supplementary Methods and Supplementary Data online.
Figure 5: (A) The substrate (left) is a lattice trefoil with 120 segments and two directly repeated reconnection sites indicated by a white sphere. The product (right) is a 2-component link obtained after one reconnection event. All substrate knots have directly repeated sites that are 60 segments apart, with a tolerance of ±6 segments, and all links have two components 60 ± 6 long so that the sum of the lengths is exactly 120. Reconnection on links is only performed between sites in different components. (B) Circos figure: all reconnection transitions in a minimal pathway from the 9 1 that satisfy Assumption1. 2-component links (resp. knots) are arranged by increasing crossing number from bottom to top in the left (resp. right) hemisphere, and are color-coded blue (resp. red). Color intensity increases with decreasing crossing number. An arc between K and L indicates at least one observed reconnection event between K and L. The thickness of the arcs corresponds to the directed transition probability between two topologies. Transitions with an observed probability < .2 are thickened to be more visible. Transitions are colored according to the probability of the most probable minimal pathway they are a member of. The first, second, and third most probable unlinking pathways from 9 1 are colored red, orange, and yellow, respectively. If no arc appears between a pair {K, L}, this means that no reconnection between them was observed. Observed transitions for all substrate topologies, including those in non-minimal pathways, are included in Supplementary Data and in Fig. S6 in the Supplementary Methods online. (C) Local reconnection move between two directly repeated sites. In the juxtaposition the reconnection sites, indicated with hashed lines, are at distance 1 and in antiparallel alignment. (D) L are T (2, n) torus knots and links (Fig.1). P min (L) is the number of minimal unlinking pathways observed for L under Assumption1. P (L) indicates the total number of minimal pathways observed for L without Assumption1. It is known that there are infinitely many minimal unlinking pathways for any T (2, 2n) link with parallel sites. 17 N HOMFLY-PT is the number of distinct HOMFLY-PT polynomials observed after one reconnection.
Pathways of DNA unlinking: A story of stepwise simplification

September 2017

·

57 Reads

In Escherichia coli DNA replication yields interlinked chromosomes. Controlling topological changes associated with replication and returning the newly replicated chromosomes to an unlinked monomeric state is essential to cell survival. In the absence of the topoisomerase topoIV, the site-specific recombination complex XerCD- dif -FtsK can remove replication links by local reconnection. We previously showed mathematically that there is a unique minimal pathway of unlinking replication links by reconnection while stepwise reducing the topological complexity. However, the possibility that reconnection preserves or increases topological complexity is biologically plausible. In this case, are there other unlinking pathways? Which is the most probable? We consider these questions in an analytical and numerical study of minimal unlinking pathways. We use a Markov Chain Monte Carlo algorithm with Multiple Markov Chain sampling to model local reconnection on 491 different substrate topologies, 166 knots and 325 links, and distinguish between pathways connecting a total of 881 different topologies. We conclude that the minimal pathway of unlinking replication links that was found under more stringent assumptions is the most probable. We also present exact results on unlinking a 6-crossing replication link. These results point to a general process of topology simplification by local reconnection, with applications going beyond DNA.

Citations (1)


... In [14,16], it is shown that the XerCD-dif-FtsK system unlinks replication DNA catenanes in a stepwise manner. The link types of the replication catenanes are torus link T (2, c). ...

Reference:

Signature and crossing number of links
Pathways of DNA unlinking: A story of stepwise simplification