black holes ( bh ) have become rather prominent in fundamental physics ever since the fundamental results in the early 70 s showing that black holes satisfy some ` thermodynamic - like laws ' , summarized in the celebrated laws of black hole mechanics , m= a , [ bhmec ] from which one can formally relate , where the relation between _ geometrical variables _ on eq .( [ bhmec ] ) can be seen as the analogue of the first law of thermodynamics if the above association between geometric and thermodynamical objects is made .this analogy is further motivated by the fact that the surface gravity of a killing horizon is constant and the area of an event horizon always grows .this observation , together with the proposal by bekenstein and hawking that bh possess a physical entropy and temperature , as confirmed by the computation of particle creation on black hole background , gave raise to a true identification between geometrical quantities and thermodynamical variables as follows : it is not unnatural to interpret that black holes must behave as thermodynamic systems , and in particular possess a non - zero temperature ( that vanishes in the classical limit ) and an entropy ( that blows up ) .quantum theory was needed in order to identify temperature and entropy with geometrical objects , by means of planck s constant , suggesting that these identifications are quantum in nature .but , in order to have a full analogy , the question of what are the underlying degrees of freedom responsible for entropy became a pressing one .in other words , how can we account for the ( huge ) entropy associated to the black hole horizons ?the standard wisdom is that only with a full marriage of gravity and the quantum will we be able to understand this issue .this is one of the main challenges that faces any candidate quantum theory of gravity .  during the past 20 yearsthere have been several attempts to identify those degrees of freedom . in particularone has to mention the success of string theory in explaining the entropy of extremal and near - extremal bh in several dimensions  .there have also been some proposals based on causal sets  and on the use of entanglement entropy of matter fields  . within loop quantum gravity  , a leading candidate for a quantum theory of gravity, there has been some progress in describing black holes ` in equilibrium ' . in particularthis implies that the objects to be studied are assumed to be isolated , in such a way that a study of its properties will guarantee that one can separate their description from that of the rest of the environment ( as one normally does in thermodynamics ) .the resulting quantum picture is that the interaction between ` bulk states ' as described by _spin networks _ as they puncture the horizon , create horizon degrees of freedom that can ( and do ) fluctuate .these degrees of freedom are , on the one hand , independent of the bulk degrees of freedom , and on the other hand , fluctuate ` in tandem ' with their bulk counterparts , as dictated by specific quantum conditions warranting the existence of the _ quantum horizon_.the original program was developed in a series of papers  and has been further studied  and also widely reviewed in  .the purpose of this contribution is to provide a bird s eye view into the field , briefly summarizing the progress made in the past 12 years , including some recent results .this contribution can also be seen as a starting point and as a reading guide for those interested in more details .  inwhat follows , we shall in particular try to answer the following questions : how do we characterize black holes in equilibrium ?that is , what are the quantum horizon states ?how do we know which states we should count ?can we learn how entropy behaves ?can we make contact , for large black holes , with the bekenstein - hawking entropy ?can we extend the formalism and consider small , planck scale bh s ?how small is small ? that is , where does the transition from the planck scale to the ` large area limit ' occurs ?  we shall not include topics such as the possibility of treating hawking radiation  or the criteria for dealing with black holes in thermal equilibrium  .this section has two parts . in the first one we review the motivation for the need of a notion of horizon that is local and not teleological as is the case of the traditional event horizon . in the second part we briefly review the main ideas behind the isolated horizons formalism .physically , one is interested in describing black holes in equilibrium .that is , equilibrium of the horizon , not the exterior .just as in the standard analysis of physical systems subject to thermodynamics considerations , one requires _ the system _ and not the whole universe to be in equilibrium .the use of globally stationary solutions to einstein s equations to study the thermodynamics of horizons is very restrictive since one is requiring the whole universe to be stationary and not just the system , i.e. the horizon .can one capture that notion via quasi - local boundary conditions ?yes ! and the answer is provided by the isolated horizons ( ih ) formalism  .the main idea is that _ some _ boundary conditions are imposed on an inner boundary of the spacetime region under consideration .the interior region of the horizon is cut out , since the isolated horizon is regarded as a boundary .is this a physical boundary ?but one can ask whether one can make sense of it , namely whether there is a consistent prescription for incorporating this hypothesis , and a consistent variational principle is possible .a second question pertains to the physical interpretation of the boundary .if ` physics ' does not end there , in the sense that in a realistic spacetime , matter and observers can fall into the interior region with a well defined evolution , what is then the justification for ` arbitrarily ' cutting this region out ?the justification is that , being null surfaces , the exterior region ( say in an asymptotic region ) will not have access to any events inside the horizon ( even if the isolated horizon does not coincide with a possible event horizon , it will lie _ inside it _ ) , the information of what happens inside is not needed for describing the physical processes in the outside region .one can then interpret the horizon , and the degrees of freedom on it , as a ` screen ' that keeps track of those aspects of the degrees of freedom that fell in but that can still interact with the outside region .for instance , the mass of the isolated horizon has certain information of the energy of the matter that fell in , and is responsible for the gravitational field outside the horizon .the same is true for other quantities such as charge , angular momentum , etc .these horizon charges ( multipole moments ) will carry this information , and is the input needed in formulating the theory .let us summarize the main features of ih and their quantum treatment : + i ) the boundary , the 3-d isolated horizon , provides an effective description of the degrees of freedom of the _ inside region _, that is cut out in the formalism .+ ii ) the boundary conditions are such that they capture the intuitive description of a horizon in classical equilibrium and allow for a consistent variational principle .+ iii ) the quantum geometry of the horizon has independent degrees of freedom that fluctuate ` in tandem ' with the bulk quantum geometry .+ iv ) the quantum boundary degrees of freedom are then responsible for the entropy .+ v ) the entropy thus found can be interpreted as the entropy assigned by an ` outside observer ' to the ( 2-dim ) horizon .0.15 cm   0.15 cm fig.1 left : the physical situation one expects to describe .the collapse of a stellar object creates an event horizon that settles down ( rather quickly ) and in the asymptotic future is non - expanding , giving rise to an isolated horizon .right : even if there is more matter falling in the future , there will be portions of the horizon that will be isolated .0.15 cm  just as for other approaches to black hole entropy , the lqg treatment is not free from some interpretational issues .for instance , is the entropy to be regarded as the entropy contained by the horizon ? is there some ` holographic principle ' in action ? can the result be associated to entanglement entropy between the interior and the exterior ? , etc .some of these questions have been clarified but there are still some for which we have no answer yet ( see for instance the discussion in ) .   in this part we will provide the main ideas in the definition of isolated horizons . for full detailssee .an isolated horizon is a null , non - expanding 3d - surface , equipped with some notion of translational symmetry along its generators ( it is assumed to have a congruence of null vectors generating it ) .there are three main consequences of these boundary conditions :  0.15 cm i ) the gravitational degrees of freedom induced on the horizon are captured by a connection , w_a = - ^i_ar_i where is the spin connection of the canonical theory on . thus , there is an effective reduction of the gauge symmetry from to .0.15 cm ii ) the total symplectic structure of the theory ( and this is true even when matter is present ) gets split as , _ tot=_bulk + _ hor with this is precisely the symplectic structure one would get if we were considering a chern - simons theory for the u(1 ) connection on the three dimensional manifold with a spatial section ( recall that chern simons does not require a metric , so the fact that is null is irrelevant ) .0.15 cm iii ) finally , the ` connection part ' and the ` triad part ' at the horizon must satisfy the condition , [ hor - cons ] f_ab =- e^i_abr_i , the so called ` horizon constraint ' . here is the curvature of the u(1 ) connection .it is interesting to explore the consequences of the boundary conditions in the hamiltonian framework . a detailed study of the canonical theory reveals an interesting structure . in particular, the formalism tells us what is gauge and what not . to be precise , with respect to the constraints that appear in the canonical formalism , we know that :  0.15 cm a ) the relation between curvature and triad , the horizon constraint ( [ hor - cons ] ) , is equivalent to gauss law .0.15 cm b ) diffeomorphisms that leave invariant ( i.e. that , when restricted to the horizon map to itself ) are gauge ( i.e. the vector fields generating infinitesimal diffeomorphisms are tangent to ) .0.15 cm c ) the scalar constraint must have a vanishing lapse at the horizon .thus , the gauge transformations generated by the scalar constraint ( that depend of the lapse ) , leave the horizon untouched .in particular , this implies that any gauge and diff - invariant observable _ is _ a full dirac observable .this list includes all multipole moments of the horizon .this last point is the reason behind the fact that one can sensibly talk about the quantum theory of black holes in lqg even when we have not solved the quantum dynamics in the bulk .that is , since in the quantum theory one has to implement the constraints , the fact that the lapse vanishes at the horizon implies that , from the horizon perspective , any quantum state that satisfies gauss law and is diffeomorphism invariant will by a physical state , given that the hamiltonian constraint imposes no further condition .of course , one has to make sure that the quantum horizon states ` interact ' properly with the bulk states for which the dynamics is still not fully understood .this represents one of the current challenges .loop quantum gravity is based on a canonical formulation of general relativity in terms of connections and triads ( for a brief introduction see ) .the basic canonical variables are : a^i_a su(2);e^a_i with , and real the barbero - immirzi parameter ( bi ) . loop quantum gravity defined on a manifold without boundary is based on two fundamental observables of the basic variables : , h_e(a):=p(_e a ) and , e(f , s):=_s s^ab e_iabf^i . where .the main assumption of loop quantum gravity is that these quantities become well defined operators in the quantum theory .thus , the starting point for lqg is the so called holonomy - flux algebra  .an important question is how many consistent representations of the -algebra there are . in recent years, the lost collaboration proved the following result : there is a unique representation of the holonomy - flux algebra on a hilbert space that is _ diffeomorphism invariant _ .this representation corresponds precisely to the construction of ashtekar and lewandowski .let us now give a brief description of this resulting hilbert space .first we can characterize it in terms of spin networks : _al= _ graphsh_=span ofall spin networks 0.3 cm   fig 2 . a spin network consists of a graph together with labels for the edges and for the vertices .  0.5cm a spin network , represents a particularly convenient basis for the theory .it is a state labelled by a graph , and some colorings associated to edges and vertices .the spin networks have a very nice interpretation in terms of the quantum geometry they generate .they are the eigenstates of the quantized geometry , such as the area operator , [ s]=8_pl^2 _ edges where the sum is over all the intersection points of the edges with the surface .the standard interpretation is that the edges of the graph excite the quantum geometry of the surface at the intersection points between and .the edges can be seen as quantum fluxes of area .   0.4 cm fig.3 .an artist impression of a black hole in lqc .the edges of the state on the bulk puncture the horizon endowing it with area through the labels s and with intrinsic curvature through the s .just as in the classical description of the gravitational field with an ih , the phase space could be decomposed in a bulk part and a horizon part , a basic assumption is that the total hilbert space is a tensor product of the form : = h_v _ s where , the surface hilbert space , can be built from chern simons hilbert spaces for a sphere with punctures .this represents the ` kinematical hilbert space ' .  in order to go to the physical theory , the conditions on that we need to impose are : invariance under diffeomorphisms of and the quantum condition on , the quantum equivalence of eq .( [ hor - cons ] ) : ( id_ab+^i_abr_i ) = 0 .[q - cond ] then , the theory we are considering is a quantum gravity theory , with an isolated horizon of fixed area ( and multiple - moments ). a physical state would be such that , in the bulk satisfy the ordinary constraints and , at the horizon , the _ quantum horizon condition_. .15 cm  _ entropy ._ we shall consider the simplest case of pure gravity with a non - rotating horizon . in this case , from the outset , we are given a black hole of area .the question then is : what entropy can we assign to it ? let us take the microcanonical viewpoint . to compute the entropywe shall count the number of states such that they satisfy : .15 cm   the area eigenvalue is in the interval $ ] + the quantum horizon condition ( [ q - cond ] ) is satisfied ..15 cm  the entropy will be then given by , = .the challenge now is to identify those states that satisfy the two conditions , and count them ..15 cm  _ characterization of the states ._ there is a convenient way of characterizing the states by means of the spin network basis .if an edge of a spin network with label ends at the horizon , it creates a puncture , with label .the area of the horizon will be the area that the operator on the bulk assigns to it : .is there any other quantum number associated to the punctures ?they are given by eigenstates of that are also half integers , such that .the quantum horizon condition relates these eigenstates to those of the chern - simons theory .the requirement that the horizon is a ( topological ) sphere then imposes a ` total projection condition ' on : _i m_i=0 that has to be taken into account as well .a quantum horizon state can be conveniently characterized by a set of punctures and to each one a pair of half integer , where the three previous conditions impose some restrictions on the possible values of the labels .  if we are given punctures and two assignments of labels and .are they physically distinguishable ? or a there some ` permutations ' of the labels that give indistinguishable states ?that is , what is the statistics of the punctures ?  as usual, we should let the theory tell us .one does _ not _ postulate any statistics . if one treats in a careful way the action of the diffeomorphisms on the punctures one learns that when one has a pair of punctures with the same labels s and s , then the punctures are indistinguishable and one should not count them twice . in all other cases the states are distinguishable ..15 cm  _ the counting ._ we start with an isolated horizon , with area ( assumed to be of the order of several planck areas ) and ask how many states are there compatible with the two conditions , and taking into account the distinguishability of the states .one can approach the problem in a two step process .first step : count just the different configurations and forget about .thus , given , where means the number of punctures with label ,we count the number of states : =_ j(2j + 1)^n_j with .taking the _ large area approximation _ , and using the sterling approximation , one gets as the dominant term : s= with the solution s if the s are the same . in that casewe get a different linear dependence with  . ) ] to .  as a second step one introduces the projection constraint .this has the effect of introducing a correction to the entropy area relation as an infinite series , where the first correction is logarithmic : s= - ( a ) + note that one gets , in the complete counting , the asymptotic linear dependence on area . if we want to make contact with the bekenstein - hawking formula we have to make use of the freedom in lqg provided by the bi parameter and choose .depends on the counting ] the coefficient of the logarithmic correction seems to be universal and independent of the particular counting ( for other topologies of the horizon , it might change ) .an important observation is that the formalism can be generalized to more general situations , the combinatorial problem is the same and therefore the result is that _ the same value of will yield the bh entropy_. these more general horizons include arbitrary distortion and rotation in vacuum gravity as well as coupling to electromagnetic , dilatonic , yang - mills , cosmological constant , and non - minimally coupled scalar fields .  in the following sections we shall review new developments that have occurred since 2006 .these include a new phenomena found when computing directly the number of states for small black holes , and an exact counting of states by use of methods from number theory .in this section we will describe the results found when considering small planck size horizons for which the counting of states is possible . for that onetells a computer how to count for a range of area at the planck scale . with the availability of having an exact algorithm under control, one can ask , for instance , what is the effect of considering or not the ` projection constraint ' . in the large area approximation it is responsible for the first , logarithmic , correction term .one could also ask when is the linear dependence with area first observed .that is , when do we see a transition from deep quantum effects to ` large areas ' ?let us briefly summarize the results reported in .what was found is that , without the projection constraint , the entropy approaches very fast a ` smooth ' function of area with the slope found in the analytical calculations . when including the constraint , the relation between entropy and area became oscillatory , with a well identified period , that _ on average _ , introduced the expected logarithmic correctionsthis was already identified for horizons that are as small as 100 .for details see .furthermore , it was seen that , by analyzing the ` black hole spectrum ' ( i.e. the degeneracy of states as function of area ) , both the oscillations found with a large value of as well as these structures in the ` spectrum ' possess the same periodicity .a natural question is whether there is any physical significance to this periodicity .it turns out that , if one chooses the interval : , the plot of the entropy _ vs _ area becomes stair - like , as can be seen in fig.4 :  0.25 cm   fig .the entropy as function of area shows a step - like behavior when the interval is chosen to coincide with the periodicity .0.3 cm  what one notes is that the entropy has a completely different behavior for this particular choice of interval : instead of oscillations , the entropy seems to increase in discrete steps .furthermore , the height of the steps seems to approach a constant value as the area of the horizon grows , thus implementing in a rather subtle way the conjecture by bekenstein that entropy should be equidistant for large black holes .quite remarkably , this result is robust , namely , it is _ independent _ of the counting .  while the constant number in which the entropy of large black holes ` jumps ' seems to approach : s2_0   fig .5 . the black hole spectrum shows some peaks of higher degeneracy together with some valleys .this is the origin of the step - like behavior of entropy .0.3 cm  some recent proposals have provided a heuristic understanding of the origin of these peaks and valleys . to summarize these results ,the model there proposed allows one to think of the states as organized in _ bands _ , labelled by certain combination of the total number of punctures and ` spin ' . by employing the analytic distribution that maximizes degeneracy , as originally introduced in , one can find the ` average area ' , for each band associated with this maximum degeneracy configuration , from which one can compute the change in area from peak to peak as , a = an interesting observation is that if one parametrizes this number as , one can see that must be a constant , independent of the counting ( since it only depends on the degeneracy of the states as functions of s and s ) . from the observed periodicity in the direct counting ,it was conjectured in that the value of is .interestingly , the approximate formula found in for both countings , yield a slightly different approximate values for the parameter , with , and the relative difference of the order of .this shows that the approximation is not exact and one needs a better analytical understanding of the combinatorial problem .recent progress using number theoretical considerations has turned out to be useful for the purpose of understanding the emergence of the discrete structures . in this study ,a reformulation of the counting of states and an exact counting of the number of states has been achieved recently . in this parti shall briefly summarize these results .there are two main steps involved in the counting . in the first one ,one finds a complete characterization of the area spectrum , that is , of the eigenvalues of the area operator in terms of so - called ` square - free numbers ' .then , given an allowed area - eigenvalue , one computes the number of possible ` sets ' of labels ( be them and/or ) that are compatible with that values .this sets contain also the number of punctures that have a given label assigned . in the second part of the counting process oneassigns a degeneracy to each ` set ' coming from the possible ` permutation ' of labels . at the end ,one obtains an exact number of consistent states for the given value of area .the final step , that is , the computation of the entropy can then be computed either by considering an interval as previously defined , or by summing over all values of area up until the prescribed value .let us now describe how one achieves the first step in the counting process .first , one notes that the area eigenvalues ( when measured in units of can be written as : a=_i=1^n=_k=1^k_maxn_k[ec - chida ] where are integers labelling the punctures and we have recast the sum by rearranging the punctures by their label ( is the number of punctures with label ) and summing over labels .the idea here is to employ the square free numbers as a basis for the area eigenvalues ( the numbers are ` linearly independent ' under arbitrary linear combinations with integer coefficients ) .each of the terms in the sum can be recast as an integer times a ` square free number ' ( by means of the prime decomposition of the quantities inside the square root ) .thus the sum becomes , where , for some integer .let us summarize .if we specify a square free number , we want to know for which values of integers and is the previous equation satisfied , which would tell us ( for each possible solution ) the allowed values of the labels .this equation is known as the pell equation and has an infinite number of solutions ( labelled by ) .we can then use these solutions to rewrite ( [ ec - chida ] ) as if we use the fact that the numbers are linearly independent , we can split the equation in a series of different equations of the form , where the s and the s are known , as solutions to the pell equation , and the unknowns here are the numbers .if these diophantine equations admit solutions , then belongs to the relevant part of the spectrum of the area operator , the numbers s give the spins involved , and the numbers s count the number of times that edges labelled by the spin pierce the horizon .thus , given a linear combination of square free number as the area eigenvalue , the procedure here described provides an answer to the ` degeneracy ' associated to the different pairs defining the different _ spin configurations_. the next step in order to obtain the total number of states is to count the ` -degeneracy ' , namely the different ways of accommodating the s on a given spin configuration .it is at this point that the two different countings ( gm and dlm ) provide different answers .both cases can be treated in terms of _ fusion numbers _ and fusion matrices employed in cft . for details see . just as an illustration , for the dlm counting the answercan be exactly written as with , allowing to have exact expressions for the degeneracy of states .of course , this strategy confirms the results of , but also allows to compute the spectrum of larger black holes with the same computational capacity .these results represent a starting point for more refined asymptotic analysis , by means of generating functions for the combinatorial problem , that will shed more light on the behavior of macroscopic black holes .for instance , an important question to be addressed is whether the oscillatory behavior on entropy , the _ entropy quantization _ , together with its possible implications for hawking radiation , is still present for large black holes .let us now discuss some open questions regarding quantum black holes and the progress that has been made within lqg . by the mere fact that in the isolated horizon framework one is considering only the outside region of a spacetime containing a back hole , one is not addressing the issue of the singularity .the possible singularity resolution has been analyzed in a series of papers using loop quantization techniques .the starting point of such treatments is the minisuperspace of homogeneous cosmologies on a spatial manifold with topology .these ` kantowski - sachs models ' are important given that the interior region of the global schwarzschild solution belongs to this class .it is thus natural to attempt to employ the same techniques that have been extremely useful in the treatment of ( minisuperspaces corresponding to ) homogeneous and isotropic models in cosmology ( see , for instance for a recent summary of such methods ) .those results suggest that the classical singularity inside the horizon , just as in the case of isotropic cosmologies , is avoided , and the quantum evolution continues past it , but more work is needed to reach a definite conclusion . in particular , none of the presently available models is able to overcome consistency requirements that select a unique quantization in the isotropic sector .an important open issue is how to specify black hole / horizon states from the full theory .that is , without assuming that there was a classical horizon to begin with .some progress in this direction has been made in two fronts , but still at some preliminary stage . a proposal for defining coherent states that approximate a spacetime with a black hole ,have been used to count the number of black hole states . even when potentially important , this approach is still in its early stages given that the coherent states are kinematical , and there is not a full control on the dynamical sector of the theory .another proposal for identifying black hole states from the full set of states was made within the context of symmetry reduced models in ref . . here, the idea is to specify operators on a kinematical hilbert space such that they project the kinematical states onto a ` black hole sector ' . just as in the previous case , there are still several consistence criteria that this approach must satisfy before one can make concrete predictions . in particular both approaches lack a description for dynamical processes that the horizon might undergo .  if the singularity resolution were also generic , and there existed a spacetime interpretation beyond the ` would be singularity ' , one would be lead to the ashtekar - bojowald paradigm for evaporation and ( lack of ) information loss .this picture includes the description of dynamical processes that are no longer described by the isolated horizon formalism .one needs then to consider the more general framework of _ dynamical horizons _if this picture is physically correct , there is no classical singularity and no event horizon ever forms .still , there _ is _ a horizon that forms , grows and then shrinks due to hawking radiation .information is not lost , even when , for certain observers , hawking radiation appears to be thermal ( for more details within a simple model see ) .these results are certainly intriguing and suggest a resolution of the ` information loss problem ' due to true quantum geometric effects .needless to say , more work is needed to unravel this mystery .let us summarize what we have learned from the merger of isolated horizons and loop quantum gravity .first , as we have shown , isolated horizons provide a consistent framework to incorporate black holes that are physically in equilibrium , as classical objects . as we have argued , one can consistently quantize the theory , as described by the ih phase space , employing both the methods of quantum geometry that are useful in the bulk , together with techniques from chern - simons theory on a sphere .a detailed study of the action of the constraints allows us to give a full characterization of the quantum horizon degrees of freedom that contribute to the entropy .it is found that the entropy is _ finite _ , without the need of a regulator nor a cut - off , and that its dominant term is linear in area for large horizons in planck units .furthermore , the formalism allows us to translate the entropy counting into a purely combinatorial problem for which one can attempt algorithmic _ brute force _computations , as well as number - theoretic treatments .a very important feature of this formalism is that one can incorporate and count the entropy of a whole class of different black holes , where one can include arbitrary distortion and rotation in vacuum gravity as well as coupling to electromagnetic , dilatonic , yang - mills and non - minimally coupled scalar fields . in all these cases the combinatorial problem to be solved is _ the same _ ( even when its translation into physically relevant quantities might vary ) and therefore , entropy is _ always _ proportional to area in the large area limit ( or with the expected contribution from the scalar field in the non - minimally coupled case ) .  as we have explored , when one considers the problem of a direct counting of the number of states , and thus being forced to consider small horizons , several unexpected features appear for these planck size black holes . while one recovers the asymptotic linear dependence on area and the logarithmic correction ( with the right coefficient ) , from which we can say something about bi parameter , a new behavior is observed for small horizons .it is found that there are oscillations in entropy with a constant periodicity .furthermore , when properly interpreted , this points to an effective _ quantization _ of the entropy in equidistant steps .this observed behavior suggests that loop quantum gravity can make contact , in a rather subtle manner , with both bekenstein s heuristic model , and the mukhanov - bekenstein effect . whether this scenario is realized or not remains an intriguing question .recently , attempts to understand the origin of the ` black hole spectrum ' responsible for the entropy quantization have been put forward , which have been able to give some intuitive understanding of the effect . in particular , there has been some progress to understand , from a heuristic perspective , the origin of the ` bands ' in the spectrum and their equidistant nature .a pressing question here is whether the discrete structures found at the planck scale are still present for macroscopic black holes .one would also like to understand whether the constant actually is equal to as the numerical computations and the heuristic considerations seem to suggest .if this were the case , one would need to understand its origin .as we have here tried to convey , in the past couple years there has been exciting progress in our understanding of quantum black holes within loop quantum gravity , but there are still important question that remain open regarding the detailed relation between gravity , entropy and the quantum .my understanding of black holes in loop quantum gravity grew from the interaction with numerous colleagues to whom i am grateful , including a. ashtekar , j. baez , j. daz - polo , j. engle , e. fernndez - borja , k. krasnov , j. lewandowski , c. rovelli and h. sahlmann .i would like to thank j.p .ruiz - daz for fig.3 .this work was in part supported by conacyt u47857-f grant , by nsf phy04 - 56913 and by the eberly research funds of penn state .bardeen j m , carter b and hawking s w 1973 the four laws of black hole mechanics _ commun . math .phys . _  * 31 * 161 bekenstein j d 1973 black holes and entropy _ phys_ * d 7 * 2333 ; hawking s w 1975 particle creation by black holes _ commun . math . phys . _  * 43 * 199   bombelli l , koul r k , lee j h and sorkin r d 1986 a quantum source of entropy for black holes _ phys .d _ * 34 * 373 srednicki m 1993 entropy and area _ phys . rev .lett . _  * 71 * 666 [ arxiv : hep - th/9303048 ] .rovelli c 2004 _ quantum gravity _( cambridge , uk : cambridge university press ) ; thiemann t 2001 introduction to modern canonical quantum general relativity [ gr - qc/0110034 ] ashtekar a and lewandowski j 2004 background independent quantum gravity : a status report _ class .* 21 * r53 [ gr - 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qc ] ] .i , barbero j f , diaz - polo j , fernandez - borja e and villasenor ejs 2008 black hole state counting in lqg : a number theoretical approach __  * 100 * , 211301 ( 2008 ) .[ arxiv:0802.4077 [ gr - qc ] ] ; barbero jf and villasenor ejs 2008 generating functions for black hole entropy in loop quantum gravity , _ phys . rev ._  d * 77 * , 121502 ( 2008 ) .[ arxiv:0804.4784 [ gr - qc ] ]  barbero j f and villasenor ejs _ on the computation of black hole entropy in loop quantum gravity _, [ arxiv:0810.1599 [ gr - qc ] ] .jacobson t , marolf d and rovelli c 2005 black hole entropy : inside or out ? _ int . j. theor .phys . _  * 44 * 1807 [ hep - th/0501103 ] .ashtekar a and krishnan b 2004 isolated and dynamical horizons and their aplications _ living rev.rel._ * 7 * 10 [ gr - qc/0407042 ] . corichi a , diaz - polo j and fernandez - borja e 2007 loop quantum gravity and planck - size black hole entropy , _ j. phys .ser . _  * 68 * 012031 [ arxiv : gr - qc/0703116 ] ; corichi a 2008 _ black hole entropy in lqg : a review _ , to be published .barreira m , carfora m and rovelli c 1999 physics with nonperturbative quantum gravity : radiation from a quantum black hole _ gen. rel ._  * 28 * 1293 [ arxiv : gr - 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Abstract: black holes in equilibrium and the counting of their entropy within loop quantum gravity are reviewed . in particular , we focus on the conceptual setting of the formalism , briefly summarizing the main results of the classical formalism and its quantization . we then focus on recent results for small , planck scale , black holes , where new structures have been shown to arise , in particular an effective _ quantization _ of the entropy . we discuss recent results that employ in a very effective manner results from number theory , providing a complete solution to the counting of black hole entropy . we end with some comments on other approaches that are motivated by loop quantum gravity .


a deoxyribonucleic acid ( dna ) molecule is a linear heteropolymer whose monomeric subunits , called nucleotides , are denoted by the four letters a , t , c and g. the sequence of these nucleotides in a dna molecule chemically encodes genetic informations . in the nucleus of an eukaryotic cell, dna is stored in a hierarchically organized structure called chromatin .the primary repeating unit of chromatin at the lowest level of the hierarchical structure is a nucleosome .the cylindrically shaped core of each nucleosome consists of an octamer of histone proteins around which 146 base pairs ( i.e., nm ) of the the double stranded dna is wrapped about two turns ( more precisely , 1.7 helical turns ) ; the arrangement is reminiscent of wrapping of a thread around a spool .there are equispaced sites , at intervals of base pairs ( bp ) , on the surface of the cylindrical spool .electrostatic attraction between these binding sites on the histone spool and the oppositely charged dna seems to dominate the histone - dna interactions which stabilize the nucleosomes . throughout this paper , the helical curve formed by the histone - dna overlapwill be called the `` footprint '' .the dna stores the genetic blueprint of an organism .if nucleosomes were static , segments of dna buried in nucleosomes would not be accessible for various functions involving the corresponding gene .however , in reality , nucleosomes are dynamic .spontaneous dynamics of nucleosomes are usually consequences of thermal fluctuations whereas the active dynamic processes are driven by special purpose molecular machines called chromatin remodeling enzymes ( cre ) fuelled by atp . various aspects of chromatin dynamics has received some attention of theoretical modelers , including physicists , over the last few years .in general , `` chromatin remodelling '' refers to a range of enzyme mediated structural transitions that occur during gene regulation in eukaryotic cells . to make dna , which is wrapped around histone octamer , accessible for various dna - dependent processes, it is always necessary to rearrange or mobilize the nucleosomes . in principle , there are at least four different ways in which a cre can affect the nucleosomes : ( i ) _ sliding _ the histone octamer , i.e. , repositioning of the entire histone spool , on the dsdna ; ( ii ) _ exchange _ of one or more of the histone subunits of the spool with those in the surrounding solution ( also called _ replacement _ of histones ) ( iii ) _ removal _ of one or more of the histone subunits of the spool , leaving the remaining subunits intact , and ( iv ) complete _ ejection _ of the whole histone octamer without replacement . our theoretical work here is closely related to _sliding_.  in the next sectionwe describe a scenario in which either a cre motor ( or , other atp - dependent motors that translocate along dsdna ) traverse the `` footprint '' . because of the stochasticity of the underlying mechano - chemical kinetics , the _ footprint traversal time _ ( ftt )is a fluctuating random variable . extending an earlier model developed by chou , we analytically calculate the _ mean _ ftt ( mftt ) of the cre motor .  to our surprise, we found that the atp - dependence of the various atp - driven activities of cres have not been studied systematically in the published literature .in particular , we address the question : how does the mftt of a cre motor vary with the variation of the concentration of atp ?this rate is not necessarily directly proportional to the rate of atp hydrolysis by the cre because the mechanical sliding of the nucleosome need not be tightly coupled with the hydrolysis of atp by the cre .therefore , we develope here an analytical theory predicting the atp - dependence of the atp - dependent footprint - traversal by cre . we hope our result will stimulate systematic experimental investigations on the atp - dependence of atp - dependent cres .chromatin is not a frozen static aggregate of dna and proteins .spontaneous thermal fluctuations can cause a transient unwrapping and rewrapping of the nucleosomal dna from one end of the nucleosome spool ; the corresponding rates for an isolated single nucleosome are , typically , and , respectively . in other words , once wrapped fully , the nucleosomal dna remains in that state for about before unwrapping again spontaneously ; however , it waits in the unwrapped state only for about before rewrapping again spontaneously .surprisingly , the accessibility of the nucleosomal dna is only modestly affected if instead of a single nucleosome the experiment is repeated with an array of homogeneously distributed nucleosomes .moreover , folding of an array of nucleosomes makes the linker dna about times less accessible .furthermore , on the nucleosomal dna , the farther is a site from the entry and exit points , the longer one has to wait to access it by the rarer spontaneous fluctuation of sufficiently large size .thus , nucleosomal dna far from both the entry and exit sites is practically inaccessible by spontaneous thermal fluctuations .can a nucleosome slide _ spontaneously _ by thermal fluctuations thereby exposing the nucleosomal dna ?interestingly , spontaneous repositioning of nucleosome on dna strands is a well known phenomenon .how can one reconcile accessibility of nucleosomal dna by such repositioning with the difficulty of access by unwrapping from either end of the spool ? if the dna were to move unidirectionally along its own superhelical contour on the surface of the histone , at every step it would have to first _transiently _ detach simultaneously from all the binding sites and then reattach at the same sites after its contour gets shifted by bp ( or multiples of bp ) .but , the energy cost of the simultaneous detachment of the dna from all the binding sites is prohibitively large because the total energy of binding at the sites is about  .but , why ca nt the cylindrical spool simply roll on the wrapped nucleosomal dna thereby repositioning itself ?if the nucleosome rolls by detaching dna from one end of the spool , can not it compensate this loss of binding energy by simulataneous attachment with a binding at the other end ?if such an energy compensation were possible , detachment from only one binding site would be required at a time .but , the cylindrical spool has a finite size on which only a finite number ( ) of binding sites for dna are accomodated . therefore , by rolling over the dna the spool would not offer any vacant binding site to the dna with which it can bind .this rolling mechanism would successfully lead to spontaneous sliding of the nucleosome only if the histone spool were infinite with an infinite sequence of binding sites for dna on its surface .we now describe a plausible mechanism for spontaneous sliding of a nucleosome . in the process of normal `` breathing '' ,most often the spontaneously unwrapped flap rewraps exactly to its original position on the histone surface . however , if the rewrapping of a unwrapped flap takes place at a slightly displaced location on the histone spool a small bulge ( or loop ) of dna forms on the surface of the histones .since the successive binding sites are separated by , the length of the loop is quantized in the multiples of  .such a spontaneously created dna loop , can diffuse in an unbiased manner on the surface of the histone spool . in the beginning of each step dna from one end of the loop detaches from the histone spool , but the consequent energy loss is made up by the attachment of dna at the other end of the loop to the histone spool before the step is completed .consequently , by this diffusive dynamics , the dna loop can traverse the entire length of the binding sites on the histone spool of a nucleosome which will manifest as sliding of the nucleosome by a length that is exactly equal to the length of dna in the loop .the diffusing dna bulge can be formed by a `` twist '' , rather than bending , of dna .spontaneous sliding of a nucleosome , however , is too slow to support intranuclear processes which need access to nucleosomal dna .it is now widely agreed that atp - dependent active remodeling of nucleosome can account for the fast sliding of nucleosomes .nevertheless , bulging dna loop is expected to play a key role in the remodeling process .our model describes how a cre motor can wedge itself at the fork between the histone spool and a transiently detached segment of dsdna and , by exploiting the spontaneously diffusing loop by an atp - dependent ratcheting , traverse the footprint in a directed manner .because of the intrinsic stochasticity of the mechano - chemistry of the cre and that of the diffusive motion of the dna loop , the overall motion of the cre is noisy and the time it takes to traverse the footprint is random .the main question we address in this paper is the following : if is the mftt , what is the dependence of on the concentration of atp in the surrounding aqueous medium ? to our knowledge , in the published literature neither systematic experiments nor any analytical theory has addressed this question . in this paper , by extending chou s model of cre , we capture the role of atp explicitly and derive an analytical expression for the dependence of on the concentration of atp .a cre may be regarded as a molecular motor where input energy is derived from atp hydrolysis and the output is mechanical work .the directed movement of the cre may be caused either by a _ power stroke _ or by a _brownian ratchet _mechanism .the kinetics of the atp - dependent cre motor is formulated in our model in terms of a set of master equations .the kinetic scheme can be interpreted in terms of both power stroke and brownian ratchet mechanisms .one interesting question in the context of cre is whether the cre translocates along the dna by moving around the nucleosome , or whether the cre anchors on the histone octamer and `` pumps '' dna by pulling around the octamer . from the perspective of physicists, these two alternative scenarios can be viewed as merely a change of frame of reference- one is fixed with respect to the dna whereas the other is fixed with respect to the cre .therefore , we describe the operation of the cre with respect to a reference frame with respect to which the cre translocates along the dna ; but , the model can be reformulated by a coordinate transformation so as to capture the alternative scenario where the cre pumps the dna .we model a mono nucleosome where a dsdna is wrapped one - and - three - fourth turn around a disc - shaped spool made of histone proteins ( see fig.[fig - spool](a ) ) . following chou , we consider the scenario where the cre `` wedges itself underneath the histone '' .the sites of histone - dna contact along the dna chain is represented as a one - dimensional lattice . therefore , the lattice constant is , typically , bp ( ( see fig.[fig - spool](b ) ) .the total number of lattice sites is equal to the total number of histone - dna contact in a single nucleosome .   in this subsectionwe present a summary of chou s ideas which we need in the next subsection where we extend chou s model . herewe consider the simple situation when no cre is present and the kinetics of the system is governed solely by spontaneous thermal fluctuations . because of these fluctuations , from either end of the histone - dna contact region , small segments of dna momentarily unwrap from the histone spool at a rate . for energetic reasons ,the most likely length of such a segment would be one lattice spacing , i.e. , about .following chou , we call such unwrapped segments a `` flap ''. the rate of the reverse transition , in which re - binding of the dna flap with the histone , takes place at a rate .a flap need not re - make the original histone - dna contact . instead , by pulling in an extra segment of the dna , its next segment can bind with the last binding site on the histone spool , with rate thereby forming what chou referred to as a loop. while located at either end of the lattice , a loop can revert to a flap at a rate .the rates and are well approximated by  where is the energy cost of bending the dna into the shape of the loop .a loop can step forward or backward . in the _ absence _ of any cre ,the rates of the forward and backward steppings of the loop are equal ( denoted by ) , provided the size of the loop remains unaltered ( see fig.[fig - diffn1 ] ) ; in each forward step it unwraps one segment of dna from the histone in the direction of its hop and re - wraps another equally long segment behind it .therefore , one can approximate by   when a loop , after entering the lattice from one end , makes an eventual exit from the other end , it completes the `` sliding '' of the histone spool by a distance along the dna in the opposite direction .therefore , from the perspective of the sliding histone spool , its effective rate of hopping by a step of size along the dsdna strand is the same as the rate at which a dna loop of length traverses the lattice of sites from one end to the other .suppose denotes the probability that the loop is located at ( ) .following chou s arguments , based on master equations for , one gets  in the absence of a cre , the traversal of a dna loop of length from left to right is as likely as that from right to left .therefore , the histone spool can slide forward or backward , with equal rate , by a step of size .as we ll see in the next subsection , peeling off of the dna from the histone spool by a cre motor keeps decreasing the effective value of which , in turn , increases the effective sliding rate .next , we consider the effect of dna loop diffusion on the atp - dependent translocation kinetics of a cre . the model and results presented in this subsection are extensions of chou s work by incorporating explicitly a brownian ratchet mechanism for cre motors .  as in ref., we assume that the step size of the cre motor is identical to the length of the thermally generated dna loop .therefore , the mechanical movements of the cre motor can be described as that of a `` particle '' on the one - dimensional lattice on which the equi - spaced sites denote the histone - dna contact points .we denote the position of the cre motor on this lattice by the integers .we now extend chou s model by exploiting a superficial similarity with the garai - chowdhury - betterton ( gcb ) model for the brownian ratchet mechanism of monomeric helicase motors .a dna helicase unwinds a dsdna and translocates along one of two strands . at any arbitrary instant of time, the configuration of the system looks very similar to that shown in fig.[fig - spool](b ) except that the surface of the dna spool and the dsdna would be replaced by the two strands of the dsdna itself .the lattice constant is in the case of a helicase whereas it is about bp in fig.[fig - spool](b ) . in the brownian ratchet mechanism , momentary local unwinding of a segment ,typically , bp long , takes place at the fork by spontaneous thermal fluctuation ; the opportunistic advance of the helicase merely prevents closure of the segment .similarly , in the brownian ratchet mechanism of the cre , the cre is assumed to `` wedge '' itself just in front of the dna - histone fork .the cre motor can move forward only if the segment in front of it is unwrapped by thermal fluctuation .the mechano - chemical cycle of the cre is captured in our model exactly the same way in which that of the helicase was formulated in the gcb model .we assume that sequence of states in each mechano - chemical cycle of a cre can be combined into two distinct groups which we label by the integers and ( see fig.[fig - omega1221 ] ) .the allowed transitions and the corresponding rate constants are shown in fig.[fig - model1 ] .the physical processes captured by these rate constants can be motivated by a comparison with the abstract brownian ratchet mechanism , illustrated in fig.[fig - brrat ] .atp hydrolysis by the cre drives its transition from the state to the state at a rate . let us assume that the motor experiences two different types of potentials in the states and .let us further assume that initially the periodic potential , with asymmetric sawtooth - like period , is kept on for sometime and during this time the motor settles at a position that coincides with one of the minima of this potential .now if this potential is switched off then the probability distribution of the position of the motor will spread as a symmetric gaussian . after sometime this gaussian profile is broad enough to overlap with the next well ( shadded region in the fig.[fig - brrat ] ) , in addition to the original well .now if the sawtooth potential is again switched on then , with a non - zero probability ( that is proportional to the area of the shaded region ) the motor will find itself in the next well .our model accounts for this possibility with the transition associated with the rate constant .there is also a finite probability that the particle stays back in its original well ; this is captured by the transition with the rate constant .the cre motor would step forward at the rate if the next site in front is cleared .but if the next site is not cleared and it has to wait for the unwrapping of the dna segment by thermal fluctuation .consequently , its _ effective _ hoping rate is reduced from the free hopping rate by a factor that depends on both and .  when a diffusing loop reaches in front of the motor it momentarily creates a flap of two bond segments .three different transitions are now possible ( see fig.[fig - fate ] ) : ( i ) the motor s position remains unaltered while the two open segments close , ( ii ) the motor moves forward by one step while one segment of the flap closes ; ( iii ) the motor moves forward by two steps and the flap can not close .the rate for the process ( i ) is irrespective of the `` chemical '' state of the motor .however , the rates of the processes ( ii ) and ( iii ) depend on whether the motor was in the `` chemical '' state or .if the motor is in the state , the rate of the process ( ii ) is given by and that of the process ( iii ) is given by .therefore , with the normalization constant where the symbols , and are the probabilities of the processes ( i ) , ( ii ) and ( iii ) above when the motor is in the `` chemical '' state . similarly , with the normalization constant are the corresponding probabilities , when the motor is in the `` chemical '' state .\(a ) +  and , respectively.,title="fig : " ] + ( b ) +  and , respectively.,title="fig : " ]  suppose , is the maximum number of histone - dna contacts possible in the nucleosome .let denote the instantaneous position of the motor . is the distance between the motor and the far end of histone - dna contact .the master equations for the probabilities are as follows :      we define as the probability that the histone - dna contacts are intact at time , irrespective of the position of the cre motor . from equations ( [ eq - mn1])-([eq - mn10 ] ) , summing over , we get the following equations : for n(n+1 ) +\nonumber \\ & & \omega^{f}_{12}p_{2,n+1}+\omega_{12}p_{2,n}-\omega_{21}p_{1,n}\end{aligned}\ ] ] and +\nonumber \\ & & \omega_{21}p_{1,n}-\omega_{12}p_{2,n}-\omega^{f}_{12}p_{2,n } \end{aligned}\ ] ] for n = n and for 3 n n and for n and for n and we define the _ survival probability _ to be the probability that the cre has not yet reached the far end of the footprint till time , given that initially ( at ) there were intact contacts between the histone spool and the dna on the footprint in front of the cre motor .obviously , is the solution of the equations for with the initial condition .interestingly , the time - evolution of can be re - cast as  where the transition rates and depend on the value of as follows :  for + + for n = n + + for + + for + + for + +  the master equations ( [ eq - s1])-([eq - s2 ] ) together , effectively , correspond to the kinetic scheme shown in the fig .[ fig - model2 ] . using this scheme ,the mftt for the single cre motor can be calculated analytically by extending the theoretical framework developed in ref. for calculating the mean first - passage time of random walks .following pury and caceres , the mftt is given by since and , integrating the equations ( [ eq - s1 ] ) and ( [ eq - s2 ] ) with respect to , we get + a^-_{n}[t_{1,n-1}-t_{1,n } ] \nonumber \\  & + & r_n[t_{2,n } - t_{1,n } ] \label{eq - t1}\end{aligned}\ ] ]   defining equations ( [ eq - t1 ] ) and ( [ eq - t2 ] ) can be expressed as   now , in the special case equations ( [ eq - t3 ] ) and ( [ eq - t4 ] ) become  next , multiplying equation ( [ eq - t6 ] ) by and eq .( [ eq - t7 ] ) by , and then adding the resulting equations , we get eq . ( [ eq - t8 ] ) can be re - written as where , we can rewrite eq .( [ eq - t9 ] ) as follows   for a _ fully wrapped _histone , the mftt is given by using ( [ eq - t10 ] ) in ( [ eq - t11 ] ) we , finally , get \label{eq - t12}\end{aligned}\ ] ]  since it is not easy to get an intuitive feeling for the implications of the expression ( [ eq - t12 ] ) , we anaylyze its special simpler forms in some limiting cases . in the limit of extremely slow motor ,i.e. , , as expected , the expression ( [ eq - t12 ] ) for the mftt _ diverges_.  for ensuring high - speed of the cre motor , we need simultaneously and . if , for simplicity , we make the additional assumption that is the slower of the two , i.e. , , we have and , and . hence , in this limit , for and , therefore , which is identical to the corresponding limiting value of reported in ref. .this is a consequence of the fact that in the limit of extremely fast motor , because of the assumption of very large value of , the 2-state model reduces to an effectively 1-state model .we make a numerical estimate of in this limit by computing an approximate value of . defining as the flap binding constant, we can rewrite the equation ( [ eq - liminfty ] ) as range of typical values of has been used earlier by chou . using this range of values for , one can estimate , provided a typical value of is known .therefore , we now estimate the typical numerical values of following schiessel and coworkers .suppose , nm ) be the length of the dna that wraps around the histone spool .let be the contour length of the loop induced by spontaneous thermal fluctuations where ( see fig .[ fig ] ) is the exposed arc length on the histone spool that was covered by the dna segment prior to the loop formation and is a small segment of the linker dsdna that has been pulled into the loop .we assume that the life time of a loop is much shorter than the average time required to form a loop . following schiessel et al . we write down the rate of loop formation as comparing eqs .( [ eq - al_est1 ] ) and ( [ eq - al_est2 ] ) we obtain since characterizes the rate of unbiased diffusion of the loop around the histone spool  where is the corresponding diffusion constant . from stokes - einstein relation , where  and is the effective viscosity of the aqueous medium . combining all the results and substituting these into eq .( [ eq - kb - est1 ] ) we finally obtain the estimation can be completed only if an estimate of is available .following ref . ( eq .( 2a ) of ) , we get where is the bending elastic constant of the semi - flexiable dna chain , is the adsorption energy per unit length and is the radius of the histone spool . using the reasonable values quoted in ref. , namely , and we obtain from eq .( [ eq - lp ] ) . using this estimate of , together with pn - nm , , and centipoise, we obtain from eq .( [ eq - kb - est2 ] ) .  with the above estimated value of and from eq .( [ eq - liminfty2 ] ) we get the estimates for and for .such small values of , estimated from eq.[eq - liminfty ] , arise from the fact that the approximate form ( [ eq - liminfty ] ) is valid only in the limit of extremely fast motor .therefore , this limiting formula provides only a lower bound and does not correspond to real cre motors under physiological conditions .   of linker chainbeing incorporated into the nucleosome , with length of the exposed surface .( adapted from fig . 2 of ref . ) . ]   plotted against the normalized motor speed for different values of with . ]  in fig.[fig - survtm ] we plot the normalized mftt as a function of the normalized motor speed for and a few fixed values of the parameter .for any fixed value of , the normalized mftt decreases monotonically with the increase of the normalized motor speed and saturates to the value given by equation ( [ eq - liminfty ] ) in the limit . moreover , for a given value of , as the flap binding constant increases the mftt increases .\(a ) +  plotted against the normalized motor speed for different values of with ( a ) , and ( b ) .,title="fig : " ] + ( b ) +  plotted against the normalized motor speed for different values of with ( a ) , and ( b ) .,title="fig : " ]  in fig.[fig - survtm - hw12 ] , is plotted against for ( a ) , , and ( b ) , , each for a few distinct values of .the mftt decreases as increases .this is a consequence of the fact that depends on the atp concentration . for small reduces the amplitude of peeling time .  in fig.[fig - survtm - hw21 ] we demonstrate that for large value of , which effectively speeds up the motor , reduces the magnitude of the mftt .\(a ) +  plotted against the normalized motor speed for different values of with ( a ) , and ( b ) .,title="fig : " ] + ( b ) +  plotted against the normalized motor speed for different values of with ( a ) , and ( b ) .,title="fig : " ]  although the qualitative trends of variations of with in our model is similar to that in chou s model , wide range of variation of is possible in our model by controlling which , in turn , can be controlled by the atp concentration .plotted against the atp concentration [ atp ] for , and two different values of . ]  in order to explore the dependence of on the concentration of atp , we first assume that  \label{eq - atp1}\ ] ] assuming a typical value , we have plotted the normalized mftt against the atp concentration for two different normalized values of the unhindreed motor speed keeping the other parameters fixed . with the increase of atp concentration , the mftt decreases and , gradulally saturates . when atp concentration is sufficiently high , the step with rate constant is no longer rate - limiting . we also find that , for a given atp concentration , the higher is the value of the shorter is the mftt .the linear dependence of on atp concentration , as envisaged in ( [ eq - atp1 ] ) , may be valid only at sufficiently low concentration of atp .in general , may follow the usual _ michaelis - menten equation _ for the rate of enzymatic reactions ( because represents the rate of atp hydrolysis catalyzed by the cre motor ) . in that case itself would saturate with the increase of atp concentration , instead of increasing linearly with [ atp ] .in this paper we have studied the process of atp - dependent chromatin remodeling . for simplicity ,we have considered only a single nucleosome consisting of a dsdna strand wrapped one and three - fourth turns around a cylindrical spool made of histone proteins .we have extended chou s model by assigning two distinct `` chemical '' states to the cre and postulating a minimal mechano - chemical kinetic scheme for capturing the effects of atp hydrolysis explicitly .our theoretical framework has been developed exploiting a close analogy with the unzipping of a double - stranded dna by a helicase .we have written down the master equations for the postulated kinetic scheme .this model of footprint traversal by atp - dependent cre can be easily interpreted as an implementation of a brownian ratchet mechanism . from an analytical treatment of this stochastic kinetic model ,we have derived analytical expression for the mftt of the atp - dependent cre .we make explicit analytical predictions on the dependence of the mftt on ( i ) the unhindred speed of the cre , as well as on ( ii ) the concentration of atp . in principleour theoretical predictions can be tested by carrying out _ in - vitro _ experiments with a single nucleosome .* acknowledgements : * dc thanks michael poirier and tom chou for useful discussion and correspondence , respectively .dc acknowledges support of the visitors program of the max - planck institute for the physics of complex systems in dresden where parts of this work were carried out during two separate visits .dc also thanks frank jlicher for discussions in the initial stages of this work and anirban sain for a critical reading of the manuscript .this work is also supported , in part , by iit kanpur through the dr .jag mohan chair professorship ( dc ) and by a research grant from csir , india ( dc ) .ag thanks ugc , india , for a senior research fellowship .99 a. wolfee , _ chromatin : structure and function _ , ( academic press , 1998 ) . j. widom , annu .. biomol.struc .* 27 * , 285 ( 1998 ) .h.schiessel , j. phys . condens. matter , * 15 * , r699 ( 2003 ) . c. lavelle and a. benecke , eur .j. e * 19 * , 379 ( 2006 ) .kornberg and y. lorch , cell * 98 * , 285 ( 1999 ) .k.j . polach and j. widom , j. mol .biol . * 254 * , 130 ( 1995 ) .anderson , a. thastrom and j. widom , mol . cell .* 22 * , 7147 ( 2002 ) .fan , x. he , r.e .kingston and g.j .narlikar , mol .cell * 11 * , 1311 ( 2003 ) .a. flaus and t. owen - hughes , biopolymers * 68 * , 563 ( 2003 ) .a. flaus and t. owen - hughes , febs j. * 278 * , 3579 ( 2011 ) .halford , a.j .welsh and m.d .szczelkun , annu .. struct . * 33 * , 1 ( 2004 ) .a. saha , j. wittmeyer and b.r .cairns , nat .cell biol .* 7 * , 437 ( 2006 ) .clapier and b.r .cairns , annu .* 78 * , 273 ( 2009 ) .l. r. racki and g. j. narlikar , curr .opin , genet .* 18 * , 137 ( 2008 ) .h. schiessel , j. widom , r.f .bruinsma and w.m .gelbert , phys .* 86 * , 4414 ( 2001 ) .kulic and h. schiessel , phys .lett . * 91 * , 148103 ( 2003 ) .i.m . kulic and h. schiessel , biophys .j. * 84 * , 3197 ( 2003 ) .h. schiessel , eur .j. e * 19 * , 251 ( 2006 ) .f. mohammad - rafiee , i.m . kulic and h. schiessel ,biol . * 344 * , 47 ( 2004 ) .r. blossey and h. schiwssel , febs j. * 278 * , 3619 ( 2011 ) .w. mbius , r.a .neher and u. gerland , phys .* 97 * , 208102 ( 2006 ) . j. langowski , eur .. j. e * 19 * , 241 ( 2006 ) .a. lense and j.m .victor , eur .j. e * 19 * , 279 ( 2006 ) . c. vaillant ,b. audit , c. thermes and a. arneodo , eur .j. e * 19 * , 263 ( 2006 ) .p. ranjith , j. yan and j.f .marko , pnas * 104 * , 13649 ( 2007 ) .g. lia , e. praly , h. ferreira , c. stockdale , y.c .tse - dinh , d. dunlap , v. croquette , d. bensimon and t. owen - hughes , mol .cell * 21 * , 417 ( 2006 ) .cairns , nat .struct . and mol* 14 * , 989 ( 2007 ) .g. li , m. levitus , c. bustamante and j. widom , _ rapid spontaneous accessibility of nucleosomal dna _ , nat .struct . mol .* 12 * , 46 - 53 ( 2004 ) .poirier , m. bussiek , j. langowski , j. mol . biol . * 379 * , 772 - 786 ( 2008 ) . m.g .poirier , e. oh , h.s .tims and j. widom , nat .biol . * 16 * , 938 ( 2009 ) .tims , k. gurunathan , m. levitus and j. widom , j. mol .* 411 * , 430 ( 2011 ) .

Abstract: atp - dependent chromatin remodeling enzymes ( cre ) are bio - molecular motors in eukaryotic cells . these are driven by a chemical fuel , namely , adenosine triphosphate ( atp ) . cres actively participate in many cellular processes that require accessibility of specific segments of dna which are packaged as chromatin . the basic unit of chromatin is a nucleosome where 146 bp 50 nm of a double stranded dna ( dsdna ) is wrapped around a spool formed by histone proteins . the helical path of histone - dna contact on a nucleosome is also called `` footprint '' . we investigate the mechanism of footprint traversal by a cre that translocates along the dsdna . our two - state model of a cre captures effectively two distinct chemical ( or conformational ) states in the mechano - chemical cycle of each atp - dependent cre . we calculate the mean time of traversal . our predictions on the atp - dependence of the mean traversal time can be tested by carrying out _ in - vitro _ experiments on mono - nucleosomes .


in the past few years a considerable effort has been devoted to the solution of einstein s equations in numerical simulations of strong - field , highly dynamical sources of gravitational radiation .this effort is partly motivated by the development and construction of gravitational wave detectors such as ligo , virgo , geo and tama .knowing the theoretical waveform produced by the most likely astrophysical sources of gravitational radiation will not only increase the probability of a successful detection but , most importantly , will allow for the extraction of astrophysically significant information from the observations .the binary black hole `` grand challenge '' _ alliance _ , is a major example of this effort , in which a multi - institutional collaboration in the united states was created in order to study the inspiral and coalescence of a binary black hole system , one of the most significant source of signals for the interferometric gravity wave detectors .present three - dimensional ( 3d ) numerical relativity simulations face a number of fundamental and in some cases unsolved problems , including : coordinate choice , most suitable form of einstein s equations , singularity avoiding techniques , gravitational wave extraction and outer boundary conditions .while a robust solution to the generic problem is still awaited , some interesting results have already been obtained , for instance , in the evolution of a generic 3d black hole , in the translation of a 3d black hole across a numerical grid , and in the extraction of gravitational wave information and imposition of outer boundary conditions .  determining the asymptotic form of the gravitational waves produced in a dynamical evolution of einstein s equations is an important goal of many numerical relativity simulations .this goal , however , necessarily requires accurate techniques which compute waveforms from numerical relativity simulations on 3d spacelike hypersurfaces with finite extents . in an ideal situation with unlimited numerical resources, the computational domain can extend into the distant wave zone , where the geometric optics approximation is valid and the gravitational waves approach their asymptotic form . with present - day computational limitations , however , the outer boundary of typical numerical relativity simulations lies rather close to the highly dynamical and strong - field region where backscatter of waves off curvature can be significant . as a result , additional techniques need to be implemented in order to `` extract '' such information from the strong - field region and `` evolve '' it out to a large distance .in two recent papers , we have presented a new method for extracting gravitational wave data from a 3d numerical relativity simulation and evolving it out to an arbitrary distant zone .our method has been developed within the alliance in order to match a generic full 3d cauchy solution of nonlinear einstein s equations on spacelike hypersurfaces with a linear solution in a region where the waveforms can be treated as perturbations on a spherically symmetric curved background .this `` perturbative module '' is used not only to extract gravitational wave data from the cauchy evolution but , at the same time , to impose outer boundary conditions ( a parallel development is also underway in the alliance to match interior cauchy solutions to exterior solutions on characteristic hypersurfaces ) . indeed , while the problem of radiation extraction is important for computing observable waveforms from numerical simulations , imposition of correct outer boundary conditions is essential for maintaining the integrity of the simulations themselves , as incorrect outer boundary conditions are often a likely source of numerical instabilities .one of the most important requirements for any radiation - extraction and outer - boundary module is that it provides for stable evolution of the interior equations and minimizes the spurious ( numerical ) reflection of radiation at the boundary .this requirement is particularly important for the `` grand challenge '' investigation in which the computations need to be performed on a gravitational wave emission timescale , which is much longer than the orbital one .  in this paperwe present the application of our cauchy - perturbative matching method to a standard testbed : the evolution of 3d linear and mildly nonlinear gravitational waves .the plan of this paper is as follows : in section [ tpm ] we briefly review the main features of the approach and recall the essential elements of its numerical implementation .we then concentrate on the two major aspects of this work : in section [ stb ] we present the _ `` short term '' _ properties of the cauchy - perturbative matching and show its ability to provide an accurate and numerically convergent approximation to the gravitational waveform that would be observed in the wave zone surrounding an isolated source . in section [ ltb ] we turn to the _ `` long term '' _ properties of our approach and present a number of different implementations which lead to a _stable _ numerical evolution , long after the bulk of the gravitational waves have left the computational grid .as discussed in ( hereafter paper i ) , the cauchy - perturbative matching method involves replacing , at least in parts of the 3d numerical domain , the solution of the full nonlinear einstein s equations with the solution of a set of simpler linear equations that can be integrated to high accuracy with minimal computational cost .= 10.0 cm  1.0truecm 1.0truecm  in order to do this , it is necessary to determine the region of spacetime where a perturbative approach can be applied . in general , the 3d numerical grid ( indicated as n in ) will comprise an isolated region of spacetime where the gravitational fields are strong and highly dynamical . in this region, indicated as in , the full nonlinear einstein equations must be solved . outside of , however , in what we will refer to as the perturbative region , a perturbative approach is not only possible but highly advantageous .anywhere in the portion of covered by n we can place a two - dimensional ( 2d ) surface which will serve as the surface joining numerically the highly dynamical strong - field region and the perturbative one . here , we have chosen this surface to be a 2-sphere of radius , indicated as * e * in .it is important to emphasize that the 2-sphere * e * _ need not _ be in a region of spacetime where the gravitational fields are _ weak _ or the curvature is _small_. in contrast to previous investigations which matched einstein s equations onto a minkowski background , the matching is here made on a schwarzschild background . as a result ,the only requirement is that the spacetime outside of approaches a schwarzschild one . even in the case of a binary black hole merger, it will be possible to find a region of spacetime , sufficiently distant from the binary black holes , where this requirement is met .  in a practical implementation of the cauchy - perturbative method, a numerical code provides the solution to the full nonlinear einstein equations everywhere in the 3d grid n except at its outer boundary surface * b*. at the extraction 2-sphere * e * , a different code ( i.e. the perturbative module ) `` extracts '' the gravitational wave information and transforms it into a set of multipole amplitudes which we have here chosen to depend only on the radial and time coordinates of the background schwarzschild metric ( see paper i and section [ tbe ] for details ) .  in this way ,two of the three spatial dimensions of the problem are suppressed and the propagation of gravitational waves on a curved background is reduced to a one - dimensional ( 1d ) problem . during each timestep , information about the gravitational field read - off at * e *is propagated by the perturbative module out to the 2-sphere * a * in the asymptotic flat region of spacetime .this is done by solving a set of coupled 1d linear differential equations ( one for each of the multipoles extracted at * e * ) on the 1d grid l ( not shown in ) covering the perturbative region and ranging between and . from a computational point of view, this represents an enormous advantage : with a few straightforward transformations , the computationally expensive 3d evolution of the gravitational waves via the nonlinear einstein equations is replaced with a set of 1d linear equations that can be integrated to high accuracy with minimal computational cost .although linear , these equations account for all of the effects of wave propagation in a curved spacetime and , in particular , automatically incorporate the effects of backscatter off the curvature ( only the wave - wave effects are omitted ) .  as a result of our construction ( and as shown in ) , the perturbative region is entirely covered by a 1d grid l and only partially by a 3d grid in the complement to in n. the overlap between these two grids is essential .in fact , the knowledge of the solution on allows the perturbative module to provide boundary conditions at the outer boundary surface * b * and , if useful , dirichlet data on every gridpoint of n outside the strong region .as we will further discuss in section [ pb ] , this freedom to specify boundary data on a 2-surface of arbitrary shape as well as on a whole 3d region of n represents an important advantage of the perturbative module over similar approaches to the problem of gravitational wave extraction and imposition of boundary conditions .our treatment of schwarzschild perturbation theory is based on the third - order einstein - ricci hyperbolic formulation of einstein field equations .a principal advantage of this approach is that gauge - invariant wave equations arise simply from a linear reduction of the full equations without complex changes of variables .thus , the matching of the perturbative solutions to the fully nonlinear ones becomes rather straightforward .once the perturbative equations are derived , these are completely general and can be applied to numerical codes solving einstein s equations in either an explicitly hyperbolic form or , as in the present case , in the standard form  .we split the gravitational quantities of interest into background parts ( denoted by a tilde ) and perturbed parts .these are the three - metric , the extrinsic curvature , the lapse function and the shift vector , where the tilde denotes background quantities . assuming a schwarzschild background ,    we then have , while the perturbed parts have arbitrary angular dependence .  using this background , we linearize the hyperbolic equations and reduce the wave equation for to a linear wave equation for involving also the background lapse .we then separate the angular dependence in this equation by expanding in terms of tensor spherical harmonics  ( we use the notation of  )    note that are functions of only and , for clarity , angular indices for each mode are suppressed .similarly , the odd - parity multipoles and and the even - parity multipoles , , , and also have suppressed indices for each angular mode .there is an implicit sum over all angular modes in ( [ eq : kappa_expand ] ) .the six multipole amplitudes are not independent .we use the linearized momentum constraints to eliminate the odd - parity amplitude and the even - parity amplitudes and . as a result , for each mode ] we obtain one odd - parity equation for :   \biggr\ }      ( a_\times)_{_{\ell m } } = 0 \ , \label{oddwave}\end{aligned}\ ] ]  and two coupled even - parity equations for and : ( a_+)_{_{\ell m } } + \hskip 6.0truecm   \nonumber\\    \biggl[\frac{4}{r } \widetilde{n}^2 \left(1 -\frac{3m}{r}\right )    \partial_r + \frac{2}{r^2 }    \left(1 - \frac{m}{r } - \frac{3m^2}{r^2}\right )    \biggr ] ( h)_{_{\ell m } } = 0 \ ,  \label{evenwave1 } \end{aligned}\ ] ] ( h)_{_{\ell m } }    - \frac{2 m}{r^3 } \left(3 - \frac{7 m}{r}\right )    ( a_+)_{_{\ell m } } = 0 \ . \label{evenwave2}\end{aligned}\ ] ]  the independent multipole amplitudes , , and the corresponding wave equations ( [ oddwave])([evenwave2 ] ) for each mode are at the basis of the cauchy - perturbative matching .[ here is defined in terms of the trace of , i.e. where is the standard scalar spherical harmonic . ]   as discussed in section [ intro ] , with a few straightforward modifications , this method can be applied to a generic 3d numerical relativity code which solves the cauchy problem of einstein s equations in either the standard or hyperbolic form .in addition to the standard time integration of the extrinsic curvature and of the spatial metric , three new procedures are performed during each timestep .  _( a ) _ the gravitational radiation information contained in and is transformed into the independent multipole amplitudes , , and their time derivatives for all of the relevant modes ( see paper i for details ) .the maximum mode at which the angular decomposition is truncated depends on the basic features of the problem under investigation .however , a simple comparison of the relative amplitudes of the different multipoles it is usually sufficient to provide information about maximum mode necessary ._ ( b ) _ the values of multipole amplitudes and their time derivatives computed at the extraction 2-sphere on a given timeslice are imposed as inner boundary conditions on the 1d grid l and evolved [ using the radial wave equations ( [ oddwave])([evenwave2 ] ) for each mode ] forward to the following timeslice this provides the solution , for the new time level , on the whole perturbative region .since the outer boundary of l is located , by construction , well out in the wave zone , a simple radial outgoing wave sommerfeld condition can be imposed there .  _( c ) _ from the values at the new time level of , , and of their time derivatives , it is possible to `` reconstruct '' the values of or and thus to impose outer boundary conditions on the 3d grid n. the details of how this is done depend on the formulation of einstein s equations solved in n. in the computations discussed here , we have used the 3d `` interior code '' of the alliance adopting an formulation of einstein s equations . in this case , only the outer boundary data for are necessary , since the interior code can calculate at the outer boundary by integrating in time the boundary values for .it is important to emphasize the great flexibility of the cauchy - perturbative approach in providing outer boundary value data .once again , the ultimate goal is that of providing , during each timestep , boundary values of the relevant quantities at the 2-surface * b * delimiting the 3d grid n. since we can compute the new values at any point of n which lies in the perturbative region , not only we can provide boundary data on a 2-surface of arbitrary shape , but , ( if necessary ) on the the whole portion of n outside of the extraction 2-sphere * e*. this represents a great advantage for which no approximation is required and , as we will discuss in detail in section [ ltb ] , represents an essential prescription in order to obtain the stability of the code on very long timescales .in paper i we presented tests of our code based on the propagation of linear waves on a minkowski background ( i.e. with ) . in those tests , we simulated the cauchy evolution of the nonlinear interior code by providing an analytic solution on the 3d grid .this was necessary in order to evaluate the accuracy and the convergence properties of the module independently of any error which may develop due to the numerical evolution .as a result of those investigations , we were able to show the module s ability to extract the gravitational wave information , to evolve this information out to large distances and to impose self - consistent and convergent dirichlet outer boundary conditions . while extremely useful , those tests could not address a number of important questions which are strictly related to the use of numerical data coming from the solution of the full einstein s equations .in particular : ( i ) what is the influence of the location of the extraction 2-sphere * e * on the accuracy of the extracted gravitational wave data ?( ii ) do dirichlet boundary conditions on * b * provide a long term stability ?( iii ) what are the most convenient boundary conditions to impose ?( iv ) are there numerical techniques that would improve the application of the cauchy - perturbative matching ? in this paperwe provide an answer to all of these questions and discuss the properties of a cauchy - perturbative matching in the more realistic study of linear and mildly nonlinear waves .  as in paperi , we have here computed the propagation of , ( unless otherwise stated ) even - parity linear waves , initially modulated by a gaussian envelope with amplitude and width parameter  ( section [ vott ] will discuss variations on this type of initial data by using higher modes and higher amplitudes ) .being time - symmetric at the initial time , these waves have ingoing and outgoing parts . at each time level , the extrinsic curvature and 3-metric are computed using the interior code of the alliance solving the full einstein s equations with a geodesic slicing condition ( i.e. , ) on a 3d vertex - centered grid , with extents  ] and defined so that ( i.e. at the inner edge of the blending layer ) and ( i.e. at the outer edge of the blending layer ) . could be the perturbative solution of einstein s equations in the blending region ( note that the following arguments will not depend on being a perturbative solution .in fact , in gmez s first application of the blending boundary conditions , was the value given by an analytic solution . )      in the simplest case in which the perturbative value for the extrinsic curvature is zero , the result of the blending is then a wave equation ( in this case , in fact , ) with a variable phase propagation speed , , which is unity at the inner radius of the blending region and `` smoothly '' ( in a discretized sense ) goes to zero at the outer edge of the blending region :            as a result , the wave packet will have unit outgoing and ingoing phase velocities at the inner edge of the blending region , decreasingly smaller ingoing phase velocities in the blending region and only outgoing phase velocity at the outer edge of the blending region and outside of it .  finally , it can be seen that the blending approach , especially the blending to the sommerfeld condition , just described , has a close relation to the techniques described in reference .however , for the strongly nonlinear black hole simulations , the more manageable approach of blending with _ analytic solutions _ , as in the first example above , has been successful in some cases .this method is computationally much simpler than explicitly modifying the equations .n. t. bishop , r. gmez , p. r. holvorcem , r. a. matzner , p. papadopoulos and j. winicour , phys .lett . * 76 * , 4303 ( 1996 ) ; n. t. bishop , r. gmez , l. lehner , j. winicour , phys .d. * 54 * , 6153 ( 1996 ) .

Abstract: we present results from a new technique which allows extraction of gravitational radiation information from a generic three - dimensional numerical relativity code and provides stable outer boundary conditions . in our approach we match the solution of a cauchy evolution of the nonlinear einstein field equations to a set of one - dimensional linear equations obtained through perturbation techniques over a curved background . we discuss the validity of this approach in the case of linear and mildly nonlinear gravitational waves and show how a numerical module developed for this purpose is able to provide an accurate and numerically convergent description of the gravitational wave propagation and a stable numerical evolution .