document: the kinetics of polymer loop formation has been studied for several decades and recently has attracted renewed attention due to the particular importance in biology . 
   the dna loop formation is a basic process that underlies genetic expression , replication , and recombination  @xcite . 
   for example , in _ e. coli _ the _ lac _ repressor ( laci)-mediated loop is crucial for the repressive regulation of _ lac _ genes . 
   the hairpin loop formation is the elementary step in protein folding  @xcite and structure formation in rna folding  @xcite . 
   a cell is crowded with a multitude of subcellular structures including globular proteins and rnas  @xcite , with which dna is constantly interacting . a dna fragment about to loop is often subject to temporally fluctuating forces due to its dynamic environment including the other part of the chain . 
   recently , the power spectrum of the fluctuating force exerted on cytoskeleton was measured to be an order of magnitude larger than that expected from thermal equilibrium condition  @xcite . 
   this indicates that the cell interior is an active and nonequilibrium medium . 
   the advance of single molecule experiment techniques provides detailed information on the dna loop formation . 
   finzi and gelles  @xcite observed laci - mediated dna loop formation and dissociation by monitoring nano - scale brownian motion of the micron - sized particle attached to one end of the dna . 
   et al . _ 
   @xcite showed that in _ gal _ repressor and dna - bending protein hu mediated looping , mechanical constraints such as tension and torsion play a pivotal role . 
   gemmen  _ et al . _ 
   @xcite studied effects of tension in the presence of two - site restriction enzymes which can cut the dna upon binding on two sites simultaneously . 
   they found that the cleavage activity decreases approximately 10-fold as the tension increases from 30 fn to 700 fn . 
   they also found that the optimum loop size decreases with the tension , which is qualitatively in agreement with theoretical predictions  @xcite . 
   more recently , chen _ et al . _ 
   @xcite studied effects of tension in femtonewton range on the kinetics of laci - mediated dna looping . 
   they found that small tension of 100 fn scale on the substrate dna can not only increases the looping time  @xcite but also found that the looping time is greatly reduced in the presence of a fluctuating tension  @xcite . 
   these results suggest the ubiquitous roles of the static and temporally fluctuating tensions in regulation of the dna loop formation . yet 
   , there appears to be no unifying conceptual or theoretical framework that explains a variety of experiments including these .    theoretically , on the other hand , yan _ 
   et al . _ 
   @xcite developed a transfer matrix method to calculate semiflexible polymer end - to - end distance distribution function and loop formation probability ( or @xmath0-factor ) . 
   they studied various effects of nonlinear elasticity arising from dna bending protein - induced kinks  @xcite or thermal - fluctuation - induced bubbles on the @xmath0-factor . 
   their study provides a valuable insight to understand dna bending on short length scale  @xcite , which has attracted much attention recently @xcite . 
   they also studied effects of tension on the @xmath0-factor  @xcite , which is related to the free energy barrier for loop formation  @xcite , thus to the loop formation rate . 
   similar results are obtained by using an elastic theory of a semiflexible polymer  @xcite . however , since the loop formation rate is not proportional to the @xmath0-factor alone but depends on the free energy given an arbitrary chain end - to - end distance , it is hard to quantitatively compare these theories to the experiment  @xcite . 
   independently , blumberg _ et al . _ 
   @xcite studied effects of static tension on protein - mediated dna loop formation by modeling the dna conformation to be either one of two states , looped and unlooped states . in their appealing calculations of free energy change associated with the transition under the tension , they considered not only the stretching free energy of dna but also dna alignment constraint imposed by protein binding . 
   they found that for the loop size larger than 100 base pair distance ( @xmath1 nm ) , a tension of 0.5 pn can increase the looping time by more than two order of magnitude . 
   there is room for improvement in their approach , however , on the evaluation of the free energy that can be valid for short end - to - end distance of the chain as well as a description of detailed kinetic process using the mean first - passage time approach .    in this paper , in an effort to understand the basic physical mechanism of the biopolymer looping in a coherent manner , we perform brownian dynamics simulation of semiflexible polymers treated as extensible wormlike chain , combined with one - dimensional theory of barrier crossing over the free energy of loop formation . for analytical understanding , 
   we use , as an example , the mean - field wormlike chain model  @xcite , which is shown to be a good approximation for the free energy for the chain lengths we consider here . with static tensions , we find that the looping time , defined as the mean first - passage time to cross the free energy barrier , steeply increases with the applied tension @xmath2 , in an agreement with our simulation results but distinct from the previous theoretical result  @xcite . for the case of time - dependent tension , we consider dichotomically fluctuating tension , where the looping times are found to be reduced , consistent with the experiment  @xcite . 
   most importantly , we find so - called the resonant activation , where the looping time is the minimum at an optimal flipping time of the dichotomic force . in this exploratory study 
   , we neglect the alignment constraint on the loop formation , which is minor effect for the chain lengths we consider here  @xcite .    in the following section 
   , we describe our polymer model and simulation method , whose results are discussed in sec . 
   [ sec : results ] . 
   we summarize our results in sec . 
   [ sec : conclusion ] . 
   we consider the semiflexible polymer looping in the presence of static and fluctuating tension . 
   the polymers are modeled as semiflexible chains of @xmath3 beads of diameter @xmath4 , with the interaction potential @xmath5 . 
   here @xmath6 , where @xmath7 and @xmath8 are the stretching and bending energy @xmath9 @xmath10 where @xmath11 is the stretching stiffness , @xmath12 is the natural bond length , @xmath13 is the bending stiffness , @xmath14 is the position of the @xmath15th bead , and @xmath16 is the angle of the @xmath15th bond . 
   .[fig : schematic],width=321 ]    the dynamics of @xmath3 beads ( @xmath17 ) are described by the overdamped langevin equation @xmath18 where @xmath19 is the friction coefficient of the bead , and @xmath20 is the gaussian random noise with mean @xmath21 and variance @xmath22 . here 
   , @xmath23 denotes ensemble average , @xmath24 and @xmath25 are the cartesian coordinate indices , @xmath26 is the boltzmann constant , and @xmath27 is the absolute temperature . 
   the additional force @xmath28 is the tension which is applied only to the two end segments ( @xmath29 and @xmath3 ) along the chain end - to - end direction , @xmath30 ( see fig . 
   [ fig : schematic ] ) . 
   the bending stiffness of the short semiflexible chains we consider allows us to neglect the excluded volume effect . 
   furthermore , we neglect the hydrodynamic interactions , which is found to be small in the short chains we consider here  @xcite . 
   we use the parameters @xmath12 , @xmath31 , and @xmath32 to fix the length , energy , and time scales , respectively . 
   the dimensionless parameters in our simulation are @xmath33 , @xmath34 , @xmath35 , and @xmath36 . in our model , we consider @xmath37 as 10 nm , so with @xmath36 the persistence length @xmath38 of the chain is 50 nm . 
   the diameter of the bead is set to be 5 nm to fit the hydrodynamic friction coefficient of a cylinder with 2 nm diameter and 10 nm length , which is about @xmath39 @xmath40 @xmath41 with the water viscosity @xmath42 @xmath40 @xmath43 @xmath41 at room temperature . 
   the time unit is then 1.14  @xmath44s . 
   the equations of motion are integrated by using a second - order stochastic runge - kutta algorithm  @xcite with the time - step @xmath45 . in our simulation 
   , we consider that the looping occurs whenever the chain end - to - end distance @xmath46 is shorter than a cutoff distance @xmath47 . 
   the average were taken over at least 2,000 and 5,000 independent runs for static and fluctuating tension cases , respectively . 
   we first study the effects of static tension on semiflexible polymer looping . here 
   we consider the bead number @xmath48 , 18 , and 24 which respectively correspond to the chain lengths @xmath49 , 170 , and 230 nm , similar to the loop size considered in recent experiments  @xcite . 
   figure  [ fig : n12_static1 ] shows the looping time @xmath50 as a function of tension @xmath2 for the chain of @xmath48 and @xmath51 . 
   we apply the tension @xmath2 from -81 to 162 fn , comparable in magnitude to the typical entropic force on the double - stranded dna , @xmath52 @xmath53 80 fn . 
   the @xmath2 is much smaller than the piconewton scale forces those are typically involved in active molecules  @xcite e.g. , molecular motors in a cell . as @xmath2 changes over the scale as small as 100 fn , the looping time @xmath54 dramatically changes ; for example , @xmath54 increases about 5 times as the tension @xmath2 increases 120 fn .     as a function of tension @xmath2 for the chain of @xmath48 . 
   [ fig : n12_static1],width=321 ]    to understand this behavior , we consider semiflexible polymer looping as a one - dimensional barrier crossing process  @xcite , which is described by the langevin equation    @xmath55    here , @xmath56 is the free energy of the chain given the chain end - to - end distance @xmath57 , where @xmath58 is the radial distribution function , and @xmath59 is a random force due to thermal fluctuation given by a gaussian and white noise that satisfies @xmath60 and @xmath61 . the shape of @xmath62 obtained by the @xmath3-beads simulation is shown in fig . 
   [ fig : n12_mfwlc ] ( solid line with circles ) for the chain of @xmath48 . in this one - dimensional description , 
   the looping time is the mean first - passage time ( mfpt ) for the variable @xmath57 to reach the cutoff distance @xmath47 starting from the initial chain end - to - end distance @xmath63 . 
   it is given by  @xcite    @xmath64    where @xmath65 is the relative diffusion coefficient of two end beads , @xmath66 . 
   then the looping time is @xmath67 , where @xmath68 represents the average over the initial equilibrium distribution . 
   the mfpt in the presence of tension @xmath2 , also can be calculated by using the free energy , @xmath69 , where @xmath70 is the free energy without tension .    for the @xmath58 
   , we use the mean - field wormlike chain model ( mf - wlc )  @xcite as an example . 
   the radial distribution function @xmath58 of this model is @xmath71^{-\frac{9}{2 } } \exp[-\frac{3l}{4 l_{p } } \frac{1}{(1-(r / l)^2)}]$ ] . 
   this formula yields a reasonable approximation to our simulation result for @xmath70 , except for @xmath72 , as shown in fig . 
   [ fig : n12_mfwlc ] . 
   the large deviation between two curves in the region near @xmath73 is because our model allows the chain extensibility while mf - wlc does not . 
   however , according to the single molecule experiment using optical tweezers  @xcite , the dna is extensible , manifesting overstretching transition subject to a strong tension , thus showing a significant deviation from the inextensible wlc . 
   the optimal value @xmath11 to fit the force - extension curve is found to be @xmath74 2700  @xcite . however , the simulation using @xmath11 as large as @xmath75 demands computing time much longer than that using @xmath35 . unlike the free energy , the normalized looping time @xmath76 , the looping time in the presence of tension @xmath2 relative to the looping time in its absence , is expected to be quite insensitive to the value of @xmath11  @xcite . 
   for this reason we adopt @xmath35 , which was also employed in a number of studies on the looping  @xcite .     of @xmath48 chain obtained from the simulation ( solid line with circles ) and from the mean - field wormlike chain model 
   @xcite ( solid line).[fig : n12_mfwlc],width=321 ]    to understand the effect of tension on the looping time , the mf - wlc provides a useful analytical model . 
   figure  [ fig : normalized_looping_static ] shows @xmath77 as a function of @xmath2 for the chain lengths @xmath3=12 , 18 , and 24 . here 
   the symbols are simulation results and the dashed lines are from mfpt calculations ( eq .  [ eq : mfpt ] ) . for the range of @xmath2 we study here , 
   two results are in an excellent agreement . 
   this figure also shows that the normalized looping time increases exponentially with @xmath2 . 
   the reason is that , for the short chain lengths considered here , the free energy of the loop formation increases with @xmath78 , so @xmath54 increases approximately in exponential with @xmath2 . 
   in contrast , the theory of ref .  @xcite considered that , for low tension ( @xmath79 fn ) , @xmath54 increases exponentially with @xmath80 because of gaussian force - extension relation they used .     as a function of tension @xmath2 for @xmath48 , @xmath81 and @xmath82 . 
   the symbols are simulation data and the dashed lines are from mean first - passage time calculations using mean - field wormlike chain model  @xcite.[fig : normalized_looping_static],width=359 ]    the looping time in the presence of a tension @xmath2 can be written in dimensionless form @xmath83 where @xmath84 . 
   @xmath85 is a function of a dimensionless scaling variable @xmath86 . 
   it means for very large @xmath87 , a minute tension @xmath2 can dramatically change the looping time . 
   indeed , fig . 
   [ fig : normalized_looping_static ] shows that , for longer chains , the normalized looping time changes more sensitively with @xmath2 . if @xmath3 is 100 , corresponding to @xmath88 @xmath44 m , a change of the force as small as 4 fn can affect the looping time . 
   this is a consequence of the cooperativity of the long polymer chains arising from the chain connectivity , which is previously addressed in a study of polymer translocation through membrane  @xcite . 
   the sensitivity of looping on tension is an emergent behavior that manifests beyond the complexity of real dna loop formation  @xcite , e.g. , the details of chain conformations outside of the loop , orientation constraint , and associated proteins . 
   finally , we compare our result with the recent experimental data  @xcite for the dna loop size @xmath89 nm . as shown by the circles in fig . 
   [ fig : cutoff_5 ] , the looping time rises steeply as a function of tension @xmath2 relative to the one with @xmath90 fn , which is the minimum tension used in the experiment . 
   we also plot the corresponding relative mfpt by a solid curve . with the cutoff distance chosen to be the bead diameter , @xmath47=5 nm 
   , our result is in good agreement with the experiment  @xcite . 
   because our model does not consider many details of dna loop formation , this agreement could be somehow fortuitous but encouraging . 
   a theory including effectively the complexity of dna loop formation and further controlled experiments of various loop sizes are needed for quantitative comparison . 
   ( @xmath90 fn ) as a function of tension @xmath2 with the cutoff distance @xmath91 nm for @xmath89 nm obtained by the mfpt calculation using mf - wlc model  @xcite ( solid line ) . 
   the circles are experimental data including error bar from ref . 
   @xcite.[fig : cutoff_5],width=321 ]      now suppose that tension temporally fluctuates due to the nonequilibrium noise inherent _ in vivo _ systems which can be generated by a variety of constituents of a cell , e.g. , protein like rna polymerase . also a dna fragment about to loop is influenced by the other part of the chain whose conformation is constantly fluctuating . as a simple example of nonequilibrium fluctuations , we consider a dichotomic noise , with which the tension @xmath92 flips between two level of the forces @xmath93 and @xmath94 with a mean flipping time @xmath95 . 
   the @xmath92 is a noise with zero mean and its time correlation function is @xmath96 . 
   we generate dichotomic noise @xmath92 using the algorithm described in  @xcite . in the initial equilibration time , @xmath92 is given with either @xmath93 or @xmath94 with an equal probability 1/2 , and in a small time - step @xmath97 , @xmath92 can flip to the other value with the probability @xmath98 , which makes the mean flipping time be @xmath95 .     in the presence of dichotomic tension as well as static tensions for the chain of @xmath48 . 
   the dichotomic force amplitude is @xmath99 fn , and the static tensions amplitudes are @xmath100 ( circle ) and @xmath101 fn ( rectangular ) . 
   the looping time shows a minimum at optimal flipping time @xmath102 and @xmath103 for @xmath100 and 161 fn , respectively . 
   ( b ) the probability @xmath104 that the dichotomic force is negative ( i.e. , inward direction ) at the instant of looping as a function of @xmath95 . 
   [ fig : ran12_static],width=321 ]    figure  [ fig : ran12_static ] @xmath105 shows the normalized looping times , @xmath106 , as a functions of @xmath95 for the chain length @xmath3=12 in the presence of fluctuating tension @xmath92 added to the static tension @xmath2 . 
   we consider a value of dichotomic noise amplitude , @xmath99 fn and two values of static tensions amplitudes , @xmath100 ( circles ) and 161 fn ( squares ) , which are similar to those used in a recent experiment  @xcite . 
   for very short @xmath95 , the dichotomic forces are averaged out , so the looping times converge to the values without dichotomic force . 
   they gradually decrease with @xmath95 until the minimum at @xmath107 and @xmath103 for @xmath100 and 161 fn , respectively . for very long @xmath95 , 
   the dichotomic force rarely changes in a typical looping time , so the looping time goes to the average of the looping time with tension ( @xmath108 ) and the looping time with tension ( @xmath109 ) . 
   the average is dominated by the looping time with ( @xmath108 ) , so that the looping time sharply rises with @xmath95 . 
   related to this , we study the probability @xmath104 that the dichotomic tension @xmath92 is negative ( i.e. , is in inward direction ) at the instant of looping as a function of @xmath95 ( fig . 
   [ fig : ran12_static ] @xmath110 ) . 
   it has a maximum at @xmath111 and @xmath112 300 for @xmath100 and 161 fn , respectively , which means that , for @xmath95 near the @xmath113 , most of looping occurs when @xmath92 is in inward direction , and the looping time becomes the minimum . 
   in the limits of very short or very long @xmath95 , @xmath92 is positive or negative with equal probabilities at the moment of looping . 
   the maximum of @xmath104 is larger for larger @xmath2 . evidently the minimum looping time is closely associated with the maximum of the @xmath104 . 
   the minimum of @xmath54 is found to occur when the flipping time @xmath95 is comparable to the diffusion time of a brownian _ particle _ in the free energy @xmath62 , @xmath114 , where @xmath115 is a numerical value of order unity that increases with the tension @xmath2  @xcite . 
   this phenomenon is an extension of the resonant activation ( ra ) originally found in the single brownian particle crossing over a fluctuating barrier  @xcite . 
   indeed , we can regard the polymer looping in the presence of dichotomic tension as the process of a brownian _ particle _ crossing over a fluctuating _ free energy _ barrier .    to study how the dichotomic force affects the ra phenomena 
   , we consider different value of @xmath116 . 
   figure  [ fig : n12_t ] shows the normalized looping time with @xmath117 ( circles ) , 121 fn ( squares ) _ in the absence _ of static tension ( @xmath118 ) . 
   the looping time has resonant minimum at the optimum flipping time @xmath119 which is smaller than the case with static tension @xmath2 . at the optimum flipping time , the looping time @xmath120 is smaller for larger @xmath116 , while , at very long @xmath95 , @xmath120 is larger for larger @xmath116 . therefore at certain @xmath95 in between , there is a crossing point .     in the presence of dichotomic tension for the chain of @xmath48 . 
   the dichotomic force amplitudes are @xmath99 ( circle ) and @xmath121 fn ( rectangular ) . 
   the looping time shows a minimum at optimal flipping time @xmath119 . [ 
   fig : n12_t],width=321 ]    in contrast to the ra of single particle , the fluctuating barrier heights and thus the looping time depends sensitively on the chain length @xmath3 . 
   we obtain the normalized looping time @xmath122 for different chain lengths @xmath87 in the presence of dichotomic force of amplitude @xmath117 fn and static tension @xmath100 fn ( fig . 
   [ fig : compare ] ) . while the optimal flipping time @xmath123 increases with @xmath87 as implied by the relation @xmath124 , the minimum of @xmath125 decreases with @xmath87 . 
   this is because , similar to the static tension case , the fluctuating barrier height will be a function of @xmath126 , so that the longer chain tends to be more susceptible to the tension @xmath116 , and have the lower relative looping time .     in the presence of dichotomic tension as well as static tensions for the chains of @xmath48 ( circle ) , 18 ( rectangular ) , and 24 ( triangle ) . 
   the dichotomic force amplitude is @xmath99 fn , and the static tension amplitude is @xmath100 fn . 
   as @xmath3 increases , the optimal flipping times @xmath113 increase and the minimum values of normalized looping time @xmath127 decrease.[fig : compare],width=321 ]    a recent experiment  @xcite has shown that a fluctuating tension on dna greatly reduces the looping time for small @xmath95 , in consistency with our results . 
   they attributed the phenomenon to an increase of the effective temperature . 
   this may be reasonable for the small @xmath95 , where the dichotomic noise adds to the thermal noise , but could not lead to the nonmonotonic resonant behavior emerging over the entire range of @xmath95 . 
   we have studied the effects of static or time - dependent tension on the semiflexible polymer looping using brownian dynamics simulation . for the case of static tension , we have found that a minute tension as small as 100 fn can dramatically change the looping time , especially for long chains . 
   this sensitivity is a consequence of the cooperativity of the chain arising from chain connectivity . for the case of time - dependent tension , we considered dichotomically fluctuating tension , where the tension in average changes its sign in time @xmath95 . 
   the looping time has a resonant minimum at an optimum flipping time @xmath123 , which is a nontrivial extension of the resonant activation ( ra ) of single brownian particle . 
   the effect of time - dependent tension is also more significant for longer chains . 
   our results are consistent with recent experiments for both static  @xcite and a fluctuating tension  @xcite cases . 
   although we neglect the details of chain conformation outside of the loop and orientation constraints , etc . , our model could be a basic step to understand the loop formation process @xmath128 where biopolymers are constantly subject to forces . 
   this study suggests a possibility that a biopolymer can self - organize optimally utilizing the ambient nonequilibrium fluctuations . 
   further experiment using dichotomically fluctuating tension with various @xmath95 is called for to establish the ra phenomena in polymer loop formation . 
   as the chain gets shorter , we observe small difference between the looping time with hydrodynamic interaction studied in  @xcite and those without hydrodynamic interaction obtained by us using the path integral hyperdynamics method  @xcite .
summary: biopolymer looping is a dynamic process that occurs ubiquitously in cells for gene regulation , protein folding , etc . in cellular environments , biopolymers are often subject to tensions which are either static , or temporally fluctuating far away from equilibrium . 
   we study the dynamics of semiflexible polymer looping in the presence of such tensions by using brownian dynamics simulation combined with an analytical theory . 
   we show a minute tension dramatically changes the looping time , especially for long chains . considering a dichotomically flipping noise as a simple example of the nonequilibrium tension 
   , we find the phenomenon of resonant activation , where the looping time can be the minimum at an optimal flipping time . 
   we discuss our results in connection with recent experiments .