A nonequilibrium strategy for fast target search on the genome
AA nonequilibrium strategy for fast target search on the genome
F. Cagnetta, D. Michieletto,
1, 2, 3 and D. Marenduzzo SUPA, School of Physics and Astronomy, University of Edinburgh, Edinburgh EH9 3FD, United Kingdom MRC Human Genetics Unit, Institute of Genetics and Molecular Medicine,University of Edinburgh, Edinburgh EH4 2XU, UK Department of Mathematical Sciences, University of Bath, North Rd, Bath BA2 7AY, United Kingdom
Vital biological processes such as genome repair require fast and efficient binding of selected pro-teins to specific target sites on DNA. Here we propose an active target search mechanism basedon “chromophoresis”, the dynamics of DNA-binding proteins up or down gradients in the densityof epigenetic marks, or colours (biochemical tags on the genome). We focus on a set of proteinsthat deposit marks from which they are repelled – a case which is only encountered away from ther-modynamic equilibrium. For suitable ranges of kinetic parameter values, chromophoretic proteinscan perform undirectional motion and are optimally redistributed along the genome. Importantly,they can also locally unravel a region of the genome which is collapsed due to self-interactions and“dive” deep into its core, for a striking enhancement of the efficiency of target search on such aninaccessible substrate. We discuss the potential relevance of chromophoresis for DNA repair.
Within the crowded nucleus of eukaryotic cells, itis vital that selected proteins and enzymes can locatetheir target on chromatin—the complex of DNA wrappedaround histone octamers [1]—within minutes [2, 3] of aspecific stimulus. DNA lesions, for instance, occur sev-eral thousands of times a day in every cell [4, 5]: therequirement for speed of the relevant repair machinery isthus not negotiable.Passive mechanisms are unlikely to offer the requiredefficiency: a protein exploring a human chromosome—average size (cid:39) base pairs (bp)—via 1D diffusionalong the DNA ( D < s − ) would take over years to find a single target. Purely 3D diffusionwithin the human nucleus, whose typical size is ∼ µ m ,is equally impractical. Its limits are apparent by us-ing Smoluchowski’s prediction for diffusion-limited re-action rates [6], k (cid:39) πD a (cid:39) M − s − , calcu-lated with D = 10 nm s − and a ∼ , as relevant in vivo [7, 8]. This estimate leads to sufficiently shortsearching times ( – ) only for high concentrations ofsearching proteins— – per cell, or − – µ M .Some of the components of the repair machinery areindeed highly abundant [9]: it is unclear, however, howthey can access the collapsed chromatin conformationswhich are typically observed in the nucleus [10–12]. Thecombination of alternate rounds of 3D diffusion in the nu-cleus and 1D diffusive sliding on the DNA, or facilitateddiffusion [13–16], can also lead to faster search, but thespeedup is at most one order of magnitude [17–19] and re-quires a fully accessible substrate. These considerationslead us to conjecture that energy-consuming processesmay be involved in the location of DNA lesions, as wellas in other functional enzyme-DNA interactions aimedat the quick exploration of either swollen or collapsedchromatin conformations.In this Letter we introduce the concept of chro-mophoresis —the spontaneous motion of DNA-bindingproteins as a result of a self-produced pattern of chemi- cal marks —modifications, such as acetylation or methy-lation, of histone proteins that, together with DNA, formchromatin. The prefix chromo- is chosen because, in ourmodel, DNA-binding proteins move along colour gradi-ents (as marked histones can be thought as having a dif-ferent colour than unmarked ones, Fig. 1), and also sug-gests that phoresis occurs along chromatin. As the marksdeposited provide a layer of inheritable information be-yond the DNA sequence, they are referred to as epige-netic . In the context of epigenetics, biophysical mod-els normally consider a positive feedback loop betweenthe released epigenetic marks and the protein dynamics,leading to accumulation and pattern formation [20–23].However, assuming an energy input allows also for nega-tive feedbacks, whereby a protein deposits a mark fromwhich it flees.An exemplary instance of negative feedback, which isalso involved in DNA repair, is PARylation: the addi-tion of Poly ADP-ribose (PAR) chains on histone pro-teins [4, 9]. PARylation is know to decrease the localaffinity to chromatin-binding protein; and to facilitatethe recruitment of reparing enzymes at the lesion [24, 25].Its role in the location of the lesions, instead, is still underdebate. In general, the scenario we propose is reminis-cent of chemorepulsion in active matter, where it leads tocoordinated motion [26–28]. As we shall show, negativechromophoresis also yields nontrivial patterns and it pro-vides a generic nonequilibrium mechanism for fast targetsearch on chromatin. Intriguingly, the mechanism workseven on a collapsed globule, where the target may not beimmediately accessible to diffusive searching proteins. Model –
We consider a 3D model for protein chro-mophoresis on a chromatin fibre. The latter is built, fol-lowing a well-established description of eukaryotic chro-mosomes in vivo [21, 29–33], as a flexible bead-and-springpolymer of length M . Each bead represents a set of nu-cleosomes, and we set the bead size σ to − kbp, or − nm. Chromophoretic proteins are represented as a r X i v : . [ q - b i o . S C ] A p r A ProteinDNATargetModi fi edDi ff usion B Figure 1.
Chromophoretic search. A
Proteins bind anddiffuse along a fluctuating chromatin substrate (grey) search-ing for a target (red). B Proteins deposit epigenetic marks(cyan) along the substrate at rate k on . The repulsion betweenproteins and the deposited marks results in directed motionand nontrivial collective behaviour. N spherical beads with viscous friction coefficient γ andare assumed, for simplicity, to have also size σ . The sys-tem is immersed in a heat bath at temperature T , andthe equations of motion are solved by using LAMMPS inBrownian Dynamics mode [34] (see SI). In the absenceof any chromophoretic process, proteins bind to the fi-bre non-specifically with affinity (cid:15) , modelled as a trun-cated Lennard-Jones (LJ) potential. Provided (cid:15) is com-parable to k B T , proteins can slide on chromatin, withan effective diffusion coefficient D (Fig. 1A, see alsoRef. [18]). Larger (cid:15) instead leads to cluster formation viathe bridging-induced attraction [30].The proteins we consider here deposit an epigeneticmark on the fibre bead they are bound to at a rate k on (Fig. 1, B). This mark, in turn, abrogates the attractionof the protein to the marked beads. Marks are sponta-neously lost at rate k off , modelling random or active re-moval. This model harbours a negative feedback, as themark released by the proteins raises, rather than lower,the potential energy describing fibre-protein interaction.Hence, unlike the case of positive feedback [12, 21], thissystem cannot be described by an effective equilibriummodel. To understand the dynamics that can originatefrom these microscopic rules, we first consider a simpler1D model that neglects spatial structure and fluctuationsof the chromatin fibre.
1D approximation –
As a first approximation, thechromatin fibre can be treated as a straight line. Thepotential landscape generated by the LJ interaction withthe fibre determines the protein dynamics, and is sub-stantially easier to compute for a 1D substrate (see Fig. 2and SI). In the absence of any mark, a protein sits be-tween two adjacent fibre beads so as to minimise thepotential energy (Fig. 2A). The protein can escape thewell in the direction orthogonal to the fibre (verticalaxis in Fig. 2). The escape rate r esc can be computed,for (cid:15) (cid:38) k B T , as a Kramers problem [35], and scales as ∼ e − (cid:15)/ ( k B T ) . Additionally, a barrier (cid:15)/ obstructs theprotein motion parallel to the fibre (horizontal axis in A BC tt+dtt+2dt Figure 2.
Simplified 1D model. A
Potential landscape seenby a protein binding to an unmarked (grey beads) region ofthe chromatin fibre. B Potential after deposition of a mark(blue bead). C Potential after the protein hops one bead tothe right and deposits another mark.
Fig. 2), i.e. the thermal diffusion between adjacent po-tential wells. Kramers’ theory (see SI) yields the effec-tive hopping rate for the symmetric random walk theprotein performs on the fiber as, q = A (cid:15) e (cid:15)/ kBT , with A (cid:39) . / π a numerical factor depending on the poten-tial curvature. Note that, unless otherwise stated, weset k B T = γ = σ = 1 in what follows, so that dimensionalformulas for rates can be obtained by multiplying thosewe give here by k B T / ( σ γ ) (see SI).The diffusion coefficient of the 1D diffusive sliding isthen D D = q . In our model, however, a bound proteindeposits an epigenetic mark on one of the neighbouringchromatin beads which, by silencing the LJ attraction,reshapes the potential landscape (Fig. 2B) and drivesthe model away from equilibrium. While the barrierover the marked bead remains unaltered, the one overthe unmarked one is tilted, becoming a declivity of size (cid:15) . From Kramer’s theory, the rate at which the proteinslides down the declivity is q + = B(cid:15) , with B (cid:39) . / π .As q + /q (cid:39) (cid:0) e (cid:15)/ k B T (cid:1) , we expect the protein to slidedownhill with near-one probability (recall (cid:15) (cid:38) k B T forKramers’ theory to hold).If the mark-deposition rate k on is large enough, theprotein is likely to mark the underlying bead, thus end-ing up in the configuration depicted in Figure 2C. Asthere are two marked beads upstream of the protein,the barrier over the marked bead changes: the rate ofsliding downhill remains q + while that of hopping back-wards changes to q − = C(cid:15)e − (cid:15)/k B T , with C (cid:39) . / π (cid:38) .Therefore, in a typical microscopic sequence of events, achromophoretic protein binds to the substrate, then itrandomly marks one of the two beads on either side andbecomes attracted to the other. In doing so, it enters a running state, whereby it slides forward towards the un- B Figure 3.
Chromophoretic collective behaviours.
Av-erage number of chromophoretic proteins bound and movingalong the substrate for different values of total protein copynumber N and mark removal rate k off . The black dashedline marks the limiting protein number L/l trail discussed inthe text: it provides an upper bound for (cid:104) N on (cid:105) . The insetshows the two-point correlation function in the direction ofthe protein motion (to the right in the figure). marked portion of the fibre at rate q + ∼ (cid:15) or hops back-ward onto the marked segment at rate q − ∼ (cid:15)e − (cid:15)/k B T . Abackward hop would end the running state, forcing theprotein off the fibre, hence renormalising the escape rateto r (cid:48) esc = r esc + q − (cid:39) q − . The relevant lengthscale ofthis process is the “run length”, i.e. the chromatin seg-ment that the protein explores before detaching. This isgiven by l run ∼ v/r (cid:48) esc , where v ∝ q + . More precisely, l run = B/ Ce (cid:15)/k B T (cid:39) / e (cid:15)/k B T .Mark evaporation does not change the picture, un-less occuring at rate k off > q + , which we never considerhere [36]. It is required, instead, that k on (cid:29) q + , though3D simulations show the running state exists down to k on ∼ q . Reinstating dimensional factors, this trans-lates into k on > D /σ , or k on > s − for D ∼ − µ m /s, a bead size σ = 30 nm and (cid:15) (cid:38) k B T , as for pro-teins on chromatin. This rate of post-translation mod-ification is compatible, albeit slightly faster, than thatof typical modifications: for instance, k on (cid:39) min − − s − for acetylation or phosphorylation [37].The unidirectional motion of a single protein acceler-ates target search substantially, by enlarging the distancecovered while bound to the substrate. In addition, mul-tiple chromophoretic proteins bound on the same fibreinteract with each other via the trails of epigenetic marksleft on the substrate. This effect is manifest in the paircorrelation function, i.e. the average density profile seenby a running protein (Fig. 3, inset). The downstreampeak at short distance is due to collisions with proteinsmoving in the opposite direction, and the upstream dipto the epigenetic trail and consequent protein depletion.Due to this forward-backward asymmetry, the resultingeffective interaction breaks the action-reaction principle(e.g., a particle in the wake of another is repelled bythe latter without affecting its motion), underscoring thenonequilibrium nature of the model. Snapshot swollen
Figure 4.
Kymographs of chromophoretic proteins .Kymographs showing the epigenetic mark dynamics. A M = 1000 , (cid:15) = 2 , k off = 0 . , k on = 10 and N = 20 proteins which, when not on the fiber, re-bind to it atrate . . B
3D model with with M = 1000 , (cid:15) = 4 , k off = 0 . , k on = 1 and N = 20 proteins in a L = 50 cubic box. C Snap-shot from 3D simulations showing chromophoretic proteins(red) and epigenetic marks (cyan) on chromatin (grey).
The inter-particle interactions are thus controlled bythe epigenetic mark dynamics: this provides an avenueto set up a cooperative search strategy, which is un-available to conventional facilitated diffusion. Due tothe trail-mediated exclusion between proteins, the av-erage number of proteins bound at any time does notexceed
L/l trail , where l trail ∼ v/k off is the average traillength. We therefore expect chromophoresis to suppressstochastic fluctuations in the relative distance betweenneighbours (see SI), leading to hyperuniform spreadingalong the substrate [38]. A direct consequence of thisis a faster search, as each protein only needs to scan arange ∼ l trail and is unlikely to bind to a segment whichhas already been scanned. Simulations confirm that theaverage number of bound proteins is controlled by k off /(cid:15) (Fig. 3). Biologically, k off can be modulated in responseto endogenous or external stimuli for many epigeneticmarks [37].
3D Model –
We now discuss the case where the chro-mophoretic dynamics occurs on a 3D fluctuating chro-matin fibre. We first focus on parameters for which thefibre is swollen (Fig. 4, Suppl. Movie 1). The proteinsdynamic can be quantified via kymographs, showing thelocal epigenetic state of the polymer (grey=unmarkedor cyan=marked) overlaid with protein positions (black)versus time. Both 1D and 3D systems display the samedynamical features, such as collisions and trail-mediateddissociations (Fig. 4A,B).The eukaryotic genome in vivo , however, is not aswollen fibre but is understood as a confined and mi-crophase separated polymer [12, 21, 39–45]. In particu- c o v e r e d f r a c t i o n o f t h e fi b e r f r a c t i o n o f c o v e r e d fi b e r t i m e s p e n t o n t h e fi b e r A BC D
Figure 5.
Chromophoretic search on a collapsed sub-strate. A
Average fraction of the fibre visited in a singlebinding-unbinding, or “diving”, event as a function of k off for k on = 0 . and (cid:15)/k B T = 5 . Results are averaged over sev-eral diving events and − independent simulations. B Fraction of time spent on the fibre over the total simulationtime τ /T for different observation times T . In A and B thered dot-dashed line highlights a critical k off marking the valueat which the fraction of covered fibre is maximal and thereis a transition in the behaviour of residence time τ /T . C,D
Snapshots of two chromophoretic proteins (red) “diving” intoa globule while pushed by their own trail (cyan). lar, a locally collapsed state is a typical representationof a heterocromatic , or transcriptionally silent, genomicregion [11, 21]. Whilst standard facilitated diffusion stud-ies are normally carried out in swollen conditions [18, 46],lesions and single- or double-stranded DNA breaks mayoften be buried within collapsed and inaccessible hete-rochromatic regionsThus, to explore a regime of target search relevantfor DNA lesion repair in vivo , we perform simulationswith a number of protein bridges that fold the polymersubstrate into a collapsed globule [47–49]—modelling, forinstance, multivalent HP1 proteins associated with het-erochromatin [50]. Once the chromatin fibre is foldedby these abundant bridges, we release a trace amount ofchromophoretic searchers. The two species of proteinsinteract sterically and, for simplicity, each has the samebinding affinity for unmarked chromatin. We also assumethat neither bind to the epigenetic mark deposited bythe chromophoretic species. Inspection of the simulationsshows that, strikingly, chromophoretic searchers can dis-rupt bridging-induced collapse and locally open the chro-matin fibre. It is notable that similar phenomenology isobserved during DNA repair as large chromatin regionssurrounding DNA breaks swell [51].In order to quantify the efficiency of chromophoretic search, we monitor the fraction of beads that are visitedeach time a chromophoretic protein binds the substrate.Remarkably, we discover that there is a non-monotonicbehaviour as a function of k off (Fig. 5), which can be ex-plained as follows. For k off → ∞ the epigenetic marksevaporate immediately and the searchers only stick to thesurface of the polymer globule: this limit is analogous toconventional facilitated diffusion, where a buried targetwould remain inaccessible to the searchers. In the oppo-site regime, k off → , the fibre swells but the searchersfail to remain attached for long because of the large frac-tion of non-sticky epigenetic marks. In both these limits,therefore, the fraction of beads visited for each bindingevent tends to (panel A of Fig. 5).In the regime of intermediate k off we instead observea qualitatively different phenomenon: searchers can beseen “diving” into the globule during simulations (Fig. 5and Suppl. Movie 2), by creating a local opening madeof marked beads that slowly turn to unmarked. Duringthe turnover time, the searcher is likely to be driven fur-ther inside the globule (i.e., to dive) as, on average, theprotein sees a gradient of unmarked beads towards theinterior. This gradient is actively maintained by the de-position of epigenetic marks, and fuels the descent of thechromophoretic searchers into the core of the globule.Once a searcher has dived deep enough into a globuleit may remain trapped for a long time due to the largedensity of unmarked beads which it can stick to. Duringthis time it can explore a large fraction of the polymercontour length by constantly churning the inside of thepolymer globule. As a result, the optimum turnover rate k c marks a cusp in the fraction of fibre visitided per diveas a function of k off . We further find that for k off > k c searchers spend a very long time attached to the fibre, butcannot make much progress inside the core due to stericeffects, whereas for k off < k c the time spent on the fibreafter binding is finite (i.e., it tends to zero for sufficientlylong simulations, Fig. 5B). Conclusions –
In summary, we have proposed a novelnonequilibrium mechanism for target search within thegenome. Inspired by the deposition of epigenetic markson chromatin and consequent response of certain proteinsto the gradient of such marks, we dub this mechanism“chromophoresis”. In this work we focussed on a neg-ative feedback between marks and proteins, so that theproteins are repelled from the mark they deposit. We dis-cover that, if mark deposition is sufficiently fast, a singlechromophoretic protein can perform unidirectional mo-tion on chromatin, while multiple proteins interact viaepigenetically-mediated repulsion, as a result of whichthey spread out along the fibre with suppressed 1D den-sity fluctuations. Thus, we found chromophoresis to pro-vide a generic pathway for accelerated target search, es-pecially in cases where the chromatin fibre collapses intoa globular configuration, as in a large fraction of the hu-man genome. Under this condition, we proved the exis-tence of an optimal evaporation rate of epigenetic marksfor which the exploration of the fibre is fastest. Close tothe optimal condition the proteins can locally untanglethe globular chromatin and dive into its core, which isinaccessible to simple passive searchers performing facil-itated diffusion.In addition to the intriguing physics, chromophore-sis is potentially relevant in the context of chromatinPARylation. Proteins of the PARP family, which are thechromatin-binding proteins responsible for PARylation,are recognised as a key part of the repair machinery: assuch, they need to locate DNA lesions [4], which mightbe buried within collapsed chromatin globules. PARyla-tion has been shown to swell chromatin in vitro [52] andis thought to affect the dynamics of PARP itself, as wellas of other proteins, promoting their detachment fromthe fibre [24]. 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