Dynamics of the Protein Search for Targets on DNA in the Presence of Traps

Transcription

1 pub.ac.org/jpcb Dynamic of the Protein Search for Target on DNA in the Preence of Trap Martin ange,, Maria Kochugaeva,, and Anatoly B. Kolomeiky*,, Department of Chemitry, Rice Univerity, Houton, Texa 775, United State Center for Theoretical Biological Phyic, Rice Univerity, Houton, Texa 775, United State Johanne Gutenberg Univerity, Mainz 5522, Germany ABSTRACT: Protein earch for pecific binding ite on DNA i a fundamental biological phenomenon aociated with the beginning of mot major biological procee. It i frequently found that protein find and recognize their pecific target quickly and efficiently depite the complex nature of protein DNA interaction in living cell. Although ignificant experimental and theoretical effort were made in recent year, the mechanim of thee procee remain not well-clarified. We preent a theoretical tudy of the protein target earch dynamic in the preence of emipecific binding ite which are viewed a trap. Our theoretical approach employ a dicrete-tate tochatic method that account for the mot important phyical and chemical procee in the ytem. It alo lead to a full analytical decription for all dynamic propertie of the protein earch. It i found that the preence of trap can ignificantly modify the protein earch dynamic. Thi effect depend on the patial poition of the target and trap, on ditance between them, on the average liding length of the protein along the DNA, and on the total length of DNA. Theoretical prediction are dicued uing imple phyical chemical argument, and they are alo validated with extenive Monte Carlo computer imulation. INTRODUCTION All major biological procee are governed by protein DNA interaction. Many of them begin after a protein molecule firt earche and then bind to a hort egment of DNA with a pecific equence, which i known a a pecific binding ite. Thi allow protein to effectively tranfer the genetic information contained in DNA by initiating a cacade of biochemical procee relevant for the ucceful functioning of biological cell. Clearly, the protein earch for target on DNA i one of the mot important phenomena in nature. Thi ubject wa invetigated for many year, 2 5 but ignificant progre wa achieved in recent year with a development of advanced experimental and theoretical method However, many detail of the mechanim of the protein earch for target on DNA till remain not well-undertood One of the mot facinating obervation in thi field i that many protein can find and recognize their pecific binding ite much fater than expected if the earch would take place only via 3D bulk diffuion. 3,5,6,27,28 Thi urpriing reult i called a facilitated diffuion, and it i frequently argued that thi happen due to the protein earch being a combination of 3D and D mode. 3,5,6,27,28 More pecifically, the propoed picture aume that the protein molecule aociate nonpecifically to DNA, can ome length of DNA by liding, diociate from DNA, and repeat thee action everal time until the target i located. Several experiment upport thi point of view. 7,,,8 Thee obervation alo ugget that the pecific nature of the protein DNA interaction hould have a tronger effect on the protein earch dynamic. Indeed, in real ytem there are many equence that have tructure and chemical compoition imilar to the pecific ite. 2,33 The protein molecule can be trapped in thee emipecific ite for large period of time, and it i not clear then how a fat earch can be accomplihed. However, the majority of theoretical model for the facilitated diffuion ignore thi effect, auming that the nonpecifically bound protein lide along the homogeneou DNA chain with the ame diffuion contant. 6,28 There are everal theoretical tudie that take into account the poibility of trapping. 2,32 They argue that the bound protein can fluctuate between everal conformation while till being aociated with the DNA chain, and thi lead to the avoidance of thee emipecific binding ite. However, the molecular mechanim of thi avoidance are not clear, and there i no experimental proof for thi. In thi paper, we preent a theoretical invetigation of the protein earch for target on DNA with emipecific binding ite that are viewed a trap. Our analyi ue a dicrete-tate tochatic approach 5,6 that explicitly take into conideration major phyical and chemical procee in the ytem. It allow u to obtain a full analytical decription for all dynamic propertie in the protein earch by utilizing a method of firtpaage procee. The application of the dicrete-tate tochatic method to the protein earch without trap uncovered three dynamic regime, depending on the relative value of the important length cale in the ytem. 6 For the protein liding length λ larger than the length of the DNA chain, the protein i involved in a D earch proce with a randomwalk dynamic. When the liding length i larger than the target Received: July 28, 25 Revied: Augut 27, 25 Publihed: September 2, American Chemical Society 24

2 The Journal of Phyical Chemitry B ize but maller than the length of DNA, the earch mechanim combine D and 3D motion. For even maller liding length the diffuion along the DNA chain i not poible, and the protein earche for the target only via 3D motion. By generalizing and extending thi method to the ytem with emipecific ite, we how that the trap have a trong effect on the earch dynamic. Surpriingly, there are many counterintuitive obervation when the preence of the trap might accelerate the earch. We alo tet our theoretical prediction with Monte Carlo computer imulation. THEORETICA METHODS The dicrete-tate tochatic approach 5,6 can be generalized for analyzing the protein earch with an arbitrary number of target and trap on DNA. To capture the main feature of the proce, we conider a impler model where a ingle protein molecule earche for one pecific binding ite on a ingle DNA chain, which alo ha one emipecific binding ite, a preented in Figure. To implify thing even further, we aume that Figure. General cheme for the protein target earch on DNA with a trap. There are 2 nonpecific, pecific, and trap binding ite on the DNA chain. The target i at ite m, and the trap i at ite m 2.A protein molecule can lide along the DNA chain with rate u, or might diociate into the olution with the rate k off. The bulk olution i labeled a a tate. From the olution the protein can aociate to any ite on DNA with the total rate k on. trap are irreverible; i.e., if the protein molecule bind to the emipecific ite it will never diociate. Thi i a trong aumption, but becaue the earch time in many ytem are quite hort and experiment can be done only for finite period of time, thi hould approximate the protein earch dynamic reaonably well. We conider a ingle DNA molecule with 2 nonpecific binding ite and two pecial ite: one of them placed at m i a pecific target for the protein, while another one (at m 2 )ia emipecific trap: ee Figure. For convenience, we alway aume that m < m 2, i.e., the trap i alway to the right of the target (Figure ). But the pecific order of the target and trap, obviouly, doe not affect the phyic of thi phenomenon. In in vitro experiment the protein molecule move much fater in the bulk olution than on DNA, and we aume that it can reach any ite on DNA with the ame probability. A total protein aociation rate to DNA i equal to k on, while the bound protein can diociate into the olution with a rate k off, a hown in Figure. In addition, the DNA-bound protein can lide along the chain with rate u in both direction (Figure ). Thi rate can be viewed a a D diffuion contant for moving on DNA. The protein earch alway tart from the olution that we label a a tate. There are two poible final outcome of the earch proce. The protein molecule can find the target, and thi i a ucceful event. Or, the protein might fall into the trap and never leave it: thi i not a ucceful event. Thu, the probability to reach the target in thi model i alway le than one. 24 The protein earch for the target can be aociated with a firt-paage proce of reaching the pecific binding ite, and thi provide a direct way of evaluating the dynamic of the ytem. 6 We introduce a function F n (t) which i defined a a probability to reach the target at time t for the firt time, while not being trapped to the emipecific ite, if initially (at t ) the protein molecule tart at the tate n (n,,..., ). It i important to note that thi i a conditional probability for the protein molecule that are not captured by the trap. The temporal evolution of thee probabilitie can be decribed via a et of backward mater equation 5,6 d Fn ( t) uf [ n+ ( t) + Fn ( t)] + kofff( t) (2 u + k ) F( t) off n () for 2 n and n m or m 2. At DNA boundarie the dynamic i lightly different and d F ( t) d F( t) uf () t + k F () t ( u + k ) F() t 2 off off uf () t + k F () t ( u + k ) F () t off off In addition, for the bulk olution (n ) we have d F( t) k Fn () t konf () t on n Alo, there are additional contraint in the ytem, which can be written a Fm( t ) δ( t) Fm ( t) 2 (5) The phyical meaning of thee expreion i the following. If the protein molecule tart at t at the target ite m, the earch proce i uccefully finihed immediately. But if the protein at any time bind to the trap ite (m 2 ), it will never find the target. To olve eq 5, we reformulate the problem in the language of aplace tranformation, i.e., with F t n() e Fn() t. 6 Then the et of backward mater equation can be tranformed into a et of impler algebraic expreion ( + 2 u + koff) Fn() u[ Fn+ () + Fn ()] + kofff () (6) for 2 n and n m or m 2, and ( + u + k ) F( ) uf( ) + k F ( ) (2) (3) (4) off 2 off (7) ( + u + k ) F ( ) u F ( ) + k F ( ) off off (8) kon ( + k F F on) ( ) n( ) n (9) Fm() Fm () 2 () The olution of thee equation can be found by auming a n general form of the olution a Fn( ) Ay + B. 6 Thi yield

3 The Journal of Phyical Chemitry B Figure 2. Contour map for the mean earch time to reach the pecific binding ite a a function of the poition of the target m and trap m 2. Parameter ued for calculation are k on u 5 and k off 3. The length of the DNA chain i (a) and (b). with kon( koff + ) S( ) F () ( + k + k ) + k k S ( ) on off on off 2 () m + m ( + y)( y 2 ) S() 2m m m ( y)( + y )( + y 2 ) (2) S2() 2+ m + ( + y)[2( y m2 m ) + ( y 2 m 2m )( y 2( m + y 2) )] 2m + m m m ( y)( + y 2( ) )( + y 2 )( + y 2 ) (3) and u + koff y ( + 2 u + koff) 4u () 2u (4) Explicit analytical expreion for the firt-paage probability function in the aplace form provide u with a complete decription for all dynamic propertie in the protein earch. More pecifically, a function Π [with y( ) y] S() Π F ( ) S () 2 m + m + 2( m ) 2 2 ( y )[ + y ] 2+ m m2 m2 m 2m + 2( m2) 2( y ) + ( y )[ y + y ] (5) i the overall probability (at all time) for the protein molecule to reach the target tarting from the bulk olution. It i generally le than becaue of the poible falling into the trap. For a ymmetric ditribution of the target and the trap with repect to the middle of the DNA chain, when m 2 m +, it follow from eq 5 that the probability to reach the pecific ite i alway Π /2. A mean firt-paage time to reach the target, T, which we alo identify a the average earch time, i given by T F() Π (6) It i important to note that thi i a conditional mean firtpaage time, which mean that the average i taken only over the ucceful trajectorie that lead to the target. The earch trajectorie that end up in the trap are ignored for the calculation of the mean earch time. The explicit expreion for T can be written a T k + off kon( S2()) d S2() +Π k k S () d S() on off 2 (7) It can be hown that the firt term on the right ide of eq 7 correpond to the earch time for the ytem with two target (at the poition m and m 2 ), 34 while the econd term correct thi reult by accounting for the fact that the ite at m 2 i the irreverible trap. From thi point of view, the earch time can be preented a d S 2() T(target/trap) T(2 target) + Π d S() (8) The reaon that our ytem with the target and the trap i related to the earch on the DNA chain with 2 target i due to the fact that target and trap are pecial poition on DNA that guide the dynamic. In the ytem with two target, there are two probability fluxe for going from the olution to thee pecial ite, and the mallet time (or the larget flux) determine the overall earch time. In our model with the target and trap we alo have two probability fluxe to the pecial ite, but only one of them, to the target, define the earch time. RESUTS AND DISCUSSION Spatial Ditribution of Target and Trap. The firt quetion we would like to addre i the effect of the patial ditribution of target and trap in the protein earch dynamic. The average time to reach the target a a function of the poition m and m 2 for different DNA length are preented in Figure 2. One can ee that there are optimal poition for the target and for the trap (Figure 2a), namely, m /4 and m 2 3/4, for which the mean firt-paage time are minimal. Thee are exactly the ame optimal poition for the ytem with two target. 34 The following argument can be ued to explain thi effect. Becaue at thee condition the earch i taking place motly through D diffuion, if the tarting poition of the protein on DNA i the ite n >(m + m 2 )/2, on average, it will not reach the pecific binding ite. Thi mean that the problem with the target at the ite m and the trap at the ite m 2 i identical to the earch on DNA of the length (m + m 2 )/2 with only target at m. In thi cae, the mot optimal poition for the target i in the middle of the DNA egment, 6 i.e., m (m + m 2 )/4. Thi lead to the relation m 2 3m. Now, thee optimal poition mut alo be ymmetric with repect to the middle of the chain becaue the exchange of the location of the target and the trap hould not affect the outcome. Thi yield m + m 2. It can be eaily hown then that putting the target and the trap to ite m / 4 and m 2 3/4 atifie thee requirement. 242

4 The Journal of Phyical Chemitry B However, the mot optimal ditribution i not oberved for all condition. Increaing the length of DNA (ee Figure 2b) completely change the picture. Now, any poition of the target and the trap along the DNA chain, a long a they are not at the boundarie (m and m 2 ), lead to the ame earch time. Thi i taking place becaue for < λ < (where the protein liding length i given by λ u/ k off ) the earch follow the liding regime: the protein molecule can the length λ on the DNA before diociating, and it repeat thi earching cycle many time (/λ on average) before reaching the target. After each diociation, the protein doe not have any memory of what part of DNA it jut canned. With equal probability it can bind to any ite on DNA. A a reult, the abolute poition of the target and trap are not important anymore for the earch optimization. Dynamic Phae Diagram. In the next tep, we invetigate how the preence of the emipecific ite influence the mechanim of the earch in different regime. The reult are preented in Figure 3. Firt of all, the general feature of the dynamic phae diagram do not change with the addition of the trap ite. There are till 3 earch regime depending on the relative value of the canning length λ u/ k off, the DNA length, and the target ize, which i taken to be equal to. 6 When λ > we have the phae in which the protein molecule bind to DNA and move along the chain until it encounter the pecific binding ite. Thi i the D random-walk earch regime with the expected quadratic caling on the earch time a a function of the DNA length. 6 For < λ < the ytem i in the liding regime, where the protein molecule bind nonpecifically to DNA, can a ditance λ, and diociate, and thi earch cycle, on average, i repeated everal time until the target i found. Thi earch mechanim can be viewed a a combination of 3D and D motion. In thi phae, the caling of the earch time i linear with becaue the number of earch cycle i proportional to the DNA length, i.e., T / λ. 6 For even maller canning length, λ <, the earch become purely 3D becaue the protein bound to DNA cannot lide along the chain. Again, the linear caling of the earch time with i oberved becaue, on average, the protein ha to viit ite before it aociate to the target. 6 One can alo ee from Figure 3 that adding the trap decreae the earch time for the ytem that originally had only a ingle target. However, it come with a price of lowering the probability to reach the target: ee Figure 4. For the mot optimal poition of the target and trap thi probability i alway equal to /2 (Figure 3a and 4). Thi can be eaily explained if we notice that the mot optimal ditribution i ymmetric (m /4 and m 2 3/4), which mean that exactly half of the trajectorie are ucceful and another half end up in the trap. For other patial ditribution the probability to reach the target depend on the earch regime. In the random-walk dynamic phae (λ > ), thi can be explained by purely geometric factor becaue of the onedimenional nature of the earch proce. A we already dicued above, if the protein tart the earch at the poition n >(m 2 + m )/2 (ee Figure ), then on average it will be trapped ince the ditance to the emipecific ite i horter. So the probability to reach the target in thi regime can be etimated a Π m + m 2 2 (9) Figure 3. Dynamic phae diagram for the protein earch on DNA with one target at poition m, with two target at poition m and m 2 and with the target and the trap at poition m and m 2. Parameter ued for calculation are k on u 5 and : (a) m /2, m /4, and m 2 3/4; (b) m /4, m /4, and m 2 /2; and (c) m /2, m /2, and m 2. For the mot optimal poition m /4 and m 2 3/4 thi yield Π /2, while for m /4 and m 2 /2 thi give Π 3/8, and for m /2 and m 2 we obtain Π 3/4. Thee calculation fully agree with the reult preented in Figure 4 in the limit of very large canning length λ. For very mall λ <, when the earch follow the 3D mechanim, the patial poition of the target and trap are not important. In thi cae, exactly half of the trajectorie will be ucceful, yielding Π / 2. For the intermediate liding regime ( < λ < ), the probability to reach the target obviouly ha the lower and 243

5 The Journal of Phyical Chemitry B Figure 4. Probability to reach the target a a function of the canning length for different ditribution of the target and trap ite. Parameter ued for calculation are k on u 5,, and k off i changing. Symbol are from Monte Carlo computer imulation. m + m upper bound between /2 and 2 (which could be larger 2 or maller than /2), and the explicit value of Π depend on the relative contribution of D and 3D fluxe into the target ite. It i alo important to compare the protein earch dynamic on DNA with two target with the earch in the ytem that ha only one target and one trap at the ame poition, a hown in Figure 3. In the jumping and liding regime (λ < ), there i no difference in the earch time between two ytem becaue the location of the pecial ite doe not influence the earch mechanim. The protein motly reache the pecific binding ite from the bulk olution. For the ytem with two target, one-half of earch trajectorie will go to one of the target, and the econd half will finih at the econd pecific ite. The ymmetry require that both of thee et of trajectorie have the ame mean time becaue the target are inditinguihable. For the protein earch on DNA with the target and the trap the dynamic i imilar: half of all trajectorie will end up at the trap, and they will not be counted. But another half of trajectorie that reach the pecific binding ite have the ame mean earch time a in the cae of two target on DNA. The ituation i different in the random-walk phae (λ > ). Here, the earch time for the target and trap ytem could be the ame a for the two target ytem (Figure 3a). Thi i the cae for the ymmetric location of the pecial ite. The preence of the trap could alo low down the earch (Figure 3c), or urpriingly, it could even accelerate the earch (Figure 3b). It i intereting to note that, in the target and trap ytem where the earch i fater, the probability to find the target i lower (ee Figure 4). Thu, thi effect can be alo explained by the geometric argument, a dicued above. The trap effectively remove trajectorie with longer earch time. How Trap Accelerate the Search. We have already hown that the addition of the emipecific ite trongly affect the protein earch dynamic for pecific binding ite. To quantify thi effect we introduce two auxiliary function, r and r 2, which are defined a the ratio of the earch time on DNA with one or two target and for the ytem with the target and trap for the fixed value of the poition m and m 2, repectively T ( target) T (2 target) r r2 T (target/trap) T (target/trap) (2) Thee acceleration function for variou et of parameter are preented in Figure 5 and 6. Figure 5. Acceleration parameter r a a function of the trap location for the fixed poition of the target. Parameter ued for calculation are k on u 5. (a) The target i at m /4 and k off 3. The target i at m /2 for the reference ingle target ytem. (b) The target i at m,, and k off.. The target i at m for the reference ingle target ytem. Dahed line correpond to r 4. Symbol are from Monte Carlo computer imulation. Figure 6. Acceleration function r 2 a a function of the poition of the trap for the fixed poition of the target. Parameter ued for calculation are, m,k on u 5, and k off.. Dahed line correpond to r 2. Symbol are from Monte Carlo computer imulation. Firt, we analyze how the trap influence the earch on DNA with only one target a preented in Figure 5. One can ee that it i uually fater to find the target if there i a emipecific ite in the ytem. For not very long DNA, there i an optimal poition of the trap that provide the hortet earch time (Figure 5a,b). The acceleration can even reach very high value if the target i far away from the middle of the chain and the 244

6 The Journal of Phyical Chemitry B trap i cloe to the target: ee Figure 5b. Thee obervation can be explained uing the following argument. For thi et of parameter, the protein reache the pecific binding ite motly via D liding along the DNA chain ( λ). Introducing the trap ite into the ytem ha two oppoite effect: it remove many long-time trajectorie from the earch, lowering the mean earch time. However, it alo decreae the flux into the target ite from one ide of DNA which make the earch longer. Balancing thee two effect lead to the optimal poition of the trap. Thee argument alo ugget that the maximal acceleration can be achieved if the target it at the boundary, i.e., m (ee Figure 5b). In the limiting cae of m and m 2 the acceleration i equal to 4 (Figure 5b). Thi i becaue thi ytem of the target and trap on DNA with the length can be mapped into the earch on DNA of the length /2 with the target and without the trap. Since the D earch in the random-walk regime ha a quadratic caling of the earch time (T 2 ), the earch time acceleration become r ( /2) Increaing the length of DNA hift the ytem to the liding earch regime, and the maximal acceleration in the earch i equal to 2 (Figure 5a). The patial poition of the target and trap are not important becaue of 3D + D earch mechanim, a explained above. In thi regime, the trap effectively remove half of all earch trajectorie, which i equivalent to lowering the number of earch cycle alo in 2 time. Becaue of the linear caling for the earch time in thi dynamic phae, thi yield r 2. The comparion of the protein earch dynamic on DNA that ha one target and one trap with the ytem with two target i hown in Figure 6. Here the effect of the trap ite i more complex. For relatively hort ditance between the target and the trap, the earch i much fater than that for the cae of two target. Thi reult i unexpected, but it can be explained uing the geometric argument. The trap effectively remove a ignificant part of DNA from the earch, and becaue for thi et of parameter (Figure 6) the earch i motly onedimenional, thi lead to large acceleration. One could alo think about two cloely located target a one effective new target, and all our argument why it i fater to earch by adding the trap to the ytem with one target can be applied now. However, moving the trap ite away from the target lower thi effect and tarting from ome ditance the earch in the ytem with two target i fater becaue the number of pecific ite i larger. One can ee in Figure 6 that for m 2 / >.67 the acceleration parameter goe below the unity. Thu, our calculation clearly how that the patial ditribution of the target and trap control the earch dynamic. SUMMARY AND CONCUSIONS We preented a theoretical invetigation on the role of emipecific binding ite in the protein earch for target on DNA. Our approach i baed on the dicrete-tate tochatic method that connect the earch proce with the firt-paage event. The advantage of thi approach i that it provide a full analytical decription for all dynamic propertie in the ytem. We determined that the protein earch dynamic i governed by everal important length cale uch a the DNA length, the average liding length of the protein along the DNA chain, the ditance between the target and trap, and the ditance to the DNA end from the pecific and emipecific ite. It wa found that there i the optimal patial ditribution of the target and 245 trap that for hort DNA lead to the mallet earch time, while for long DNA the earch i not affected by exact poition of pecific and emipecific binding ite. Thi wa explained by exploring the dynamic phae diagram which how three different regime for the protein earch depending on the relative value of the relevant length cale in the ytem. We alo analyzed the probability of reaching the target, and it wa found that it varie for different dynamic earch regime. Furthermore, we invetigated the acceleration in the earch due to the preence of the trap ite. Adding the emipecific ite in mot cae decreae the earch time for the ytem with only one target. For the ytem with two target on DNA the ubtitution of one them with the trap lead to more complex behavior. For a hort ditance between the pecial ite the earch i accelerated, while for large ditance the earch become lower. Thee phenomena are explained by noting that the ignificant fraction of the earch trajectorie i removed from the earch due to falling into the trap. Our theoretical approach provide a imple and clear picture of the complex biological procee during the protein earch for the pecific binding ite on DNA. At the ame time, it hould be noted that the preented theoretical method i not exact, and it involve everal approximation. The conformational freedom of DNA chain and the interegment tranfer procee are neglected. We alo aume that the protein move fater in the bulk olution than on DNA. In the real biological cell all thee aumption probably are not valid, but it i not clear how thi would affect the overall earch dynamic. But the mot eriou iue in our work i the aumption of the trap irreveribility. In reality, the protein molecule cannot be aborbed by thee trap for an infinite amount time, and they will be eventually releaed. In addition, ome of thee trap are not very trong. It will be critically important to addre thee iue in more advanced theoretical and experimental tudie. AUTHOR INFORMATION Correponding Author * tolya@rice.edu. Note The author declare no competing financial interet. ACKNOWEDGMENTS The work wa upported by the Welch Foundation (Grant C- 559), by the NSF (Grant CHE-36979), and by the Center for Theoretical Biological Phyic ponored by the NSF (Grant PHY ). We alo would like to thank Prof. J. 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