Insight into the Unwrapping of the Dinucleosome
Fatemeh Khodabandeh, Hashem Fatemi, Farshid Mohammad-Rafiee1
IInsight into the Unwrapping of the Dinucleosome
Fatemeh Khodabandeh, Hashem Fatemi, and Farshid Mohammad-Rafiee Department of Physics, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan 45137-66731, Iran (Dated: January 28, 2020)Dynamics of nucleosomes, the building blocks of the chromatin, has crucial effects on expression,replication and repair of genomes in eukaryotes. Beside constant movements of nucleosomes by ther-mal fluctuations, ATP-dependent chromatin remodelling complexes cause their active displacements.Here we propose a theoretical analysis of dinucleosome wrapping and unwrapping dynamics in thepresence of an external force. We explore the energy landscape and configurations of dinucleosomein different unwrapped states. Moreover, using a dynamical Monte-Carlo simulation algorithm, wedemonstrate the dynamical features of the system such as the unwrapping force for partial and fullwrapping processes. Furthermore, we show that in the short length of linker DNA ( ∼ −
90 bp),the asymmetric unwrapping occurs. These findings could shed some light on chromatin dynamicsand gene accessibility.
INTRUDUCTION
Chromatin has a crucial role in the essential eukary-otic biological processes such as replication, transcrip-tion, recombination, and other gene regulations. Thebasic subunit of chromatin is the nucleosome core par-ticles (NCP), which consists of 147 base pair (bp) DNAwrapped around the core of histone proteins [1]. In chro-matin, these NCPs are separated from each other by vari-able lengths of linker DNA ( ∼ −
100 bp) [2, 3]. Sincea typical gene is composed of tens to hundreds of nucle-osomes, the accessibility to gene contents is vital for thelife cycle of cells.
In vivo and in vitro experiments showed that the genecontents of chromatin are accessible due to the structuralfluctuations of the nucleosomal DNA [4–6]. This dynam-ical behavior becomes more important and interestingwhen active translocation motors such as RNA poly-merases or remodelers exert effective forces and torqueson the nucleosomes [7, 8]. In these situations, one mayask about the effect of force on the nucleosome dynamics.Using optical and magnetic tweezers, one can apply atunable force on a single nucleosome or a nucleosome ar-ray. In the experiments on the nucleosome arrays, an ex-ternal force has been applied, and the unwrapping tran-sition at high force [9, 10] and low force limit [11] hasbeen examined. In these studies, the details of mononu-cleosome unwrapping transition were not clear. Recently,some experiments have been done for studying the un-folding dynamics of the 30-nm chromatin fiber at the lowforce regime [12]. Although the elastic properties of the30-nm chromatin fiber depend on its architecture [13], ithas been shown that when the stretching force is smallerthan ∼ ∼ − ∼ a r X i v : . [ q - b i o . B M ] J a n N N flank , , F F Wednesday, July 25, 2018
FIG. 1: The schematic picture of a dinucleosome with twoflanking DNA. The nucleosomes are denoted by N and N .The plot is corresponding to the situation that the first andthe second nucleosomes are in states n = 1 and n = 6,respectively. The applied force is in the direction of z . for short enough linker DNA lengths, the dinucleosomesystem prefers to unwrap asymmetrically. MATERIALS AND METHODS
We consider two nucleosomes on a long DNA underan external stretching force and study the wrapping andunwrapping dynamics of the nucleosomes. We assumethat the two flanking DNAs are long enough and thereis a linker DNA between the two histone octamers withthe length L linker . The schematic picture of the prob-lem has been shown in Fig. 1. In order to study thewrapping-unwrapping dynamics of the nucleosomes, weneed to have the energy landscape of the problem. Wecalculate the energy of the dinucleosome in the force andtorque balance condition using the model that has beendescribed in Ref. [23]. In this model, the DNA is consid-ered as an elastic slender rod with the bending rigidityof κ . We assume that any changes in the twist of DNAcan be washed out in the boundaries of flanking DNAs[23, 24]. X-ray crystallography experiments reveal thatthere are 14 binding sites in nucleosomes, where the mi-nor grooves of the DNA face inwards to the histone pro-teins, and the DNA length between two adjacent bindingsites is about 10 bp [25, 26]. We introduce n i as thenumber of opening binding sites for the nucleosome i .In general, the total energy of the dinucleosome dependson the external force, F , the length of the linker DNA, L linker , the lengths of two flanking DNAs, L flank , and L flank , , and the number of opening binding sites of thenucleosomes, n and n , and can be written as E tot = E Nuc , ( n ) + E Nuc , ( n )+ E linker ( F, L linker , n , n )+ E flank , ( F, L flank , , n ) + E flank , ( F, L flank , , n ) . (1)As can be seen in Fig. 1, there are two segments ofthe DNA wrapped around the histone octamers, whichwill be called “nucleosomal DNA”, and three segmentsof the DNA which are free. The energy of the nucleoso-mal DNA consists of three parts: (1) the elastic energy of the deformed DNA, (2) the DNA-histone adsorptioninteractions, and (3) DNA-DNA electrostatic repulsiveinteractions. The two first contributions can be consid-ered as effective adsorption energy per binding sites andare denoted by ε ads . The DNA-DNA repulsion energyper unit length is indicated by ε es . We note that thisterm is considered when the wrapping nucleosomal DNAhas more than one turn. Therefore, the total energy ofthe nucleosomal DNA for the nucleosome i can be writtenas [21] E Nuc , i ( n i ) = an i [ ε ads + ε el Θ(7 − n i )] , (2)where a = 10 bp is the mean length of the DNA betweentwo adjacent binding sites and Θ( x ) shows the step func-tion that is zero for x <
0, and 1 for x ≥
0. It shouldbe noted that in the physiological conditions, the DNAprefers to wrap around the octamer and therefore ε ads has a negative value. Furthermore, when n i is less than7, there is an effective repulsive energy due to DNA-DNAelectrostatic interactions and ε el should have a positivevalue.There are two contributions to the energy of the linkerand the flanking DNAs: the elastic bending energy andthe work of the external force. Considering these two con-tributions, the energy of each linker and flanking DNA iswritten as E DNA = (cid:90) L κ (cid:18) d ˆ tds (cid:19) ds − (cid:90) L (cid:126)F · ˆ t ds, (3)where L denotes the length of the considered naked DNA,which can be the linker or the flanking part, and ˆ t is thelocal tangent unit vector of the DNA at the arclength of s . As we have discussed in Ref. [23], there is a constrainton the local tangent vector of the DNA at the positionswhere the DNA exits the histone octamer. Therefore,when the nucleosomal DNA is partially unwrapped, thelocal tangent vectors of the free DNAs at the exit pointsof the histone octamers are changed. Considering theforce and torque balance conditions of the initial equi-librium state, the mentioned changes at the boundariesof the free DNAs cannot hold the equilibrium conditionsand so the orientations of the nucleosomes change in or-der to go to the equilibrium conformation.Therefore, when the nucleosomal DNAs are partiallyunwrapped, the system goes from ( n = 0 , n = 0) to anew state of ( n , n ). Due to this transition, the totalenergy of the system is changed. The orientation of theoctamers and the conformation of the DNAs can be de-termined using the method described in Ref. [23]. Thetotal energy of the system at the new state of ( n , n ) isdetermined by Eqs. (1)–(3) under the force and torquebalance conditions. It is possible to have more than onesolution that fullfill the force and torque balance con-ditions for a given state of ( n , n ). In this paper, wechoose a solution corresponding to the global minimumenergy for the system.Knowing the energy landscape of the problem in termsof ( n , n ), enables us to estimate the opening and clos-ing rates of the binding sites of the nucleosomes. Weassume that the dominant mechanism for the openingand closing of the binding sites of the nucleosomes isone-by-one. It means that at each time step we can haveone of the following events: ( n , n ) → ( n ± , n ) or ( n , n ) → ( n , n ± n , n ) → ( n + 1 , n ) or ( n , n ) → ( n − , n ),respectively. After defining k i uw and k i rw as the un-wrapping and rewrapping rates of the binding sites, re-spectively, at zero force for nucleosome i , one can writethe transition rates as [21] k (1) uw ( F, n , n ) = k (1)0 uw e − λ F β [∆ E ∗ ( n +1 ,n ,F ) − ∆ E ∗ ( n ,n ,F )] (4a) k (1) rw ( F, n , n ) = k (1)0 rw e (1 − λ F ) β [∆ E ∗ ( n +1 ,n ,F ) − ∆ E ∗ ( n ,n ,F )] (4b) k (2) uw ( F, n , n ) = k (2)0 uw e − λ F β [∆ E ∗ ( n ,n +1 ,F ) − ∆ E ∗ ( n ,n ,F )] (4c) k (2) rw ( F, n , n ) = k (2)0 rw e (1 − λ F ) β [∆ E ∗ ( n ,n +1 ,F ) − ∆ E ∗ ( n ,n ,F )] , (4d)where β ≡ / ( k B T ), λ F denotes a load distribution factorthat will be considered as a fitting parameter [27], and∆ E ∗ is defined as∆ E ∗ ( n , n , F ) ≡ E min ( n , n , F ) − E min ( n , n , F = 0) . (5)In Eq. (5), E min ( n , n , F ) denotes the total energy ofthe system (Eq. (1)), when the dinucleosome is at equi-librium under the external force F , and the number ofbinding sites of the nucleosome i is n i . It is worth men-tioning that the unwrapping and rewrapping rates at zeroforce, k i uw and k i rw , can be estimated using the detailedbalance equation as [28] k (1)0 uw k (1)0 rw = e − β [ E min ( n +1 ,n ,F =0) − E min ( n ,n ,F =0)] , (6) k (2)0 uw k (2)0 rw = e − β [ E min ( n ,n +1 ,F =0) − E min ( n ,n ,F =0)] . (7)Using these transition rates, we simulate the dynam-ics of the system employing the Gillespie algorithm [29].We use the parameters suggested in Ref. [21], namely γ F = 0 . k ( i )0 rw = 10 s − , (cid:15) es = 0 . B T / nm, and (cid:15) ads = 0 .
78 k B T / nm. Using this set of parameters andthe model described in Ref. [21], the simulation resultsfor the dynamical behavior of a mono-nucleosome underexternal tension are in very good agreement with the ex-perimental data. RESULTS
The energy landscape of a mono-nucleosome is sym-metric in the phase space of the number of opened bind-ing sites from the left and right of the nucleosome, whichmeans the symmetric and asymmetric unwrapping of
Monday, July 30, 2018
FIG. 2: (color online) The representation of ∆ E ≡ E min ( n , n ) − E min ( n = , n = 0) vs. the number of openedbinding sites of the nucleosomes for F = 3 pN and L linker =20 bp. n Li and n Ri represent the number of opened bindingsites from the left and right of the nucleosome i . The colorbar in the right panel shows different colors corresponding tothe energy in the unit of k B T . Two examples of asymmetricregions in the unwrapping energy landscapes are shown bytwo ellipses. the nucleosome have the same energy. One may thenask what happens in the dinucleosome case, where thelength of the linker DNA may change the energy land-scape of the system. As it is shown in Fig. 2, for shortlengths of the linker DNA like L linker = 20 bp, the energylandscape of the dinucleosome unwrapping is asymmet-ric. The inset plots of the figure, represent the free en-ergy landscape of the first nucleosome unwrapping, whilethe second nucleosome remains intact in a fixed num-ber of opened binding sites. Two examples of asymmet-ric regions in the landscape of the unwrapping energy (a)(b) Tuesday, August 7, 2018
FIG. 3: (color online) ∆ E ≡ E min ( n , n , L linker ) − E min ( n = 0 , n = 0 , L linker = 100 nm) in terms of the lengthof the linker DNA for F = 3 pN and different values of n .Plot (a) corresponds to n = 0 and plot (b) corresponds to n = 11, respectively. are shown by two ellipses in the figure. For example,when ( n L , n R , n R , n L ) = (0 , , , ∼
30 k B T, whereas( n L , n R , n R , n L ) = (0 , , , ∼
20 k B T. The two mentionedpoints are located in the left ellipse in the Fig. 2.As the number of opened binding sites of the nucleo-somes and accordingly the length of the linker DNA in-creases, the asymmetric behavior is washed out. Unlikethe fully wrapped dinucleosome [23], this result is due tothe dependency of the dinucleosome energy on the lengthof the linker DNA. To show this dependency more clearly,in Fig. 3, the behavior of ∆ E in terms of the length ofthe linker DNA is shown for different unwrapping cases.Plot (a) in Fig. 3 corresponds to the unwrapping of thefirst turn of the second nucleosome, when the first oneis in the fully wrapped state. As can be seen, when thelength of the linker DNA becomes longer, the differencein energy becomes smaller. We can see the same behav-ior, when the second turn of the second nucleosome startsto unwrap, see Fig. 3(b). When the length of the linkerDNA is short, due to the structural constraint, the linkerDNA should be bent dramatically; and that is why thereis a significant change in the amount of the energy, whenthe length of the likner DNA is short.Thus one expects that the four-dimensional energylandscape should be symmetric when the length of thelinker DNA is long enough, as can be seen in Fig. 4. Theplot corresponds to F = 3 pN and L linker = 200 bp. Inthis situation, the details of the nucleosomes unwrappingare not important, and the energy can be obtained as a n L2 n R n R1 n L ∆ E(k B T) FIG. 4: (color online) The representation of ∆ E ≡ E min ( n , n ) − E min ( n = , n = 0) vs. the number of openedbinding sites of the nucleosomes for F = 3 pN and L linker =200 bp. n Li and n Ri represent the number of opened bindingsites from the left and right of the nucleosome i . The colorbar in the right panel shows different colors corresponding tothe energy in the unit of k B T . function of n and n , alongside L Linker and F depen-dencies. (a) (b) (c) (d) (e)(f) (g) FIG. 5: ∆ E ≡ E min ( n , n ) − E min ( n = , n = 0) vs. thenumber of opened binding sites of the nucleosomes ( n , n )for F = 8 pN and L linker = 200 bp. The conformations ofthe nucleosomes are shown for different partially unwrappedstates of nucleosomes. As we mentioned before, the local conformation of thenucleosomes and their relative orientation respect to eachother, depends on the linker DNA length, the external n n (n + n ) -505101520 (cid:1) E ( k B T ) Path IPath II (cid:1)
E(k B T) (a)(b) FIG. 6: (a) The free energy landscape of the dinucleosome,∆ E ≡ E min ( n , n ) − E min ( n = 0 , n = 0), versus the numberof opened binding sites of the nucleosomes. The color barin the right panel shows different colors corresponding to theenergy in the unit of k B T . Two possible paths for unwrappingare shown by the purple (path I) and black (path II) arrows.(b) ∆ E versus the total number of opened binding sites, n + n , for the two paths shown in (a). These plots correspondto F = 3 pN and L linker = 200 bp. force, and the number of opened binding sites. Fig. 5shows the energy landscape of the dinucleosome in termsof the opened binding sites. Furthermore the local con-formation of the two nucleosomes in different situationshave been shown in the figure. The plot corresponds to F = 8 pN and L linker = 200 bp. As we discussed above,we expect the energy landscape to be symmetric in termsof n and n for long length of the linker DNA, which canbe seen in Fig. 5. We also expect that for long enoughlinker DNA lengths, the dinucleosome system behaveslike two separate mono-nucleosomes. This result can betested by comparing the energy landscape of a dinucle-osome with the sum of the two mono-nucleosomes ener-gies. We have tested this argument for different lengthsof L linker and see that the intuition is fulfilled for long L linker .In Figs. 6(a) and 7(a), the energy landscape is de-picted for two different forces. When the force is low, i.e F = 1 pN, the minimum of the energy is in the state of n = n = 0, which means that the full wrapped nucle-osomes remain stable. By increasing the force, the en-ergy landscape changes, and new minima with differentenergy barriers appear. The studies on the mononucleo-some reveal the nucleosome partial and full unwrapping n n -120-100-80-60-40-200
12 14 16 18 20 22 24 26 28 (n + n ) -140-120-100-80-60-40 (cid:1) E ( k B T ) Path IPath II (cid:1)
E(k B T) (a)(b) FIG. 7: The free energy landscape of the dinucleosome, ∆ E ≡ E min ( n , n ) − E min ( n = 0 , n = 0), versus the number ofopened binding sites of the nucleosomes. The color bar in theright panel shows different colors corresponding to the energyin the unit of k B T . Two possible paths for unwrapping areshown by the purple (path I) and black (path II) arrows. (b)∆ E versus the total number of opened binding sites, n + n ,for the two paths shown in (a). These plots correspond to F = 8 pN and L linker = 200 bp. happen in the forces of about F ∼ F ∼ − F = 3 pN and F = 8pN, respectively. One of the typical paths runs along the n and n axes from the local minimum to the global min-imum point (path I ), which results in partial unwrappingof the second nucleosome after the first one. In the otherpath, the system reaches the global minimum through thediameter (path II ), in which the number of opened pointsfor each of the nucleosomes are equal. In Figs. 6(b) and7(b) the energy of the system through these two paths hasbeen shown for two different forces. In these figures, thebehavior of ∆ E ≡ E min ( n , n ) − E min ( n = 0 , n = 0)is depicted in terms of the total number of opened bind-ing sites, n tot = n + n , for two different forces. We notethat in F = 3 pN and F = 8 pN, the global minima en- (n + n ) -130-110-90-70-50
19 20 21 22-95-92.5-90
F=3pNF=8.5pN (b)(a) (n + n ) -130-110-90-70-50
19 20 21 22-95-92.5-90
F=3pNF=8.5pN (b)(a)
F=8pN E ( k B T )
F <
F > L linker = 200 bp. consider a possible experimental setup, in which a con-stant loading rate is applied on the two free ends of thedinucleosome. The force-extension curve obtained fromthe simulation has been shown in Fig. 9. Four abrupttransitions in the length of DNA indicate the opening ofthe two turns of the nucleosomes. We note that for eachexperiment, the opening force can be different. When theforce is low, both two nucleosomes prefer to be in the fullwrapping state. When the stretching force increases, thefirst turn of the nucleosomal DNA of one of the nucleo-somes is unwrapped. After that, the next nucleosome isunpeeled partially. The larger the loading force gets, themore the probability of unwrapping of the nucleosomesbecomes. To see this, Fig. 10 shows the histogram offorces for the nucleosomes unwrapping corresponding tothe constant loading rate k load = 2 .
44 pN/s. The his-togram corresponds to the constant loading rate case,therefore there is a small shift in the opening curves ofthe first turn and the second turn. The smaller the load-ing rate gets, the smaller the shift in the histograms ofthe opening becomes.
DISCUSSION AND CONCLUSION
We have used the force and torque balance conditionsto find the equilibrium orientations of the dinucleosomein the presence of the stretching force, where the nucle-osomes were allowed to unwrap. For a given externalforce, the system settled down to the state correspond-ing to the minimum energy. We have seen that the dinu-cleosome conformational energy is affected by the inter-nucleosomal spacing, which is due to the orientation ofthe nucleosomes. When the linker DNA length is short,the bending energy corresponding to the linker becomesmore important, as seen in the conformations of Fig. 5.Therefore, by reducing the length of the linker DNA, thebending energy significantly increases and hence causesan asymmetric unwrapping of the dinucleosome, as dis-cussed in the result section.Asymmetric behavior has been also seen by Ngo etal. [22]. It has been shown that the direction of thenucleosome unwrapping depends on the DNA flexibility:different flexibilities on both sides results in an asymmet-ric unwrapping of the DNA from the stiffer side, whilea stochastic opening occurs for equal flexibilities. Thisphenomenon and the observed asymmetric unwrappingin the dinucleosome, occur in a way that reduces theenergy cost of the bent DNA. In experiments, differentfeatures of the unwinding such as the asymmetric andsymmetric openings have been observed with differentprobabilities [22, 30–32]. The different patterns of theunwinding of the dinucleosome can be understood fromthe energy landscape, (see Fig. 8, and the transitionrates of Eqs. (4a)-(4d)). We note that a higher energybarrier creates less chance of occurrence. We have alsoapplied a kinetic model to study the unwrapping dynam-ics of dinucleosome. The sequential unwrapping of thenucleosomes, as well as a delayed opening of the innerturns after the outer turns, are observed which are ingood agreement with the pulling experiments of the nu-cleosomal array [10, 13, 33].Comparison of dinucleosome and mononucleosomeopening forces shows that the presence of the second nu-cleosome slightly reduces the amount of first nucleosomeunwrapping forces (see Fig. 10). The reduction can beunderstood from the energy landscape of dinucleosome,as shown in figures 5, 6, and 7. In the dinucleosome,increasing the degrees of freedom helps the nucleosometo find a path with lower energy barriers respect to themono-nucleosome. The proposed path takes place along the path I with fluctuations of one or two binding sites.This result is in good agreement with the observationof Fitz et al. on the dinucleosome template [34]. Theyhave seen a change in the transcription dynamics of theRNA polymerase II, as well as the variation of the open-ing forces in the force-extension experiment, in the dinu-cleosome compared to mono-nucleosome. They have re-ported that the reason for these changes is the presence ofthe second nucleosome, which confirms the effectivenessof this presence.It is worth noting that the presence of the second nucle-osome changes the dynamical features of a dinucleosmerespect to a mono nucleosome, which can be understoodby looking at the energy landscapes of each case. As it isdiscussed before, in the dinucleosome there are differentpossibilities for dynamics from the initial state to the finalstate. Each path corresponds to a different energy land-scape, and the dynamical details such as the time neededfor going from one state to the final state, depend on thepath and its energy landscape. Therefore, the presencethe the second nucleosome affects the energy landscapeand may changes the dynamical aspects of the problem.This effect might be seen in the transcription rate of thenucleosomal DNA. It has been observed the transcriptionrate of RNA polymerase II through DNA becomes fasterin the presence of two nucleosomes (dinucleosome case)than for mononucleosome [34].Using cryo-EM, it has been observed that the stabil-ity of H2A-H2B dimers in the histone octamer dependson the presence of wrapped DNA in the nucleosome [32].Furthermore, the disassembly of the H2A-H2B dimerscauses further DNA unwrapping [32]. Possible exten-sions of this work may consider these effects and otherconformational changes of the histone proteins in the par-tially wrapped nucleosomes and study the dinucleosomedynamics.When the length of the linker DNA is long enough(say longer than 30 −
40 bp) , the presence of sev-eral nucleosomes does not affect on the energy landscapeand one can consider the energy landscape of an arrayof N nucleosomes as the energy landscape of N singlemono-nucleosomes. But for shorter lengths of the linkerDNA (say shorter than 20 −
30 bp), the presence of sev-eral nucleosomes affects the energy landscape. Therefore,for studying the dynamics of multi nucleosome systems,one should consider the effects of the length of the linkerDNA. Generally, obtaining the energy landscape for nu-cleosome arrays with short linker DNA is not easy, andone should solve the force and torque balance equations,mentioned and discussed in this paper and ref. [23].
ACKNOWLEDGMENTS
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