Why a Particle Physicist is Interested in DNA Branch Migration

TL;DR

This paper presents a discrete model of DNA branch migration that aligns with experimental data and explores its continuum approximation, offering insights for future research in molecular biology.

Contribution

It introduces a new discrete model for DNA branch migration and compares it with continuum diffusion, aiding experimental design.

Findings

Discrete model matches existing data

Long strand migration well approximated by diffusion

Model useful for guiding future experiments

Abstract

We describe an explicitly discrete model of the process of DNA branch migration. The model matches the existing data well, but we find that branch migration along long strands of DNA ($N \simge 40$~bp) is also well modeled by continuum diffusion. The discrete model is still useful for guiding future experiments.

Equations

$$\text{EQNdisplaylines}\frac{d\vec{C}}{dt}=\beta a^2\bar{M}\vec{C},\text{hfill}\text{hfill}(7)$$

$$\displaystyle{\EQNdisplaylines{\bar{M}={1\over a^{2}}\pmatrix{-1&~{}1&&&\cr~{}1&-2&1&&\cr&\ddots&\ddots&\ddots&\cr&&1&-2&~{}1\cr&&&~{}1&-2\cr}\,.\hfill\hfill({8}})}$$

$$\text{EQNdisplaylines}v(n)=\cos (kna+\varphi ),\text{hfill}\text{hfill}(10)$$

$$\text{EQNdisplaylines}\lambda =\frac{-4}{a^2}{\sin }^2(\frac{ka}{2}),\text{hfill}\text{hfill}(11)$$

$$\text{EQNdisplaylines}\tau =\frac{-1}{\beta a^2\lambda }.\text{hfill}\text{hfill}(13)$$

$$\text{EQNdisplaylines}k=\frac{(2m+1)}{(2N+1)}\frac{\pi }{a},\text{hfill}\text{hfill}(19)$$

$$\text{EQNdisplaylines}\vec{C}(t)=\sum _{m=0}^{N-1}{a}_m{e}^{-t/{\tau }_m}{\vec{v}}_m,\text{hfill}\text{hfill}(23)$$

$$\text{EQNdisplaylines}{a}_m=\frac{4C_0}{2N+1}\cos (\frac{\pi }{2}\frac{(2m+1)}{(2N+1)})\text{hfill}\text{hfill}(26)$$

$$\displaystyle{\EQNdisplaylines{F_{N}(t)=C_{0}\Biggl{[}1-{2\over 2N+1}\sum_{m=0}^{N-1}(-1)^{m}e^{-t/\tau_{m}}\hfill\cr\hfill{}\times{\cos^{2}({\pi\over 2}\,{2m+1\over 2N+1})\over\sin({\pi\over 2}\,{2m+1\over 2N+1})}\Biggr{]}.\qquad\hfill({28\cr}})}$$

Full text

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To appear in the Proceedings of Lattice ’96, International Symposium on Lattice Field Theory, St. Louis, Missouri, USA, June 4-8 1996, Edited by M. Golterman \etal (Elsevier Science B.V., Amsterdam, 1996). \twelvepointWhy a Particle Physicist is Interested in DNA Branch Migration

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Eric Myersa and Michael F. Bruistb

\unobeylines

aDepartment of Physics, Randall Laboratory, University of Michigan,

Ann Arbor, MI 48109-1120

bDepartment of Chemistry, Vassar College, Poughkeepsie, NY 12601

We describe an explicitly discrete model of the process of DNA branch migration. The model matches the existing data well, but we find that branch migration along long strands of DNA ($N\simge 40$ bp) is also well modeled by continuum diffusion. The discrete model is still useful for guiding future experiments.

Lattice methods have proved to be very useful for studying particle interactions. As reported at this meeting, lattice gauge theory calculations now seem to be able to predict the mass of the “glueball,” and are leading to a better understanding of both chiral symmetry breaking and quark confinement. It is implicit in such calculations (and sometimes explicit) that one is to take the continuum limit, but one can, of course, apply lattice field theory methods to systems which are inherently discrete, without taking the continuum limit. DNA branch migration is a simple example.

\bottomfigure

Fig1

\caption

Schematic of single strand exchange in DNA to form the Holliday structure (b), followed by branch migration (c). \endfigure

DNA (deoxyribonucleic acid) transmits genetic information when the molecule is replicated and transmitted to daughter cells. Rare errors in replication give rise to mutations. (The adaptability of organisms with these mutations gives rise to evolution.) Mutations do not stay in the DNA segment (chromosome) in which they arose; they can transfer to other chromosomes by recombination. In recombination, two DNA segments encoding the same genes effectively break at homologous points, and then rejoin or recombine by fusing partner DNA segments. When viewed in more detail, one finds an intermediate state (a ”Holliday structure”) in which two DNAs are joined by the exchange of only one strand from each duplex\referenceHolliday, 1964 R. Holliday, \journalGenetic Research;5,282-304(1964) \endreference(\FigFig1b). The branch point which joins the two helices is mobile because of the sequence similarity of the homologous DNAs (\FigFig1c). A better understanding of branch migration will lead to a better understanding of DNA recombination and the propagation of genetic variation which drives evolution.

Laboratory experiments can now replicate DNA branch migration through short defined segments of DNA with controlled initial conditions.\referenceBruist, 1996 A.W. Kirby, M.N. Gaskin, M.A. Antezana, S.J. Goodman, E. Myers, and M.F. Bruist, submitted to J. Molecular Biology \endreference\referencePanyutin, 1995 I.G. Panyutin, I. Biswas, and P. Hsieh, \journalEMBO J.;14,1819-1826(1995) \endreferenceOur goal is to develop a model of these experiments which preserves the discrete nature of DNA. We focus on the motion of the branch point and ignore the helical structure of DNA. (A related model has been described by Fujitani and Kobayashi.\referenceFujitani, 1995 Y. Fujitani and I. Kobayashi, \journalPhysical Review E;52,6607-6622(1995) \endreference) This can be thought of as a series of linked chemical reactions in a “compartment model.”\referenceBentley and Cooke D.L. Bentley and K. L. Cooke, \booktitleLinear Algebra with Differential Equations, (Holt, Rinehart and Winston, 1973) \endreferenceImagine that all of the molecules with branch points in a particular position are kept in one compartment, and that when the branch point on a given molecule moves to a neighboring position that molecule is removed from its compartment and added to the appropriate neighboring compartment. If there are $N$ possible positions for the branch then there will be $N+1$ compartments, the last being for molecules which have separated. Ignoring this last compartment, let the $n\hbox{-{th}}$ component of the vector $\vec{C}(t)$ represent the concentration of molecules having the branch in the $n\hbox{-{th}}$ position at time $t$, and let $\beta$ represent the rate of transport in the forward or backward directions (assuming these are equal—it is easy to generalize to the case where they are not). The dynamics of this system is then described by a set of $N$ coupled first order linear differential equations, which can be written in matrix form as

[TABLE]

where the $N\times N$ matrix $\bar{M}$ is

[TABLE]

All blank entries are zero, and we have introduced $a$ as the spacing between successive positions along the DNA.

The similarity of \Eq8 to the lattice second derivative is obvious, and so it should come as no surprise that the continuum limit of this model is the diffusion equation. However, we seek solutions to \Eq7 before the continuum limit is taken. The procedure for finding these solutions is similar to the continuum case. First, one can remove a time-dependent factor by writing $\vec{C}(t)=e^{-t/\tau}\vec{v}$, where $\vec{v}$ is an eigenvector of $\bar{M}$. If the $n\hbox{-{th}}$ component of $\vec{v}$ is written as $v(n)$, then the eigenvectors of $\bar{M}$ have the form

[TABLE]

where $n=0,1,\ldots,N-1$, and where $k$ and $\phi$ are yet to be determined constants. The eigenvalues are

[TABLE]

from which it follows that

[TABLE]

In the continuum case the undetermined constants $k$ and $\phi$ would be determined by boundary conditions, but there are no “boundaries” to a matrix. Nevertheless, it is useful to imagine that there are extra compartments at either end of the system. \Eq10 is only an eigenvector because it fits the +1, -2, +1 differencing pattern along the diagonals of $\bar{M}$. This pattern must be maintained for the eigenvalue equation to be valid. To do so at the upper left corner of $\bar{M}$ we imagine an extra point at $n=-1$ such that $v(-1)=v(0)$, which forces $\phi=\half ka$. This says that the lattice derivative is zero, so in analogy to the continuum case we call this a “lattice Neumann” boundary condition. To maintain the differencing pattern at the lower right corner of $\bar{M}$ we imagine an extra point at $n=N$, such that $v(N)=0$ (this is the final compartment that we ignored earlier). This “lattice Dirichlet” boundary condition forces the unknown constant $k$ to be

[TABLE]

where $m=0,1,2\ldots N-1$. There are exactly $N$ distinct eigenvalues, which we will label $\vec{v}_{m}$.

\bottomfigure

Fig2

\caption

Data from an experiment measuring the fraction of branched DNA converted to separated final product, along with the best fit curve from the model, \Eq28, for $N=40$. \endfigure

The most general solution of \Eq7 is a linear combination of all of the solutions,

[TABLE]

where the $a_{m}$ are constants, and where $\tau_{m}$ is related to the $m\hbox{-{th}}$ eigenvalue via \Eq13. The values of the $a_{m}$ are determined from the initial conditions, using the fact that the different eigenvectors $\vec{v}_{m}$ are orthogonal. In the experiments we are modeling,\citeBruist, 1996 the initial conditions are such that all molecules begin with the branch point at one position, adjacent to a reflective barrier created by a divergence in the DNA sequences. These initial conditions are represented by $\vec{C}(0)=C_{0}\delta_{n,0}$. Substituting this into \Eq23, taking the dot product with $\vec{v}_{m^{\prime}}$ and using the orthogonality of the $\vec{v}_{m}$ yields

[TABLE]

The experiments we model only measure the concentration of separated molecules at a given time, not the number of molecules having the branch point in a given position. To compute this observable we take the initial concentration, $C_{0}$, and subtract from it the sum over all of the branched molecules left in the system. This gives

[TABLE]

Aside from the number of steps, $N$, there are only two free parameters in this expression, $C_{0}$ and $\beta$ (which is related to the $\tau_{m}$ by \Eq13).

\figure

Fig3

\caption

Difference between the best fit discrete and continuum measures of the final product. \endfigure

\Fig

Fig2 shows measurements of the concentration of final product (separated molecules) from an experiment using a DNA junction in which the branch point must migrate 40 base pairs (bp) in one direction before coming to the end of the strands. The errors include systematic errors, not just statistical variations. The curve in the figure is a non-linear least-squares fit to \Eq28 with $N=40$. The fit is quite good. One notable characteristic of branch migration is the “lag” at the beginning of the experiment—no material is produced as final product for several minutes after the experiment has begun. The theoretical curve reproduces this behavior well.

Is it possible to see the discrete nature of the motion of the branch point? In particular, can we tell the number of base pairs traversed in a single step of branch migration? Unfortunately, the answer is no. The data in \FigFig1 are also fit well by \Eq28 with $N$ anywhere between 4 and 40, as well as by the analogue of \Eq28 obtained from continuum diffusion. \FigFig3 shows the difference between $F_{N}(t)$ from \Eq28 and $F(t)$ obtained from continuum diffusion (assuming the same $C_{0}$). Except for very short molecules, the differences are smaller than the size of present experimental errors. Nevertheless, \FigFig3 shows that the discrete nature of branch migration will be most noticeable within times less than $\tau_{0}$. This is useful knowledge for designing future experiments. We therefore expect that our discrete model will lead to a better understanding of the underlying dynamics of branch migration. \nosecheadREFERENCES\ListReferences