Facilitated diffusion framework for transcription factor search with conformational changes
aa r X i v : . [ q - b i o . S C ] J u l Facilitated diffusion framework for transcription factor searchwith conformational changes
J´erˆome Cartailler * and J¨urgen Reingruber Ecole Normale Sup´erieure, 46 rue dUlm, 75005 Paris , France. ** INSERM U1024, Paris, France. Corresponding author: [email protected] bstract
Cellular responses often require the fast activation or repression of specific genes, which dependson Transcription Factors (TFs) that have to quickly find the promoters of these genes within alarge genome. Transcription Factors (TFs) search for their DNA promoter target by alternatingbetween bulk diffusion and sliding along the DNA, a mechanism known as facilitated diffusion.We study a facilitated diffusion framework with switching between three search modes: a bulkmode and two sliding modes triggered by conformational changes between two protein confor-mations. In one conformation (search mode) the TF interacts unspecifically with the DNAbackbone resulting in fast sliding. In the other conformation (recognition mode) it interactsspecifically and strongly with DNA base pairs leading to slow displacement. From the bulk, aTF associates with the DNA at a random position that is correlated with the previous disso-ciation point, which implicitly is a function of the DNA structure. The target affinity dependson the conformation. We derive exact expressions for the mean first passage time (MFPT) tobind to the promoter and the conditional probability to bind before detaching when arriving atthe promoter site. We systematically explore the parameter space and compare various searchscenarios. We compare our results with experimental data for the dimeric Lac repressor searchin E.Coli bacteria. We find that a coiled DNA conformation is absolutely necessary for a fastMFPT. With frequent spontaneous conformational changes, a fast search time is achieved evenwhen a TF becomes immobilized in the recognition state due to the specific bindings. We find aMFPT compatible with experimental data in presence of a specific TF-DNA interaction energythat has a Gaussian distribution with a large variance.
Keywords:
Facilitated diffusion; Transcription factor; Mean first passage time; Gene regula-tion; Mathematical model; Lac repressor; E.Coli acilitated diffusion with three search modes Introduction
Transcription factors (TFs) regulate gene activation by binding to DNA promoter sites. Toenable a fast cellular response that relies on the activation or repression of specific genes, TFsperform a facilitated diffusion search where they alternate between three dimensional (3D) dif-fusion in the bulk and 1D diffusion (sliding) along the DNA (for reviews see (1–6)). Initially,facilitated diffusion was introduced to explain the experimental finding that the in-vitro associ-ation rate of the Lac-I repressor with its promoter sites placed on λ -phage DNA was around 100times larger than the Smoluchowski limit ∼ M − s − for a 3D diffusion process (7). Theo-retical considerations showed that a search that alternates between 3D diffusion and 1D slidingcan have a higher association rate compared to a pure 3D search (8–10). With dilute DNA thesearch time is dominated by the 3D excursions between subsequent DNA binding events, andsliding increases the association rate by enlarging the effective target size (antenna effect). Lateron single molecule techniques provided a direct experimental proof of the facilitated diffusionmechanism (11–14).It has also soon been realized (9, 15) that frequent bindings to the DNA are problematicbecause sliding along the DNA is slow due to strong TF-DNA interactions (13, 14). In adense DNA environment with frequent bindings to the DNA and slow 1D diffusion, the antennaeffect becomes negligible and facilitated diffusion is slower compared to a pure 3D search. Forexample, in E.Coli with a volume | V | ∼ µm , the measured search time of the Lac repressor forits promoter site is τ ∼ s (11, 13). This corresponds to an association rate k a = N Av V /τ ∼ M − s − , much lower than the Smoluchowski limit. If a TF could specifically bind only to itspromoter site and bounce off from the rest of the DNA, the search time would be extremely fastaround ∼ V / (4 πDR ) ∼ s . However, because a TF cannot already recognize its target from thebulk, frequent DNA associations are essential. Thus, the question arises: How is a fast searchpossible within a large genome despite of facilitated diffusion ?When a TF is bound to the DNA and interacts with the underlying base pairs (bps), thediffusion coefficient for sliding decays exponentially with the variance of the binding energydistribution (16, 17). With a simple facilitated diffusion model that comprises sliding along theDNA and uniform redistributions in 3D, one finds that a search time of the order of minutes isonly compatible with a variance . . k B T (18). In contrast, binding energy estimates for the Cro and
PurR
TF reveal a much larger variance around 5 − k B T (18, 19). This indicates thata simple diffusion process is not sufficient to explain the search dynamics when a TF is boundto the DNA. It has been proposed that a TF switches between two protein conformations withdifferent binding affinities to the DNA (15, 18). In the search conformation, a TF interacts onlynon-specifically with the DNA backbone leading to a smooth energy profile and fast diffusion.In the recognition conformation, a TF interacts specifically with the underlying DNA sequenceresulting in a rough energy landscape and slow diffusion. Conformational changes of the TFprotein are indeed supported by experimental observations (12, 20–24).In this work we investigate a general framework for a facilitated diffusion search with con-formational changes. We analytically derive the mean first passage time (MFPT) to bind tothe target and the conditional probability to bind before dissociation when a TF arrives at thetarget site. We further compute the ratio of the time spent in the bulk compared to attached tothe DNA, the apparent diffusion constant for sliding along the DNA, and the average sliding dis-tance before detaching. We consider a search process with Poissonian switchings between threestates (Fig. 1). State 1 and 2 (recognition and search mode) correspond to two different proteinconformations with conformation dependent TF-DNA interactions. Therefore the diffusion co- acilitated diffusion with three search modes A DNATarget E ns E State 3State 2State 1 B Figure 1:
Facilitated diffusion framework. (A) Schematic of the search process: In state 3a TF is freely diffusing in the bulk. It attaches to the DNA at a random position following aGaussian distribution centered around the previous detaching position. In state 1 and 2 a TFis attached to the DNA and diffuses along the DNA. In state 1 it specifically interacts with theunderlying DNA sequence and diffusion is slow. In state 2 it non-specifically interacts with theDNA backbone and diffusion is fast. The binding affinity to the target depends on the state.(B) Switchings between states occurs with Poissonian rates. The rate k depends on the energyof the specific interaction.efficients for sliding along the DNA and the target affinity depend on the conformation. In state3, a TF is diffusing in the bulk and it associates with the DNA at a random position following aGaussian distribution centered around the previous dissociation point. By modifying the targetaffinity in state 2 we evaluate the impact of induced switchings at the target site and the effectthat a TF misses the target when arriving at the target site. By varying the correlation distancebetween dissociation and association point we estimate how the DNA conformation and coilingaffect the MFPT. With our analytic expressions we can precisely evaluate the whole parameterspace. We analyze various search scenarios within the same framework, which is importantto accurately compare results. Other approaches partly rely on MFPT analysis, kinetic the-ory, thermodynamic equilibrium considerations, scaling arguments and other approximations,which complicates a comparison of results obtained with different methods and approximations(19, 25–30). A clean MFPT analysis for a 3 states switching model with maximal target affinityin state 1 and no affinity in state 2 has been performed in (31). Compared to (32), the authorsadditionally consider 3D excursions using a closed-cell approach as described in (9). However,due to the difficulty to compute the 3D kernel, only the asymptotic limits corresponding touniform redistributions and a rod-like DNA are discussed. The first passage time distributionfor a switching process between two 1D states has been studied in (33). Model
Model description and MFPT analysis
We start by presenting the mathematical framework and the MFPT analysis. We postpone thedescription of the biological motivation to the results part. We consider a search that switchesbetween 3 states with Poissonian switching rates k ij (Fig. 1). In state 1 and 2 a TF is attachedto the DNA of length 2 L and slides with state dependent diffusion constants D and D . Tosimplify the analysis we consider that the target is located at the center. An off-centered targetresults in a higher MFPT up to maximally a factor of 4 if the target is located at the periphery(assuming that the MFPT scales ∼ L ) (34, 35). When a TF reaches the target it binds withstate dependent affinities χ and χ . In state 1 a TF switches to state 2 with rate k . In state acilitated diffusion with three search modes
42, in addition to switching to state 1 with rate k , a TF can also dissociate with rate k andswitch to state 3 where it diffuses in the bulk. From state 3 it associates with the DNA withthe rate k at a random position drawn from a Gaussian distribution with variance σ centeredaround the previous dissociation point.Because of the Gaussian attaching distribution, we model state 3 as an 1D diffusion processalong the DNA with an effective diffusion constant D = σ k and no target affinity ( χ = 0).Thus, we finally arrive at a framework with switchings between three 1D states. The backwardFokker-Planck equation for the probability p ( x, t, n | y, m ) to find the TF at time t in state n atposition x , conditioned that it started at t = 0 in state m at position y , is (36, 37) ∂ t p ( x, t, n | y, m ) = D m ∂ y p ( x, t, n | y, m ) − χ m p ( x, t, n | y, m ) δ ( y ) − X i =1 k mi ( p ( x, t, n | y, m ) − p ( x, t, n | y, i )) , (1)with reflecting boundary conditions at y = ± L . The mean sojourn time spent in state n is τ n,m ( y ) = Z ∞ dt Z L − L dx p ( x, t, n | y, m ) . (2)From Eq. 1 we find that the τ n,m ( y ) satisfy the system of equations D m τ ′′ n,m ( y ) − X i =1 ( k m + δ mi − k mi ) τ n,i ( y ) − χ m δ ( y ) τ n,m ( y ) = − δ nm (3)with k m + = P j =1 k mj . In the Supplementary Information (SI) we exactly solve Eq. 3 andderive analytic expressions for the τ n,m ( y ). The MFPT when initially in state m at position y is τ m ( y ) = P n =1 τ n,m ( y ). We focus on the MFPT with uniform initial distribution, ¯ τ m = L R L − L τ m ( y ) dy . Because switchings between states occur fast compared to the overall searchtime, the dependency of ¯ τ m on m is negligible. Furthermore, the mean sojourn times ¯ τ i,m approximately satisfy the scaling relations (see SI Eq. 40) ¯ τ ,m : ¯ τ ,m : ¯ τ ,m = 1 : k k : k k .Hence, the MFPT with uniform initial distribution is well approximated by¯ τ ≈ ¯ τ , (cid:18) k k + k k k k (cid:19) = N (cid:18) k + 1 k + k k k (cid:19) (4)where N = ¯ τ , k is the average number of switchings between states 1 and 2.To reveal the scaling of the MFPT as a function of the DNA length L , we introduce thescaled length ˆ L = LL defined with the reference length L = 1 bp . To simplify the discussion, wefocus on a scenario with maximal affinity in state 1, χ = ∞ , in which case case the search isover when the TF encounters the target in state 1 (the analysis in the SI is performed with ageneral χ ). For a long DNA ( ˆ L µ i ≫
1) we compute in the SI N = ˆ Ll (cid:18) c , µ √ µ + c , µ √ µ (cid:19) + ˆ L l l l β (5) acilitated diffusion with three search modes l ij = L k ij D i , c , = l l (cid:18) a de + b (cid:19) , c , = l l (cid:18) a de + b (cid:19) α = l + l + l + l , β = l l + l ( l + l ) , µ / = 12 (cid:16) α ± p α − β (cid:17) a = µ ( l + l − µ ) l ( µ − µ ) , a = µ ( l + l − µ ) l ( µ − µ ) , b = l − µ l ( µ − µ ) , b = l − µ l ( µ − µ ) κ = L χ D , d = 1 l κ − (cid:18) b √ µ + b √ µ (cid:19) , e = a √ µ + a √ µ + l l κ . Search with uniform redistribution in state 3 and no target affinity in state 2
With uniform redistributions ( σ = ∞ ) we have l = 2 L /σ = 0. With κ = 0 and l = 0,Eq. 5 simplifies to N = ˆ Ll (cid:18) l − µ µ − µ √ µ + l − µ µ − µ √ µ (cid:19) , (6)in agreement with (32). We note that the term ∼ L in Eq. 5 vanished and we now have N ∼ L ,which leads to a faster search for large L . As stated before, Eq. 6 is valid for ˆ L µ i ≫ i = 1 , k →
0. For k → µ , vanishes andthe condition ˆ L µ ≫ L and k → ∼ L and not ∼ L , which isindeed the case, as can be shown by a refined analysis. However, for any fixed value k >
0, byincreasing L , N eventually scales ∼ L due to the uniform redistributions. Optimal switching scenario in state 2
When the properties of state 1 and state 3 are fixed (and D is fixed), we compute the optimalswitching rates k and k that minimize the MFPT. We introduce the parameters σ = 2 D k , σ = 2 D k + k , q = k k + k , ζ = σ σ . (7) σ is the mean square displacement in state 1, σ is the mean square displacement in state 2before switching either to state 1 or 2, q is the detaching probability, and ζ is the ratio of thedisplacements in state 1 and 2. We use the variables q and ζ instead of k and k . We have l = l qζ and l = l (1 − q ) ζ . Because diffusion in state 1 is slow compared to state 2, wehave ζ ≪
1. We further consider that the probability to switch from state 2 to 1 is much largerthan the dissociation probability, such that q ≪
1. For ζ ≪ q ≪ τ ≈ √ Lσ s ζq ! (cid:18) k + σ D ζ + qk (cid:19) . (8)For fixed σ , D , k and k , the minimum of ¯ τ with respect to ( q, ζ ) is¯ τ = √ Lσ k (1 + γζ ) (9) acilitated diffusion with three search modes γζ = q δ and γ qζ = 1, where γ = q D k σ and δ = k k γ = k k q D k σ . Interestingly,whereas the optimal rate k depends on the properties of state 1, we find for k the optimalvalue k = D D qζ = D D γ = k , independent of the properties of state 1. Search with two states only
To derive the MFPT with switchings between two states we set k →
0. With κ = ∞ and κ = 0 (state 2 now corresponds to the bulk state without binding) we find¯ τ = ˆ Ll l µ ˆ L l µ √ µ ! (cid:18) k + 1 k (cid:19) (10)with µ = l + l . For k → τ = L D . With uniform redistributions in state 2we get ( l = 0) ¯ τ = √ Lσ (cid:18) k + 1 k (cid:19) . (11)Eq. 11 as a function of k has a minimum for k = k (compare with k = k obtained with3 states). Results
We present results that we compare to experimental measurements for a dimeric Lac repressorsearch in E.Coli bacteria. We use a DNA length 2 L = 4 . × bps (13), a TF attaches to theDNA from the bulk after an average time k − = 1 . ms (13, 38), and the diffusion constant instate 2 is D = 2 µm /s ( D = 1 . × bp s ) (11, 13). We keep these values fixed throughoutthe following analysis and we focus on investigating the impact of the remaining parameters. Search scenario with conformational changes
A TF is freely diffusing in the bulk (state 3) and attaches to the DNA with a Poissonian rate k (Fig.1). We consider that the association position follows a Gaussian distribution withvariance σ centered around the previous dissociation point. σ is the correlation distancebetween subsequent detaching and attaching positions. Hence, σ is an effective parametersthat implicitly depends on the DNA configuration and on coiling. For example, a uniformre-attaching distribution obtained for σ = ∞ is usually attributed to a highly packed DNAconformation (18, 32, 35, 39–41). By varying σ we can investigate how the DNA conformationaffects the MFPT. We assume that a TF switches between a stable and an unstable proteinconformation. The lifetime of the unstable conformation ξ − is short such that a TF quicklyreturns to its stable conformation after a spontaneous conformation change. When a TF isattached to the DNA and in the stable conformation (state 2) it non-specifically interacts withthe DNA backbone and diffuses in a smooth potential well with non-specific energy E ns andfast diffusion constant D . In state 2 a TF can either dissociate from the DNA with rate k , orswitch to the unstable conformation (state 1) with rate k . The unstable conformation allowsfor additional specific TF-DNA interactions that modify the residence time k − in state 1. Weuse the Arrhenius like relation k = ξe − ∆ E , where ∆ E = E ns − E (in units of k B T ). We use acilitated diffusion with three search modes
7a maximal target affinity in state 1, χ = ∞ , in which case the search is finished when a TFreaches the target in state 1 for the first time. In state 2, the outcome at the target depends onthe affinity χ (respectively the dimensionless parameter κ ). The search is over for κ = ∞ ,in which case a TF has maximal target affinity already in state 2. In the opposite case κ = 0a TF has no indication in state 2 that it has reached the target site and there is a probabilitythat he misses the target and detaches without binding. Because switching to state 1 at thetarget site ends the search, by varying κ we can explore the impact of induced switchings atthe target site.We use constant switching rates k , k and k . This is valid for a spatially homogenousDNA and a homogenous non-specific interaction in state 2. In contrast, k and σ depend onthe specific binding energy E and are therefore not constant along the DNA. To account forthis, we first derive results using a constant E corresponding to a homogenous DNA, and ina subsequent step we average using a Gaussian distribution for E . Because of strong specificinteractions, we focus on displacements σ that are small. The lower bound for σ is reachedwhen a TF becomes immobilized in state 1. However, this does not correspond to σ = 0,because a TF at least scans the base pair it binds to. A non-zero σ also accounts for stochasticfluctuations in the DNA position due to the switching process. By noting that the MSD of themaximum displacement of a diffusion process is 2 σ , we use σ = √ to model the limiting casewhere a TF becomes immobilized in state 1 and scans only a single base pair.To facilitate the comparison with experimental data, we introduce the following parametersthat characterize various properties of a search process: τ dna = 1 k + k k k , r d d = k τ dna , σ dna = σ q + σ − qq , D dna = σ dna τ dna . (12) τ dna is the average time a TF stays bound to the DNA before detaching; r d d is the ratio ofthe time bound to the DNA to diffusing in the cytoplasm; σ dna is the mean square displacementalong the DNA before detaching; D dna is the effective diffusion constant for sliding. Search with uniform redistributions and no target affinity in state 2
We start by analyzing search processes as a function of the switching rate k with no targetaffinity in state 2 ( κ = 0) and uniform redistributions in state 3 (Fig. 2). We write k asfunction of the binding strength, k = ξe − ∆ E , and plot quantities as a function of ∆ E . Weuse the basal rate ξ = 10 s − , which is similar to the attempt frequency 10 s − used in (18), or10 s − from (39). At this stage the exact value of ξ is not important to show the behaviour as afunction of ∆ E . For example, a smaller value for ξ would shift the origin of the ∆ E -axis to theright in Fig. 2 and Fig. 3, but otherwise does not affect the graphs. Later on we will estimate amore appropriate value for ξ by considering a Gaussian binding energy distribution.We compare optimal and non-optimal searches for σ = √ (1 bp is scanned in state 1) and σ = √ (up to 3 bps are scanned in state 1). The optimal search is characterized by k = k and a rate k that is a function of ∆ E and σ (see Eq. 9). For non-optimal searches we use k = k and a rate k that is independent of ∆ E , since the properties of state 1 should notaffect the switching rate k in state 2. We use the optimal value for k computed with ∆ E = 5(hence, k = 1 . × s − for σ = √ and k = 9 . × s − for σ = √ ), which givesa fast search also with large ∆ E . We use two different values for σ = √ and σ = √ tofacilitate the comparison between optimal and non-optimal curves: in this case the optimal andnon-optimal MFPT coincide for ∆ E = 5 (Fig. 2A). acilitated diffusion with three search modes ∆ E M F P T ( sec ) A σ = √ , optimal σ = √ σ = √ , optimal σ = √ ∆ E r d d B ∆ E D d n a / D C ∆ E b i nd i ng p r ob a b ili t y P D Figure 2:
Search with no target affinity in state 2 ( κ = 0 ) and uniform redistributions( σ = ∞ ). (A) MFPT for optimal and non-optimal searches as a function of the specific bindingstrength for two values of σ ( k = ξe − ∆ E , ξ = 10 s − ). For σ = 1 / √ σ = 3 / √ κ = ∞ ). Energies are in units of k B T . The rest of the parameters are: κ = ∞ , k − = 1 . ms , D = 2 µm /s , k = k . The value of k for the non-optimal search equals the optimal valuecomputed with ∆ E = 5. (B) Ratio of the time spent associated with the DNA compared tofreely diffusing in the bulk. (C) Apparent diffusion constant for sliding along the DNA. (D)Probability P to bind to the target before dissociation when arriving at the target site.Fig. 2A shows that even an immobilized TF in state 1 can have a MFPT that is compatiblewith the experimental finding ∼ s (11, 13). The MFPT is faster for σ = √ because moreDNA is scanned during the same residence time in state 1. Interestingly, the MFPT varies onlyvery little as a function of ∆ E up to values ∆ E ∼ E corresponds to a two states process where a TF switches between state 2and 3 and has maximal target affinity in state 2 (Fig. 2A, blue curve). This is consistent withresults from (37) showing that a switching process can have a fast MFPT even if the searchercan only bind in the slow state.For an optimal search process with only two states (bulk and one sliding state), a TF spendsan equal amount of time in the bulk and associated to the DNA. This is not any more the case acilitated diffusion with three search modes k = k , which is similarto the condition for a two states process, because of switchings to state 1, a TF spends muchmore time bound to the DNA. For an optimal search we have r d d = 1 + √ δ , where δ isdefined after Eq.9. For ∆ E = 6 and σ = √ we obtain r d d ≈
10, which is similar to in vivofindings that a dimeric Lac repressor spends 90% of the search time bound to the DNA (13).The effective diffusion constant for sliding D dna decreases as specific binding becomes stronger(Fig. 2C). For an optimal search we compute D dna ≈ D /r d d . Thus, for r d d = 10 we obtain D dna ≈ . µm s , which is around 4 times larger than values estimated from single moleculetracking experiments on flow stretched DNA (13). However, such a value is in good agreementwith results from molecular dynamics simulations for a Lac dimer (42), and with an apparent1D diffusion constant D eff ∼ . µm s estimated in (13). Moreover, a large range of variabilityis observed for the 1D diffusion constant of a Lac repressor on elongated DNA estimated fromsingle molecule imaging techniques (14). Whereas D dna strongly depends on k , the slidingdistance σ dna is not affected by the residence time in state 1 (if σ remains unchanged) andis determined by diffusion in state 2 and the detaching rate k . For an optimal search with k = k , by neglecting σ , we obtain σ dna ≈ q D k ≈ bp . This value is much larger than invivo measurements for a Lac dimer around 40 bps (11, 25), but compatible with a value around240 bps obtained from molecular dynamics simulations (42). Similar to D dna , also for σ dna alarge experimental variability is observed using single molecule imaging techniques (14).Finally, when arriving at the target site in state 2, a TF can as well detach without bindingto the target (11, 25). To characterize such events, we compute the conditional probability P tobind before detaching when arriving at the target site in state 2 (see SI Eq. 61). The probabilitydepends on the affinity κ , for example, for κ = ∞ we have P = 1. For κ = 0 the probabilityis not zero and it depends on the local switching dynamics. In general, P can be expressed as afunction of the sliding distances independent of the switching rates. Thus, for constant σ , P isindependent of k or ∆ E (Fig. 2D). For the optimal search process P depends on ∆ E because k and therefore σ vary with ∆ E . Search with finite redistributions and induced switchings at the target site
We proceed and study the effect of the DNA configuration and induced switchings at the targetsite by varying σ and κ . We consider a non-optimal search with σ = √ . Lowering thecorrelation distance σ up to σ ∼ L (around 1% of the genome is correlated) has only littleimpact on the MFPT (Fig. 3A,D), but a further decrease strongly increases the MFPT (Fig. 3D).In contrast, induced switchings at the target site ( κ >
0) only moderately reduce the MFPT(Fig. 3A,B). For example, the difference in the MFPT between κ = 0 and κ = ∞ is muchsmaller compared to σ ∼ L and σ ∼ L (Fig. 3B). We conclude that a reduced coiling cannotbe compensated by induced switchings at the target site. Although a larger κ increases theprobability to bind to the target (Fig. 3C), this has only a minor impact because P has alreadya value around 50% for κ = 0. Search with a Gaussian binding energy distribution
So far we used a constant rate k corresponding to a constant binding energy E . In reality, k depends on the DNA sequence and therefore on the DNA position x . To account for this,we consider a search with a Gaussian binding energy distribution ρ ( E ) (18, 19). We further acilitated diffusion with three search modes ∆ E M F P T ( sec ) A σ = ∞ , κ = 0 σ = L , κ = 0 σ = L , κ = 0 σ = L , κ = 0 σ = ∞ , κ = 0 . σ = ∞ , κ = 0 . σ = κ = ∞ ∆ E M F P T ( sec ) σ = L B κ = 0 κ = 0 . κ = 0 . κ = ∞ ∆ E b i nd i ng p r ob a b ili t y P C κ = 0 κ = 0 . κ = 0 . κ = ∞ −3 −2 −1 0 10200400600800 log( σ /L ) M F P T ( sec ) D κ = 0 , ∆ E = 5 Figure 3:
Search with κ > and σ < ∞ . We consider the search from Fig. 2 with σ = 1 / √ κ and σ . (A) MFPT for various κ and σ . (B) MFPT with σ = L/
300 andvarious κ . (C) Probability to bind to the target before dissociation when arriving a the targetsite for various κ (note that P is independent of σ ). (D) MFPT as a function of σ for κ = 0and ∆ E = 5.consider the case where a TF is immobile in state 1 such that σ = 1 / √ N (see Eq. 6) are both independent of x . Let w ( x ) be the weight function thatmeasures how often position x is visited during a search process compared to the average. Fora uniform initial distribution, from symmetry considerations, we can deduce that w ( x ) = 1. Inthis case the MFPT is¯ τ = N (cid:18)Z w ( x ) k ( x ) dx + 1 k + k k k (cid:19) = N (cid:18)Z ρ ( E ) k ( E ) dE + 1 k + k k k (cid:19) . (13)Hence, we can compute the MFPT with the average switching rate¯ k − = Z ρ ( E ) k ( E ) dE = Z ρ ( E ) ξe E − E ns dE . (14)With E ns ≈ −
11 and a Gaussian distribution ρ ( E ) with variance σ = 5 (18, 19) we obtain¯ k ≈ ξe − ζ with ζ = σ + E ns ≈ .
5. Next we checked how the results in Fig. 2 with ∆ E = ζ = 1 . E ≈ .
5. We note that the range of the ∆ E axis in Fig. 2 depends on thevalue of ξ . Had we used a different value ξ ′ = ξe − . − ζ = ξe − ≈ s − , the origin of the∆ E axis would be shifted to the right and the results for ∆ E = 5 . acilitated diffusion with three search modes E = ζ . Thus, by using ξ ∼ s − instead of ξ ∼ s − we obtain a search scenario thatis compatible with experimental data even in presence of a Gaussian energy distribution withlarge variance. Moreover, for ξ ∼ s − we also have ξ ∼ k and we have the Arrhenius likerelation k k ≈ e − ( E − E ns ) . Discussion
We investigated a framework for facilitated diffusion with switchings between three states: abulk state (state 3) and two states with sliding along the DNA (state 1 and 2) motivated bytwo TF protein conformations. The TF-DNA interaction and the target affinity depend onthe conformation. From the bulk, a TF associates to the DNA with a Poissonian rate and aGaussian distribution centered around his previous dissociation point. We analytically computedthe MFPT, and the conditional probability to bind to the target before detaching when arrivingat the target site. We further defined and computed various other properties that characterizethe search process, e.g. sliding length, effective 1D diffusion constant or ratio of the time spentin 1D compared to 3D. We compared our results with experimental data for the dimeric Lacrepressor search in E.Coli bacteria. We investigated various properties of a search process thatwe now discuss in more detail.
Impact of the DNA conformation
It is still largely unclear how strongly the DNA conformation affects the search time (43–46). Inthe literature one can find analytic results for two opposite cases: a rod-like DNA or a maximallycoiled DNA where the re-attaching distribution is uniform. However, a systematic and consistentanalysis where the impact of coiling is gradually changed is still outstanding. In our model, theassociation rate k and the correlation distance σ are two effective parameters that implicitlydepend on the DNA conformation. For fixed σ , a higher attaching k decreases the MFPT, butonly up to a lower limit that is attained for instantaneous jumps k = ∞ (Eq. 4). The MFPTis minimal for a uniform redistribution ( σ → ∞ , Fig. 3A), a scenario that is frequently used toanalyze a facilitated diffusion process with a highly packed DNA conformation (18, 32, 35, 39–41). We find that around 1% of the DNA has to become correlated by the 3D excursions inorder to maintain such a fast MFPT (Fig. 3A). At lower correlation distances the search timeis greatly prolonged (Fig. 3D). The value of σ also determines how the search time scales asa function of the DNA length L . To show this we consider the number of switchings N thatare necessary to find the target. N increases proportional to L for σ → ∞ (Eq. 5), and sucha linear dependency is usually assumed in the literature. However, for finite σ , the leadingorder asymptotic for large L is N ∼ L and not N ∼ L . For finite L , a careful analysis isneeded to determine whether the contribution ∼ L or ∼ L is dominant. For our analysis weconsidered that σ and k are independent parameters, however, in general their values will becorrelated. For example, lets consider stretched DNA. In this case, by assuming a correlationdistance σ ∼ L , we find N ∼ L . However, because the DNA is stretched and a TF is diffusingwith diffusion constant D in the bulk, we must at least have k − ∼ σ /D . Finally, this leads toa MFPT that scales ∼ L and not ∼ L . On the other hand, with strong coiling one might have σ ∼ L with a fast rate k that is almost independent of σ , such that the MFPT scales ∼ L .We conclude that without coiling it is not possible to have a MFPT that scales ∼ L . Coiling ispermissive to obtain at the same time a large correlation distance σ and a fast attaching rate k . acilitated diffusion with three search modes Impact of induced switchings
If switchings from state 2 to state 1 are induced at the target site, the spontaneous switching k can be reduced leading to a faster search because a TF spends more time in the fast state2 (18, 26). However, it is unclear which physical mechanism would provide such a specificity.We investigated the impact of induced switchings by varying the target affinity in state 2 ( κ ).We find that a MFPT compatible with experimental data can be achieved without inducedswitchings (Fig. 22A and Fig. 23A). We estimated that in this case a switching rate around k ≈ s − is necessary. This implies conformational changes in the submillisecond range, thathave also been suggested in (12). The rate k could be further reduced by assuming a largersliding distance σ . If spontaneous switchings to state 1 are fast, the conditional probability P to bind to the target before detaching is already large for κ = 0 (Fig. 23C). In such a caseadditional induced switchings ( κ >
0) do not much affect the search time (Fig. 3B). Clearly, theimpact of induced switchings would be much larger if P would be small for κ = 0. For example,this could be achieved by lowering the target affinity in state 1 ( κ < ∞ ), or by reducing theswitching rate k . With such conditions the MFPT would be strongly increased by blockinginduced switchings. In general, for large κ = ∞ the fastest MFPT is achieved by simply notswitching to the slow state 1 ( k = 0). But in this case we return to a two states model with abulk and a fast sliding state, incompatible with a large binding energy variability. Search in presence of a Gaussian binding energy profile
State 1 is characterized by the displacement σ and the residence time k − . We analyzed thelimiting case where a TF becomes immobilized in state 1 such that it scans only the base pairto which it binds to ( σ = 1 / √ k = ξe E − E ns depends on the specific energy E in state 1 and the non-specific energy E ns in state 2. With ξ = 10 s − , E ns = − k B T and aGaussian distribution for E with variance σ = 5 k B T we obtained a MFPT around 5-6 minutes,compatible with in vivo experimental data for the dimeric Lac repressor (13). It is found thatthe Lac repressor dimer stays bound to the promoter for an average time τ b around 5 minutes(47). In our model this would correspond to a target energy E = E ns − ln( ξτ b ) ≈ − k B T ,compatible with data (18, 19). At strong noncognate DNA sites with E ∼ − k B T , a TF wouldbe trapped only for a short time k − = ( ξe − ) − ∼ s , which resolves the trapping problem(39). The speed-stability paradox strongly relies on the assumption that a TF is found withhigh probability bound to the target at thermodynamic equilibrium. However, when the MFPTis of the order of minutes, such a high probability implies that a TF blocks the promoter for avery long time. This would impede a fast cellular response, and generate the opposite problemof how a promoter can get rid of a tightly bound TF. Conclusion and prospects
In this work we presented a MFPT analysis for a facilitated diffusion search process with switch-ings between three states: a bulk state and two sliding states where the TF is attached to theDNA. The model is microscopically motivated and describes the local dynamics using effectiveparameters. Parameter values have to be extracted from more detailed models of the TF-DNAinteraction, or by fitting our analytic expressions to experimental data for the dimeric Lac re-pressor. We focused on a qualitative analysis of the model and we showed that the modelpredictions account for many features that are observed experimentally. acilitated diffusion with three search modes
13A major simplification of the current model is the fact that we reduce the impact of the3D dynamics to a Poissonian association rate k and a Gaussian re-attaching distribution withwidth σ . However, this simplification allowed to derive analytic results, which are importantto precisely analyze the parameter space. Furthermore, we generalized results with uniformredistributions corresponding to σ = ∞ . Assuming that σ is correlated to the amount of DNAcoiling, we could systematically investigate the impact of the DNA conformation. However,polymer models show that the distribution of σ is not a Gaussian but decays like a power lawat large distances (28, 29, 48). The re-entry distribution is also more complicated than a singleexponential. It will be interesting to investigate in future work how more accurate assumptionsfor the 3D dynamics based on polymer models change the results presented here. Anotherinteresting project is to compute the MFPT with L´evy flights in state 3 (41, 45).Instead of using a single state for the 3D dynamics with complex distributions for attachingtime and position, one could break down the 3D dynamics into several states with simplerdistributions and enlarge the current model by additional states. Each state would account fordifferent properties of the search process, e.g. hoppings, jumps, intersegment and intersegmentaltransfers. Author Contributions
J.R. conceived and supervised the research; J.C. and J.R. performed the analysis; J.R. and J.C.wrote the paper.
Acknowledgements
J. C. acknowledges support from a PhD grant from the University Pierre et Marie Curie.
References
1. Zabet, N., and B. Adryan. 2012. Computational models for large-scale simulations of facil-itated diffusion.
Mol Biosyst.
Phys. Chem. Chem. Phys.
Chemphyschem.
FEBS Lett.
Nucleic Acids Res.
J.Biolog. Chemistry
J. Mol. Biol. acilitated diffusion with three search modes
Biophys.Chem.
Biophys. Chem.
Biochem.
Science
Proc Natl Acad Sci U S A.
Science
Phys. Rev. Lett.
Biochemistry
Phys Rev E Stat Nonlin Soft Matter Phys.
Proc. Natl. Acad. Sci. USA
Biophys. J.
Proc. Natl Acad. Sci. USA
Proc Natl Acad Sci U S A.
Science
Cell
Mol Cell. acilitated diffusion with three search modes
Protein Sci.
Nucleic Acids Res.
Proc Natl Acad Sci USA
Biophys J.
Biophys J.
J. Phys. A: Math. Theor.
Biophys. J.
Biophys J.
Phys. Rev. E
J. Phys. A: Math. Theor.
J Phys Chem B.
Biophys. J.
J. Phys.: Condens. Matter
Phys. Rev. Lett.
Phys.Lett. A
Phys. Rev. Lett.
Phys. Biol. acilitated diffusion with three search modes
Phys. Rev. Lett.
Proc Natl Acad Sci U S A.
Biophys J.
Proc Natl Acad Sci USA
Proc Natl Acad Sci USA
Biophys. J.
Nat Genet.
Curr Opin Genet Dev. acilitated diffusion with three search modes Supplementary InformationDerivation of sojourn times and MFPT
We start from the equations for the sojourn times D m τ ′′ n,m ( y ) − X i =1 K mi τ n,i ( y ) − χ m δ ( y − y ) τ n,m ( y ) = − δ nm (15)with the switching matrix ( k m + = P j =1 k mj ) K mi = k m + δ mi − k mi = k − k − k k + k − k − k k (16)and reflecting boundary conditions at y = ± L . Because the target is located in the center at y = 0 we can restrict the analysis to the region 0 ≤ y ≤ L . By integrating Eq. 15 around y = 0we obtain ( τ n,m ( y ) = τ n,m ( − y )) D m τ ′ n,m ( y ) | y =0 + = χ m τ n,m (0) (17)which are partially reflecting boundary conditions. We remove the killing term in Eq. 15 andreplace it with these partially reflecting boundary conditions. We introduce the dimensionlessposition x = yL , the diffusion rates ν m = D m L , the dimensionless parameters κ m = Lχ m D m , thescaled switching rates l mi = k mi ν m = L k mi D m and the scaled switching matrix L mi = K mi /ν m . Thescaled sojourn times ˆ τ n,m ( x ) = ν n τ n,m ( x ) (18)satisfy the system of equations equations (0 ≤ x ≤ τ ′′ n,m ( x ) − X i =1 L mi ˆ τ n,i ( x ) = − δ nm (19)with reflecting conditions ˆ τ ′ n,m (1) = 0 at x = 1, and partially reflecting conditions ˆ τ ′ n,m (0) = κ m ˆ τ n,m (0) at x = 0. In state 3 we have a reflecting boundary condition at x = 0 and x = 1.The functions (¯ˆ τ n,m = R ˆ τ n,m ( x ) dx ) v n,m ( x ) = ˆ τ n,m ( x ) − ¯ˆ τ n,m (20)have zero mean and satisfy the system of equations v ′′ n,m ( x ) − X i L mi v n,i ( x ) = − v ′ n,m (0) . (21)The matrix L mi is singular and one eigenvalue is zero. The left eigenvector to the zero eigenvalueis ~f = (cid:18) l l l , l , l (cid:19) . (22) acilitated diffusion with three search modes X m =1 f m ˆ τ ′′ n,m ( x ) = X m =1 f m v ′′ n,m ( x ) = − f n , and after integration we find X m =1 f m v n,m ( x ) = − f n g ( x ) = − g n ( x ) , (23)where g ( x ) = (cid:18) ( x − − (cid:19) . (24)With Eq. 23 we express v n, ( x ) − v n, ( x ) as a function of v n, ( x ) and v n, ( x ) and then obtainclosed system of equations for v n, ( x ) and v n, ( x ). By introducing w n ( x ) = v n, ( x ) − v n, ( x ) (25)we find from Eq. 21 (cid:18) v n, ( x ) w n ( x ) (cid:19) ′′ − M (cid:18) v n, ( x ) w n ( x (cid:19) = − (cid:18) v ′ n, (0) w ′ n (0) + g n ( x ) (cid:19) (26)with M = (cid:18) l − βl α (cid:19) (27)and α = l + l + l + l , β = l l + l ( l + l ) . (28)We solve these equations as a function of v ′ n, (0) and w ′ n (0) and then compute v ′ n, (0) and w ′ n (0)using the boundary conditions. The eigenvalues and eigenvectors M of are µ / = 12 (cid:16) α ± p α − β (cid:17) , ~e i = (cid:18) l µ i (cid:19) , (29)with β = µ µ and α = µ + µ . With the expansion (cid:18) v n, ( x ) w n ( x (cid:19) = u ( x ) ~e + u ( x ) ~e (30)and (cid:18) (cid:19) = βl ( µ − µ ) ~e + βl ( µ − µ ) ~e , (cid:18) (cid:19) = µ µ − µ ~e + µ µ − µ ~e we derive from Eq. 26 u ′′ ( x ) − µ u ( x ) = − (cid:18) v ′ n, (0) βl ( µ − µ ) + ( w ′ n (0) + g n ( x )) µ µ − µ (cid:19) . (31) acilitated diffusion with three search modes u ( x ) is obtained by interchanging µ and µ . The function ˜ u ( x ) = u ( x ) − g n ( x ) µ − µ satisfies ˜ u ′′ ( x ) − µ ˜ u ( x ) = − c n, with c n, = v ′ n, (0) βl ( µ − µ ) + w ′ n (0) µ µ − µ + f n µ − µ . (32)With R u ( x ) dx = R ˜ u ( x ) dx = 0 we find u ( x ) = c n, (cid:18) µ − cosh( √ µ (1 − x )) √ µ sinh √ µ (cid:19) + g n ( x ) µ − µ . (33)The solution u ( x ) is obtained from Eq. 33 by interchanging ( c n, , µ ) with ( c n, , µ ), where c n, is defined in Eq. 32 with µ and µ interchanged. From Eq. 30 we obtain v n, ( x ) = c n, l µ (cid:18) µ − cosh( √ µ (1 − x )) √ µ sinh √ µ (cid:19) + c n, l µ (cid:18) µ − cosh( √ µ (1 − x )) √ µ sinh √ µ (cid:19) − l f n β g ( x ) (34) w n ( x ) = c n, (cid:18) µ − cosh( √ µ (1 − x )) √ µ sinh √ µ (cid:19) + c n, (cid:18) µ − cosh( √ µ (1 − x )) √ µ sinh √ µ (cid:19) (35)From Eq. 23 we further find (with P n f n = βl ) v n, ( x ) − v n, ( x ) = βl l v n, ( x ) − l l w n ( x ) + f n g ( x ) l v n, ( x ) − v n, ( x ) = v n, ( x ) − v n, ( x ) − w n ( x ) (36)We complete the analysis by computing the values of v ′ n, (0) and v ′ n, (0). With v ′ n, (0) = 0and ( g ′ n (0) = − f n ) X m =1 f m v ′ n,m (0) = f n (37)we can express v ′ n, (0) as a function of v ′ n, (0) v ′ n, (0) = − l l v ′ n, (0) + f n l . (38)From Eq. 32 and Eq. 38 we get c n, = v ′ n, (0) a + f n b c n, = v ′ n, (0) a + f n b (39)with a = µ ( l + l − µ ) l ( µ − µ ) , b = l − µ l ( µ − µ ) a = µ ( l + l − µ ) l ( µ − µ ) , b = l − µ l ( µ − µ ) . (40) acilitated diffusion with three search modes v ′ n, (0) we compute w n (0) using Eq. 35 and the relation¯ˆ τ n, − ¯ˆ τ n, = − v ′ n, (0) l + δ n, l (41)obtained by integrating Eq. 19. We find w n (0) = c n, µ + c n, µ − c n, ξ − c n, ξ = v ′ n, (0) l − f n β − c n, ξ − c n, ξ w n (0) = ˆ τ n, (0) − ˆ τ n, (0) − (¯ˆ τ n, − ¯ˆ τ n, ) = v ′ n, (0) κ − v ′ n, (0) κ + v ′ n, (0) l − δ n, l (42)where we used c n, µ + c n, µ = v ′ n, (0) l − f n β (43)and introduced ξ = coth √ µ √ µ , ξ = coth √ µ √ µ . (44)From Eq. 42 we find c n, ξ + c n, ξ = − v ′ n, (0) κ + v ′ n, (0) κ − f n β + δ n, l (45)and with Eq. 38 we obtain c n, ξ + c n, ξ = − v ′ n, (0) (cid:18) κ + l l κ (cid:19) + f n (cid:18) l κ − β (cid:19) + δ n, l (46)By inserting c n, and c n, from Eq. 39 we obtain v ′ n, (0)( a ξ + a ξ ) + f n ( b ξ + b ξ ) = − v ′ n, (0) (cid:18) κ + l l κ (cid:19) + f n (cid:18) l κ − β (cid:19) + δ n, l . From this we finally get v ′ n, (0) = f n de + δ n, l e (47)with d = 1 l κ − β − ( b ξ + b ξ ) e = a ξ + a ξ + 1 κ + l l κ . (48)We get the final expressions c n, = v ′ n, (0) a + f n b = f n ( a de + b ) + δ n, l a ec n, = v ′ n, (0) a + f n b = f n ( a de + b ) + δ n, l a e (49) acilitated diffusion with three search modes Sojourn times and MFPT
The scaled sojourn times areˆ τ n, ( x ) = v n, ( x ) + ¯ˆ τ n, = v n, ( x ) − v n, (0) + ˆ τ n, (0) = v n, ( x ) − v n, (0) + v ′ n, (0) κ ˆ τ n, ( x ) = v n, ( x ) − ( v n, ( x ) − v n, ( x )) + ¯ˆ τ n, = ˆ τ n, ( x ) − w n ( x ) + ¯ˆ τ n, − ¯ˆ τ n, = ˆ τ n, ( x ) − w n ( x ) + v ′ n, (0) l − δ n, l ˆ τ n, ( x ) = ˆ τ n, ( x ) − ( v n, ( x ) − v n, ( x )) + δ n, l (50)where we used ¯ˆ τ n, − ¯ˆ τ n, = δ n, l obtained by integrating Eq. 19. The sojourn times with uniforminitial distributions are¯ˆ τ n, = − v n, (0) + v ′ n, (0) κ , ¯ˆ τ n, = ¯ˆ τ n, + v ′ n, (0) l − δ n, l , ¯ˆ τ n, = ¯ˆ τ n, + δ n, l (51)¯ τ n, = − v n, (0) ν n + v ′ n, (0) ν n κ , ¯ τ n, = ¯ τ n, + v ′ n, (0) ν n l − δ n, k , ¯ τ n, = ¯ τ n, + δ n, k . (52)The MFPT with uniform initial distribution in state m is¯ τ ( m ) = X n =1 ν − n ¯ˆ τ n,m = X n =1 ¯ τ n,m . (53)Because switching between states is fast compared to the overall search time, the mean sojourntimes are almost independent on the initial state, ¯ˆ τ n, ≈ ¯ˆ τ n, ≈ ¯ˆ τ n, . By noting that ¯ˆ τ i,m f i ≈ ¯ˆ τ j,m f j we find ¯ τ i,m ¯ τ j,m ≈ ν j ν i f i f j . (54)The expression for the MFPT simplifies to¯ τ ≈ ¯ τ , + ¯ τ , + ¯ τ , = ¯ τ , (cid:18) k k + k k k k (cid:19) = N (cid:18) k + 1 k + k k k (cid:19) = N (cid:18) k k k + 1 k + 1 k (cid:19) (55)where we introduced the mean number of switchings between state 1 and 2 resp. 2 and 3 N = ¯ τ , k = ¯ˆ τ , l , N N = k k (56) N and N can be expressed as a function of the DNA length L and the mean square displace-ments σ , σ and σ . acilitated diffusion with three search modes Dependence of the search time on the DNA length
To analyze how the MFPT depends on the DNA length L , we introduce ˆ L = LL , where L = 1 bp is a reference length, and extract the dependency on ˆ L . We therefore replace l ij = L k ij D i withˆ L l ij , where l ij = L k ij D i is now evaluated with L . We proceed similarly with all the otherparameters: ν i → ˆ L − ν i , κ i → ˆ Lκ i , µ i → ˆ L µ i , α → ˆ L α , β → ˆ L β , f n → ˆ L f n , a i → a i and b i → ˆ L − b i . In terms of the rescaled parameters we get v n, ( x ) = c n, l µ L µ − cosh( q ˆ L µ (1 − x ))ˆ L √ µ sinh q ˆ L µ + c n, l µ L µ − cosh( q ˆ L µ (1 − x ))ˆ L √ µ sinh q ˆ L µ − l f n β g ( x ) (57)The parameters c n, and c n, from Eq. 49 are c n, = f n ( a de + b ) + δ n, ˆ L a l ec n, = f n ( a de + b ) + δ n, ˆ L a l e (58)with d = 1 l κ − Lβ − ( b ξ + b ξ ) e = a ξ + a ξ + 1 κ + l l κ . (59)and ξ = coth q ˆ L µ √ µ , ξ = coth q ˆ L µ √ µ . (60)For a long DNA with ˆ L µ i ≫ v n, (0) ≈ − l ˆ L (cid:18) c n, µ √ µ + c n, µ √ µ (cid:19) − f n l β . (61) v ′ n, (0) = c n, l µ + c n, l µ + f n l β (62)and from this we find for the sojourn times¯ τ n, = − ˆ L ν n v n, (0) + ˆ Lν n κ v ′ n, (0) ≈ ˆ Ll ν n (cid:18) c n, µ √ µ + c n, µ √ µ (cid:19) + ˆ Ll ν n κ (cid:18) c n, µ + c n, µ (cid:19) + l f n β (cid:18) L D n + Lχ (cid:19) (63)For k = 0 ( l = 0) we recover ¯ τ , = L D + Lχ . The number of switchings with χ = ∞ are N = ¯ τ , k ≈ ˆ Ll (cid:18) c , µ √ µ + c , µ √ µ (cid:19) + ˆ L l l l β (64) acilitated diffusion with three search modes Probability to bind to the target before detaching
When a TF reaches the target in state 2 it eventually binds with probability P or it detacheswith probability Q = 1 − P . To compute Q we consider a switching process with k = 0( l = 0) to avoid rebinding to the DNA. The mean probability to detach before binding to thetarget when initially at the target site in state 2 is Q = k τ , (0) = l ˆ τ , (0). However, wecannot use ˆ τ , (0) from our previous analysis because state 3 is missing and v n, ( x ) = 0. Morespecifically, Eq. 23 is not valid when v n, ( x ) = l = 0, and we cannot apply Eq. 37 to obtain v ′ , (0) as a function of v ′ , (0). We therefore recalculate v ′ , (0) and v ′ , (0) without Eq. 37 for l = 0 and the modified boundary condition ˆ τ , (0) = Ql . The parameters α , β , µ and µ areevaluated with l = 0, e.g. β = l l and α = l + l + l . By integrating Eq. 19 we find v ′ , (0) = − l (¯ˆ τ , − ¯ˆ τ , ) v ′ , (0) = 1 + l (¯ˆ τ , − ¯ˆ τ , ) − l ¯ˆ τ , = 1 + ( l + l )(¯ˆ τ , − ¯ˆ τ , ) − l ¯ˆ τ , (65)and from this we get ¯ˆ τ , = − l (cid:18) v ′ , (0) + l + l l v ′ , (0) − (cid:19) ¯ˆ τ , = − l (cid:18) v ′ , (0) + l l v ′ , (0) − (cid:19) . (66)Eq. 45 reads ( f = 0) c , ξ + c , ξ = − v ′ , (0) κ + v ′ , (0) κ = − v ′ , (0) κ + Ql (67)where we used v ′ , (0) κ = ˆ τ , (0) = Ql . From βl v , (0) = − c , µ ξ − c , µ ξ + c , µ µ + c , µ µ = − c , µ ξ − c , µ ξ + v ′ , (0) l α − w ′ (0)= − c , µ ξ − c , µ ξ + l + l l v ′ , (0) + v ′ , (0) βl v , (0) = l ˆ τ , (0) − l ¯ˆ τ , = (cid:18) l κ + l + l l (cid:19) v ′ , (0) + v ′ , (0) − c , µ ξ + c , µ ξ = 1 − l κ v ′ , (0) (68)Thus, we find the system of equations c , µ ξ + c , µ ξ + l κ v ′ , (0) = 1 c , ξ + c , ξ + v ′ , (0) κ = Ql (69) acilitated diffusion with three search modes c , = ˜ a v ′ , (0) + ˜ b v ′ , (0) , c , = ˜ a v ′ , (0) + ˜ b v ′ , (0) (70)with ˜ a = µ − l µ − µ , ˜ b = µ µ − µ , ˜ a = µ − l µ − µ , ˜ b = µ µ − µ (71)we obtain A (cid:18) v ′ , (0) v ′ , (0) (cid:19) = (cid:18) Ql (cid:19) (72)with the matrix A ij = ˜ a ξ µ + ˜ a ξ µ + l κ ˜ b ξ µ + ˜ b ξ µ ˜ a ξ + ˜ a ξ + κ ˜ b ξ + ˜ b ξ ! (73)The solution of Eq. 72 is v ′ , (0) = 1det( A ) (cid:18) A − A Ql (cid:19) v ′ , (0) = 1det( A ) (cid:18) − A + A Ql (cid:19) (74)From this we obtain ( P = 1 − Q ) κ = ˆ τ ′ , (0)ˆ τ , (0) = l Q v ′ , (0) = 1det( A ) (cid:18) A − A l Q (cid:19) Q = l A A − κ det( A ) . (75)For example, for k = 0 or χ = ∞ we have Q = 0. The maximum is obtained for κ = 0 Q max = l A A . (76) Q max depends on the switching rates and on κ . For example, Q max = 1 is found for κ = 0 (nobinding in state 1) or l21