A theory of the dynamics of DNA loop initiation in condensin/cohesin complexes

The structural maintenance of chromosome complexes exhibit the remarkable ability to actively extrude DNA, which has led to the appealing and popular “loop extrusion” model to explain one of the most important processes in biology: the compaction of chromatin during the cell cycle. A potential mechanism for the action of extrusion is the classic Brownian ratchet, which requires short DNA loops to overcome an initial enthalpic barrier to bending, before favoured entropic growth of longer loops. We present a simple model of the constrained dynamics of DNA loop formation based on a frictional worm like chain, where for circular loops of order, or smaller than the persistence length, internal friction to bending dominates solvent dynamics. Using Rayleigh’s dissipation function, we show how bending friction can be translated to simple one dimensional diffusion of the angle of the loop resulting in a Smoluchowski equation with a coordinate dependent diffusion constant. This interplay between Brownian motion, bending dissipation and geometry of loops leads to a qualitatively new phenomenon, where the friction vanishes for bends with an angle of exactly 180 degrees, due to a decoupling between changes in loop curvature and angle. Using this theory and given current parameter uncertainties, we tentatively predict mean first passage times of between 1 and 10 seconds, which is of order the cycle time of ATP, suggesting spontaneous looping could be sufficient to achieve efficient initiation of looping.

pseudo-topological and non-topological in the literature (7, 14-16)), a key feature for looping is that 49 DNA is constrained or bound to the complex at two contact points (Fig.1), which enable the loop to 50 grow. There are various models of how extrusion might occur, but the simplest is a classic Brownian 51 rachet mechanism (9), where ATP hydrolysis causes unbinding of the head domains, allowing the 52 DNA freedom to diffuse by Brownian motion with constrained electrostatic interactions with the 53 head domains and additional protein domains known as HEAT repeat subunits, ; the idea is that on 54 re-binding of ATP if the loop has grown by diffusion then this motion has been ratcheted. Molecular 55 motors powered by Brownian ratchet mechanisms work by relying on diffusion in some asymmetric 56 potential, here for sufficiently long loops entropy will favour the growth of loops. However, for 57 initially short loops the growth will be disfavoured by the enthalpy of bending. It is an open and 58 important question whether Brownian motion would be sufficiently rapid for SMC complexes to 59 initiate extrusion of DNA by this mechanism, or require additional force generation mechanisms to 60 drive initial loop growth. 61 From a physics perspective, the generic problem is one of the Brownian motion of a semi-flexible 62 polymer loop through an aperture whose size is of order the persistence length. Although, there has 63 been much theoretical (17,18) and empirical work (19,20) on the problem of DNA cyclisation, which 64 studies the rate that short lengths of DNA find their ends, this is not directly relevant to looping stranded DNA (31) have all been measured empirically using a range of single molecule experimental techniques. A key message from these studies is that many dynamical mechanical processes in 74 biology are far too slow to be explained if Stokes' friction with the solvent was the only source 75 of dissipation. In the case of semi-flexible or worm-like chain (WLC) polymers, measurements of 76 stretched polypeptides (29) and ssDNA (31) show that internal friction increases with tension F with 77 a power law ∼ F 3/2 predicted by a frictional worm-like chain (FWLC), which includes dissipation 78 due to internal friction which opposes bending (29). The exact origin of bending friction is unclear, 79 but will be related to steric constraints between complex molecular potentials giving rise to a local 80 roughness that at a coarse-grained level gives an effective friction that opposes bending. Although, 81 the bending friction constant of dsDNA has not been measured, we a priori expect the same behaviour 82 as empirically determined for ssDNA and unfolded polypeptides. infinitesimally, there is no change in curvature/bending of the loop and hence no bending friction.

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Further, given this phenomenon, we predict that even with relatively large initial angles in the gripping 95 state, the mean first passage time to reach an entropically dominated loop is not significantly affected, where increasing θ corresponds to increased progress of looping and a larger radius. We can now 129 characterise the looping as a Brownian walk in the variable θ in a potential landscape given by U (θ), 130 which is plotted in Fig.2a in units of k B T for d = 25nm and p = 50nm, where we see that maximum 131 energy of a loop occurs at around θ ≈ 4 radians or 230 • .

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The diffusion problem can be represented by the Smoluchowski equation: where p(θ, t) is the time-dependent probability density function of the angle θ as a function of time 134 Figure 2: a) Potential energy βU (θ) (blue line) and the effective dynamical potential βŨ (θ) = βU (θ) + ln(ζ θ (θ)) (red line) as a function of θ, where the latter incorporates the effect of internal friction in the dynamics of the loop. Note that the dynamic potential does not affect the Boltzmann distribution of angles which is still given by U (θ). The diagrams show the geometry of the DNA loop constrained to a distance d at two freely sliding attachment points, and also the relative loops sizes for different angles of θ = π (minimum friction), θ = 4 (maximum internal energy) and θ = 5.4, which corresponds to when the contour length of the loop L = 3 p and when the loop will tend to adopt non-circular conformations. b) Effective angular friction of a loop ζ θ (θ) as a function of θ, showing vanishing friction at θ = π radians due the decoupling with changes in curvature at this angle.
t and where the friction coefficient ζ θ (θ) is the effective coarse grained opposition to motion to 135 changes in the angular velocityθ, which as we will show is in general a function of the coordinate where ζ s ≈ 6πη is a solvent friction per unit length (η is the solvent viscosity), ζ B is the bending 143 friction constant and κ B = k B T p is the bending elastic constant and f (s, t) is a spatially and 144 temporally white noise term whose moments follow from the fluctuation dissipation theorem. A 145 normal mode analysis shows the relaxation τ q of different modes has a mode-and length-dependent 146 contribution to the relaxation which arises from solvent dynamics, whilst there is a mode-and length-147 independent contribution from internal frictional processes, since both internal friction and bending 148 elasticity are coupled to curvature, they give the same dispersion relation with mode number: where q is the wavenumber of the mode. Comparing these two contributions for the relaxation of where we have used the fact that L = Rθ. We see that once translated to θ space the effective 188 friction of the random walk has a very strong θ dependence, as plotted in Fig.2b, which shows in fact 189 that the friction vanishes at exactly θ = π and diverges for θ = 2π. We can understand both these  198 In practice, in the latter case once the contour length of the loop is much greater than the persis-199 tence length, the loop will adopt a very non-circular conformation before the divergence is reached.

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If we assume that once the contour length reaches a factor α larger than the persistence length, 201 non-circular looping becomes dominant then these considerations give an approximate expression for 202 the critical angle at which this occurs as For d = 25nm, p = 50nm and α = 3, this gives θ * = 5.4 radians, or θ ≈ 310 • , so this would predict 204 that loops remain reasonably circular, before degenerating into a more random configuration.  To understand this we note that the MFPT continues to increase rapidly even after the loop 225 has passed the angle of maximum energy (θ ≈ 4 radians), which can be explained because of 226 the still rapidly increasing internal friction of the loop; if we plot the effective dynamic potential 227Ũ (θ) = U (θ) + k B T ln ζ θ (θ), which appears in the integrand of Eqn.14 for the MFPT, then we see 228 that it has a maximum at θ ≈ 4.5 radians and falls to more than k B T from the maximum at θ ≈ 4 229 radians and θ ≈ 5.4 radians. The angles which are within k B T of the maximum of the effective 230 potential represents the regions over which diffusion is effectively flat, and relatively slow, due to the 231 large internal friction at these angles.

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Evaluating the MFPT for θ = θ * , we find τ ≈ 10, 000 msecs or 10secs and as the MFPT 253 plateaus for angles approaching θ > θ * this estimate will be robust to order of magnitude given

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To summarise, we present a simple theory of Brownian loop growth in SMC complexes, where 295 internal friction to bending of double stranded DNA couples to the changing geometry of the loop to 296 determine the stochastic dynamics in a non-trivial way. For condensin/cohesin to act as a Brownian 297 ratchet, loops must first grow to sufficient size to overcome the initial enthalpic barrier; the model 298 remarkably predicts that friction for loop growth will vanish at 180 • -with a corresponding plateau 299 in the MFPT -and that the MFPT for the loop to grow large enough that entropy favours its 300 growth is dominated loops that are roughly a three-quarter circle. This latter prediction conversely 301 means that loop initiation is largely insensitive to the initial angle of the bend induced in DNA in 302 the "gripping" state. Overall, we predict loop initiation times of order ∼ 1s to 10s, which is of order 303 the cycle time of ATP, suggesting loop initiation could initiate purely by Brownian motion without 304 the aid of additional active force generation mechanisms.