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Kinetics of target searching by means of two diffusion-like motions
Chemical Physics 435 (2014) 14–20
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Kinetics of target searching by means of two diffusion-like motions Konstantin L. Ivanov ⇑, Nikita N. Lukzen a b
International Tomography Center SB RAS, Institutskaya Str. 3a, Novosibirsk 630090, Russia Novosibirsk State University, Pirogova Str. 2, Novosibirsk 630090, Russia
a r t i c l e
i n f o
Article history: Received 9 December 2013 In final form 4 March 2014 Available online 12 March 2014 Keywords: Diffusion-controlled processes Diffusional search DNA repair
a b s t r a c t A theoretical approach to stochastic searching of a small target is developed, which can be applied, for instance, to searching for a damaged site on the DNA molecule by a DNA repair enzyme. It is assumed that the searching molecule moves along a chain of sites by means of stochastic jumps to the adjacent positions (one-dimensional ‘sliding’); the second motion models diffusion in three dimensions. A general expression is obtained for the flux to the damaged site. A special case is analyzed where the second motion is treated as effective 1D diffusion; the effective searching time is estimated. It is shown that the faster second motion shortens the searching time. A more realistic case where the second kind of motion leads to jumps not only to the adjacent site is also treated: it is shown that the diffusional search is facilitated even further. Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction Biological processes are frequently dealing with protein–nucleic acid interactions, notably, with specific binding of various enzymes to nucleic acids. Among them is DNA repair (i.e., removal of molecular lesions caused by metabolic activities and environmental factors), which is an important process for keeping the integrity and stability of the genome. Understanding the mechanism of searching for the particular binding site on the large DNA molecule is an intriguing problem. It is known that the search is much more efficient as compared to the simple prediction coming from the Smoluchowski theory [1]: there is experimental evidence (both in in vitro [2] and in vivo [3] studies) that the rate constant, ks , exceeds the Smoluchowski-like one, kSm ¼ 4pRD, by a few orders of magnitude. (Here R is the sum of the linear sizes of the damaged site and the enzyme and D is the mutual diffusion coefficient of the enzyme and DNA). To rationalize this unexpected result several theoretical models have been proposed, with the most well-known being the Berg–Winter–von Hippel (BWH) model [4,5], which suggests that first the enzyme binds to an arbitrary site on the DNA (non-specific binding) and searches for the damaged site in the course of subsequent stages of 1D and 3D diffusion. 1D diffusion proceeds by means of ‘sliding’ [2,6,7] of the enzyme along the DNA strand, the sliding increases the effective size of the target ⇑ Corresponding author at: International Tomography Center SB RAS, Institutskaya Str. 3a, Novosibirsk 630090, Russia. Tel.: +7 (383) 330 8868; fax: +7 (383) 333 1399. E-mail address: [email protected] (K.L. Ivanov). http://dx.doi.org/10.1016/j.chemphys.2014.03.001 0301-0104/Ó 2014 Elsevier B.V. All rights reserved.
[8]. After a sliding stage the enzyme dissociates from the DNA and diffuses in three dimensions in the solution, then it binds again to the DNA molecule and the sequence is repeated until the enzyme finds its target and binds to it (specific binding) so that the DNA repair process starts. It is often argued that the combination of 1D and 3D diffusion facilitates the process and makes the time required for searching for the damaged site considerably shorter. This is because ‘sliding’ of a large enzyme molecule along the DNA is relatively slow [3,9–12]; thus, purely 1D diffusional motion makes the search inefficient. On the other hand, 1D diffusion increases the effective target size: in this way the combination of two diffusional motions reduces the searching time. Previously, several models have been considered to describe the complex process of searching by 1D and 3D diffusional walks. These are the co-localization mechanism [13,14] (assuming that that proteins are produced near their binding sites on the DNA thereby reducing the number of searching cycles) and the correlation mechanism [15–18] (taking account of the correlations between 1D sliding and 3D diffusion). Hu et al. [19] have developed a theoretical approach, which take into account the role of DNA conformation by using an electrostatic analogy. Description of these theoretical results can be found, for instance, in a recent review by Kolomeisky [20]. Analytical solutions of the problem are usually difficult to obtain even assuming simplified geometry of DNA and protein. Solutions for realistic cases are probably impossible because the enzyme is not a spherical molecule, moreover, it can bind to DNA only in specific orientations; DNA is also a highly complex molecule having different conformations. One should note that fast searching for DNA damages can be sometimes
K.L. Ivanov, N.N. Lukzen / Chemical Physics 435 (2014) 14–20
accounted for by an alternative mechanism (charge transport through the DNA for carrying out redox chemistry at a distance) [21,22]; however, here we will not consider this mechanism and focus on subsequent diffusional motions of two different types. The aim of this work is developing a physically reasonable model of searching, which enables analytical solution for the searching time after non-specific binding of the enzyme to DNA. Although a lot has already been done in this field we will propose a model, which allows one obtaining relatively simple analytical expressions for the searching time and for the kinetics of searching for the damaged site. On one hand, we will try to make the model simple; on the other hand, we will need to take account of two types of stochastic motions. To tackle the problem analytically we will exploit the formalism of generating functions. We will find out whether the combination of 1D and 3D diffusion facilitates the search or does not and also discuss peculiarities of diffusion, which make the kinetics different from frequently assumed monoexponential function. Last but not least, our general formulas will not be limited to the specific problem of the DNA repair kinetics but will also be applicable to other cases of searching for different targets by two types of stochastic motions. 2. Methods 2.1. Theoretical model Describing subsequent stages of 1D and 3D diffusion of the repair enzyme is generally a complex problem, which can be solved analytically only assuming specific models for the enzyme motion. Here we propose a new model, which enables analytical treatment of the problem and obtaining both the effective DNA repair time, i.e., the time required to find the damaged nucleotide, and the repair kinetics. The situation treated here is presented in Fig. 1. Here we assume that the enzyme can move along the DNA strand by stochastic jumps to the nearest nucleotides, the jump rate is equal to w1 . Taking 3D diffusion into account is generally problematic; to minimize this problem we assume that 3D diffusion is a process of jumping along a parallel chain with the jump rate equal to w2 . Of course, such assumption does not consider (i) the peculiarities of 3D diffusion (which is non-redundant in contrast to redundant 1D diffusion) and (ii) the conformation of the DNA. Nonetheless, as will be shown, it can account for the effect of facilitating the 1D search, which becomes inefficient when the enzyme spends too long time on the DNA fragment as the number of sites (nuclepffiffi otides) visited by the repair enzyme grows with time only as t. In this case dissociation from the DNA with subsequent diffusion in space (being faster than ‘sliding’) and association to a new site facilitates the search, since more sites on the DNA can be visited in the unit time. Here we will thus assume that w2 w1 (free diffusion in space is much faster than the sliding process);
2.2. General results for repair flux To proceed further let us define the notations. Here we will introduce a set of values C i , which are equal to the probability to find the enzyme on the i-th nucleotide at the first chain; C 0i is then the probability to find the enzyme on the i-th site at the second chain. It is more convenient to derive equations in the matrix-form, for this reason we will work with a set of vectors
Ci ¼
wd wa k
dissociation and association will be modeled as transitions between the two chains with rates wd and wa , respectively. We will assume that both processes are slow even as compared to 1D ‘sliding’, in other words, the enzyme makes many stochastic jumps before it dissociates or re-associates. Thus, the following hierarchy of rates is assumed: w2 w1 wa ; wb . This assumption is consistent with the current view on the DNA repair processes [3]. Motion along the two chains thus models 1D and 3D diffusion, dissociation and re-association events are taken into account by the transitions between the chains. We also assume that each chain is infinite to ignore boundary effects. At t ¼ 0 the enzyme is located on the k-th nucleotide, while the n-th nucleotide is damaged. When the enzyme reaches the damaged site, specific binding occurs immediately and irreversibly; thus the search is accomplished. In this model it is therefore sufficient to calculate the time of first arrival from the initially occupied site to the damaged site. Our model allows reasonable estimates for the searching time in the case where two different stochastic motions are present although the 3D motion is accounted for in a greatly simplified way. It is worth noting that replacing 3D diffusion by effectively 1D diffusion is also done in the correlation mechanism [16,20]. In principle, one can omit the second chain and introduce 3D diffusion by assuming that the enzyme can jump not only to the nearest site but to other sites as well (see Ref. [4] and Appendix therein). However, to do so it is required to calculate the specific distribution of rates of jumping to other sites. In our treatment the second stochastic process is incorporated in the model from the very beginning and is treated as effective 1D diffusion. As will be shown in the end of the paper, our method also allows one to consider jumps between arbitrary positions, i.e., to extend the treatment to a more realistic type of motion, which captures some properties of the 3D diffusion. We will obtain analytical results for one example of a specific probability distribution of the jump length. It is worth noting that our model is bearing some similarity to the one proposed by Avetisov and coworkers [23] who treated conformational changes in frozen proteins with a hierarchy of the transition rates to describe dynamics of biomolecules on very different time scales. Cycles of 3D diffusion and 2D diffusion have also been treated in the problem of diffusional search for a small hole in a spherical cavity by Berezhkovskii and Barzykin (who developed a coarse-grained model for the search process) [24] and later by Rupprecht et al. (who obtained exact solution of the problem) [25,26].
w2 w2
n
w1 w1 Fig. 1. Model of searching for a damaged site on the DNA molecule. The repair enzyme jumps along the DNA chain (1D sliding), which is an infinite chains of sites (nucleotides); the rate of jumps to the adjacent sites is equal to w1 . 3D diffusional motion is modeled by jumps along a parallel chain; the rate of such jumps to the adjacent sites is taken equal to w2 . Transitions between the two chains (dissociation and re-association) are occurring at rates wd and wa . The searching process starts at the k-th site of the first chain; the target site is the n-th site on the first chain: upon reaching this site irreversible specific binding immediately occurs.
15
Ci
C 0i
ð1Þ
Kinetic behavior of the system is then described by the following set of equations for Ci written in the Laplace domain:
~i ¼ W ~ i1 þ C ~ iþ1 Þ din K ~ n þ dik f k ; c UÞ b C c ðC bC ðs þ 2 W c ¼ w1 0 ; U b ¼ wd wa where W wd wa 0 w2
ð2Þ
Hereafter s is the Laplace variable, tilde stands for the Laplace. .. In Eq. (2) transformed quantities; the matrices are defined by .c c -matrix stands for the stochastic jumps between the the W
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K.L. Ivanov, N.N. Lukzen / Chemical Physics 435 (2014) 14–20
b adjacent positions in each chain; the U-matrix describes the transitions between the chains, i.e., dissociation-association. For the 1 coming from k-th equation there is also a source term f 0 ¼ 0 the initial condition (at t ¼ 0 the enzyme is located on the k-th site), in the n-th equation there is an additional term in the b Cn and describes irreversible right-hand side, which is equal to K b ¼ ksb 0 with specific binding at the corresponding site. Here K 0 0 ksb being the rate constant for the specific binding process. To solve the kinetics equations it is convenient to work with the generating function (GF), FðzÞ, for the sequence of Ci s. It is defined as follows:
FðzÞ ¼
1 X
~ i zi C
ð3Þ
physical meaning of g m is as follows: it is the probability to find the enzyme at certain time m nucleotides away from the starting site at the first chain assuming that the starting site was at the first chain as well. To obtain the analytical result for the searching time and searching kinetics it becomes necessary to write down explicit ^ expression for the [1, 1]-element of the matrix GðzÞ. The expression for this element and j1 ðtÞ will be given in the following section. 3. Results and discussion 3.1. Case of jumps only to the adjacent sites In the simplest case described above where only jumps between the adjacent sites are possible we obtain the following result ^ for the [1, 1]-element of the G-matrix:
i¼1
Here z is a complex variable. In this definition we include all powers of z but not only positive ones (as is usually done). When z ¼ eih , the series from Eq. (3) is, in fact, the Fourier series expan~ i s are the Fourier harmonics sion of the GF with respect to h, i.e., C of F. Thus, the knowledge of the GF provides full information about the quantities of interest. The equation for the GF can be obtained from the original set of equations (2) by multiplying the equation for Ci by zi and summing all the equations:
1 c b ^ b Cn W U FðzÞ ¼ AFðzÞ s z2þ ¼ zk f 0 zn K z
ð4Þ
^ 11 ðzÞ ¼ G
s þ wa w2 ðz 2 þ 1=zÞ ðs þ wd w1 ðz 2 þ 1=zÞÞðs þ wa w2 ðz 2 þ 1=zÞÞ wd wa ð11Þ
Now let us calculate the quantities g m as the Fourier harmonics ^ 11 ðzÞ: of G
gm ¼
1 2p
Z p
^ 11 ðeih Þdh eimh G
ð12Þ
p
To evaluate these integrals it is convenient to substitute z as eih ^ 11 ðzÞ: in the expression for G
It is then easy to obtain the general solution of this equation:
^ 1 zk f 0 zn K b Cn FðzÞ ¼ ðAÞ
ð5Þ
In the following consideration we will be interested in the rate of specific binding, which is given by the flux from the n-th site at the first chain to the new state (specific binding state), where the repair process becomes operative. This flux can be defined in the following way:
b Cn ðtÞ jðtÞ ¼ K
ð6Þ
Thus, our final goal is evaluating Cn . Let us express this quantity ^ 1 ^ ¼ ðAÞ from the GF. To do this, let us assume that the matrix G (having the meaning of the free propagator of the system in the absence of specific binding) has already been expanded in a series with different powers of z:
^ GðzÞ ¼
1 X
s þ wa þ 2w2 ð1 cos hÞ ðs þ wd þ 2w1 ð1 cos hÞÞðs þ wa þ 2w2 ð1 cos hÞÞ wd wa 1 s2 þ ua þ 2ð1 cos hÞ ¼ w1 ðs1 þ ud þ 2ð1 cos hÞÞðs2 þ ua þ 2ð1 cos hÞÞ ud ua
^ 11 ðhÞ ¼ G
ð13Þ Here we introduced: s1;2 ¼ s=w1;2 ; ud ¼ wd =w1 ; ua ¼ wa =w2 . We ^ 11 ðhÞ even further by noting that the denominator can simplify G is a second-order polynomial of cos h with two roots, k1;2 :
k0 cos h ^ 11 ðhÞ ¼ 1 G 2w1 ðk1 cos hÞðk2 cos hÞ 1 k1 k0 1 k2 k0 1 þ ¼ 2w1 k1 k2 k1 cos h k2 k1 k2 cos h
ð14Þ
By using the following expressions for the integrals
^ i zi G
ð7Þ
i¼1
By setting equal the coefficients at zn in the left-hand side and right-hand side of expression (5) for the GF we obtain:
1 2p
pffiffiffiffiffiffiffiffiffiffiffiffiffi m Z p Z p cosðmhÞ ða a2 1Þ 1 sinðmhÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi ; dh ¼ dh ¼ 0 2p p a cos h a2 1 p a cos h
ð15Þ
we obtain that
~n ¼ G ^ nk f 0 G ^0 K ~n bC C
ð8Þ
From the vector ^jðsÞ we will need only its first component, j1 , which is equal to
~j1 ðsÞ fnk ; where f0 qffiffiffiffiffiffiffiffiffiffiffiffiffink qffiffiffiffiffiffiffiffiffiffiffiffiffink k1 k21 1 k2 k22 1 k1 k0 k2 k0 qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi þ f nk ¼ k1 k2 k2 k1 2 k1 1 k22 1
^j1 ðsÞ ¼ ksb g nk g nk ; 1 þ ksb g 0 g0
To reduce the otherwise long notations let us introduce three new quantities:
The quantity of interest then takes the form:
~jðsÞ ¼ K ~n ¼ K ^0 K ^ nk f 0 bC b ðE ^þG b Þ1 G
^ mÞ where g m ¼ ðG 11
ð9Þ
ð10Þ
Here the approximate equality is obtained assuming infinitely fast specific binding upon arrival to the n-th site, that is ksb g 0 1, which we will always use in the consideration below. Thus, for the diffusion-limited searching process we have obtained a simple general result: to evaluate all kinetic quantities of interest it is sufficient to compute only two quantities, g nk and g 0 . The
s1 s2 þ ud ua ; 4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 X¼ ðs1 þ s2 þ ud þ ua Þ2 4ðs1 s2 þ s1 ua þ s2 ud Þ; 4 s1 þ s2 þ ud þ ua : Y¼ 4
ð16Þ
D¼
ð17Þ
17
K.L. Ivanov, N.N. Lukzen / Chemical Physics 435 (2014) 14–20
Then
fnk ¼
DþX 2X
qffiffiffiffiffiffiffiffiffiffiffiffiffiffink qffiffiffiffiffiffiffiffiffiffiffiffiffiffink k1 k21 1 k2 k22 1 DX qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 2X 2 k1 1 k22 1
ð18Þ
In the notation given by Eq. (17) explicit expressions for ki ’s are:
k0 ¼ 1 þ
s2 þ u a ; 2
k1;2 ¼ 1 þ Y X:
ð19Þ
Combining all these formulas one can obtain the expression for the flux. However, expressions (16), (18) are still quite cumbersome; moreover, it is necessary to perform the inverse Laplace transformation to obtain j1 ðtÞ. As a consequence, when the rates defined in Section 2.1 are related arbitrarily, the result for j1 ðtÞ is very complex. However, in our model we assume specific relations between the rates, which are also reasonable from the experimental point of view. These are sufficient to arrive at relatively simple analytical expressions for the flux and the searching time. To proceed further let us expand the expressions for fm assuming s1 ; s2 ud ; ua 1. This corresponds to a situation where (i) diffusion along the chains is much faster than dissociationassociation and (ii) relatively long times, t > T, are treated, at which all the kinetics processes have been established (jumps along the chains and dissociation-association have occurred many times); here T w1a ; w1 . d Let us note that in expression (18)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k21;2 1 ¼ ðk1;2 1Þðk1;2 þ 1Þ 2ðY XÞ
ð20Þ
Since at s1 ; s2 ! 0 we obtain Y ! X, one of the two terms in expression (18) for fnk is dominating (in our notations this is the second term) because the denominator tends to zero. Neglecting the other term we obtain a much simpler expression for the flux
~j1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffink pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffink k2 k22 1 1 2ðY XÞ
ud þ ua 1 ðs1 s2 Þðud ua Þ þ 4 ud þ ua 4
ð22Þ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffink qffiffiffiffiffiffiffiffiffiffiffiffiffink s1 ua þ s2 ud 1 ¼ 1 s=weff ud þ ua pffiffiffiffiffiffiffi expð ssd Þ
w1 wa þ w2 wd ; wd þ wa
sd ¼
ðn kÞ weff
2
ð24Þ
3.2. Effective searching time and kinetics In Eqs. (23) and (24) we have introduced the effective time of the process, sd , which can be written as follows:
sd ¼ L2 =weff ; weff ¼ w1
wa wd þ w2 wd þ wa wd þ wa
s1D þ s3D ; w1 s1D þ w2 s3D
ð25Þ
Here L ¼ ðn kÞ is the initial distance to the damaged site (in the dimensionless units it is equal to the number of sites); it depends on the parameter c ¼ wd =wa given by the ratio of the dissociation and association rates. The physical meaning of sd is the characteristic time, at which the flux is changing; thus, it is the
sd , and the effective rate, weff , are
weff ¼ w1
s1D s3D þ w2 s1D þ s3D s1D þ s3D
ð27Þ
As compared to the purely 1D diffusion case where the searching time is s0 ¼ L2 =w1 , the time changes by the following factor:
sd 1 ¼ s0 s1Dsþ1Ds3D þ s1Dsþ3Ds3D
w2 w1
¼
1 1 þ s1Dsþ3Ds3D
w2 w1
1
ð28Þ
2 Thus, when w1 < w2 (3D diffusion is faster) so that w 1 > 0, w1 the effective searching time becomes shorter. Of course the size of this effect depends on the dissociation-association rates, which is described by the factor s1Dsþ3Ds3D in the denominator. The inverse Laplace transformation of Eq. (23) gives the following result for the flux, j1 ðtÞ, at long times:
1 2
rffiffiffiffiffiffiffiffi
s sd d exp 3 4t pt
ð29Þ
This is a function, which is equal to zero at t ¼ 0 (as initially there are no enzymes at the damaged site) going through a maximum at t max ¼ sd =6 and decaying as / t 3=2 at t ! 1. Thus, sd gives the characteristic time, at which the target site is reached. Interestingly, the mean searching time, t, is infinite for this power-law time dependence:
Z 0
This formula describes the kinetics of the searching process with weff being the effective jumping rate and sd being the searching time:
weff
sd ¼ L2
hti ¼ ð23Þ
ð26Þ
Then the searching time,
ð21Þ
which gives us that
~j1 ðsÞ
s1D ¼ 1=wd ; s3D ¼ 1=wa
j1 ðtÞ
When s1 ; s2 ud ; ua we obtain
X
effective searching time. It is worth noting that the integral of the flux over time is equal to 1, i.e., the particles do not escape by going to infinite distances and never coming back; thus, the flux indeed changes solely due to the specific binding. According to Eq. (25) at long times the motion is qualitatively 1D diffusion-like and a the effective jumping rate is given by the sum of two rates: w1 w wþw a d wd and w2 w þwa , which correspond to the effective jumping rates for d the first and second chain, respectively. Each rate has a very clear meaning: is given by the original jumping rate for the corresponding chain multiplied by the fraction of time that the enzyme resides on this chain. As it is often done [13,19,27], we can rewrite the results in terms of the durations of 1D and 3D diffusion stages, which we introduce as:
1
tj1 ðtÞdt ! 1
ð30Þ
This is a clear indication that due to the diffusion-like motion the time-behavior is considerably more complex than one would expect from the formal chemical kinetics. Diffusional ‘tails’ of the kinetics make the behavior different from exponential with a consequence that the moments of the process, e.g., hti, are not defined. The time dependence of the flux is illustrated in Fig. 2. When the time, sd , becomes shorter the flux, j1 ðtÞ, reaches its maximum earlier and the maximum is higher. This indicates that the enzyme finds its target faster. Here we do not show results for very large w2 (the highest w2 is 10w1 ) because it is difficult to show the corresponding kinetics on the same scale with that for the purely 1D-searching. For very large w2 rates the maximum shifts to even shorter times (equal to sd =6) and j1 ðtmax Þ becomes even higher. Interestingly, one can obtain the same results assuming continuous diffusion model and redefining some of the kinetic parameters. This is demonstrated in Appendix A. Thus, in this simplest case there is no principal difference between the discrete and continuous models. However, we would like to focus on the jumping model because it provides a rather straightforward extension of the results to the situation, where 3D diffusion is described by a distribution of jumping lengths.
18
K.L. Ivanov, N.N. Lukzen / Chemical Physics 435 (2014) 14–20
Let us now rewrite the expression for ½1 AðhÞ:
5
1 AðhÞ ¼
3
The full analytical treatment is still very cumbersome. However, ^ 11 ðzÞ as if one can write G
0
j1(t)
ð1 þ qÞð1 cos hÞ 1 þ q2 2q cos h
4
2
1 k0 cos h þ const 2w1 ðk1 cos hÞðk2 cos hÞ
1 0 0.0
0.5
1.0
t/
1.5
2.0
0
ð36Þ
ð37Þ
and obtain that one of the roots ki (for clarity, let it be k2 ) tends to one faster than the other, one can immediately find an approximate expression for the flux (similar to how it has been done to derive Eq. (21))
Fig. 2. Kinetics of diffusional search. Time dependence of the flux, j1 ðtÞ, shown for different values of w2 =w1 equal to 10 (solid line), 5 (dashed line) and 3 (dotted line). Results for the purely 1D-search are shown by a thin solid line. Here s1D ¼ s3D , the results are shown for a dimensionless quantity j1 ðtÞ s0 .
qffiffiffiffiffiffiffiffiffiffiffiffiffiffink pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nk ~j1 k2 k2 1 ð1 2ðY XÞÞ 2
3.3. Case of jumps to remote sites
The idea behind this derivation is the same as before; the quantities X and Y are also introduced in the same way. This procedure ^ 11 ðzÞ is indeed is valid, since for AðhÞ is given by Eq. (36) function G of the form given by Eq. (37). If we assume that s1 ; s2 ud ; ua 1 for the difference of X and Y we arrive at the expresion:
Let us assume that in the 3D-case the jumps are occurring not only between the adjacent sites but also over one site, over two sites, etc. To take this effect into account, we will assume that there is a distribution, PðkÞ, of probabilities for jumping from the i-th site to the ði kÞ-th sites. Hereafter we will always take a symmetric distribution, i.e., PðkÞ ¼ PðkÞ, the distribution will also be normalP ized 1 k¼1 PðkÞ ¼ 1. Let us modify the original set of equations (2) ~ i taking such distribution into account: for C ~i ¼ W ~ i1 þ C ~ iþ1 c UÞ b C c1 C ðs þ 2 W ~ i1 þ P1 C ~ iþ1 þ P 2 C ~ i2 þ P 2 C ~ iþ2 þ P 3 C ~ i3 þ P 3 C ~ iþ3 þ ... c 2 P1 C þW ~ n þ dik f k bC din K
ð31Þ
Here
c1 ¼ W
w1
0
0
0
;
c2 ¼ W
0
0
0 w2
ð32Þ
By applying the same mathematical procedure as before we obtain the equation for the GF:
1 c ^ c2 U b FðzÞ ¼ AFðzÞ W 1 2½AðzÞ 1 W s z2þ z ~n bC ¼ zk f 0 zn K
ð33Þ
P1 1
where AðzÞ ¼ 2 k¼1 PðkÞðzk þ 1=zk Þ. In the following analysis one can use expressions obtained above and substitute there w2 z 2 þ 1z by 2w2 ½AðzÞ 1. When z is substituted as eih , then P ^ AðzÞ becomes AðhÞ ¼ 1 k¼1 PðkÞ cosðkhÞ. The result for G11 ðhÞ then takes the form:
^ 11 ðhÞ ¼ G
s þ wa þ 2w2 ð1 cos hÞ ðs þ wd þ 2w1 ð1 cos hÞÞðs þ wa þ 2w2 ½1 AðhÞÞ wd wa ð34Þ
For an arbitrary AðhÞ function further analysis becomes problematic, at least, in the analytical form. Previously we have considered a special case of Pð1Þ ¼ 1, PðkÞ ¼ 0 ðk > 1Þ, which was relatively simple. To demonstrate effects of non-local jumps let us consider AðhÞ corresponding to PðkÞ – 0 for k > 1, in which analytical solution is still feasible. Such a case is given, for instance, by a situation where PðkÞ decays with the index k as the geometrical progression with q < 1, that is PðkÞ ¼ ð1 qÞqk1 . Then AðzÞ takes the form:
AðzÞ ¼
1q z 1=z þ 2 1 qz 1 q=z
ð35Þ
2ðY XÞ ¼
s1 ua þ s2 ud 1þq ua þ ð1qÞ 2 ud
ð38Þ
ð39Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi Thus, indeed k2 1 and 1= k22 1 has a singularity. Then one can use the same strategy as before and just redefine the searching time as
sd ¼ L2
wa þ wd s1D þ s3D ¼ L2 1þq 1þq w1 wa þ ð1qÞ w1 s1D þ ð1qÞ 2 w2 wd 2 w2 s3D
ð40Þ
For q ¼ 0 (jumps only between the adjacent sites) the result the same as previously. However, when q ! 1 (jumps to remote sites are possible) and w2 is non-zero the searching time formally tends to 0. Thus, when the jumps between remote sites are possible the searching time can be dramatically shortened. This situation corresponds, for instance, to long-range hopping along the DNA fragment or inter-segment transfer of the enzyme [4]. Of course the result sd ! 0 at q ! 1 is misinterpretation as the approximations made in order to obtain the flux, jðtÞ, are limited to relatively long times, i.e., t > T; the searching n times o cannot be shorter than T. Thus, in this case sd ¼ max w1a ; w1 . Nevertheless, increase of q d clearly leads to the reduction of the searching time. One can interpret Eq. (40) in terms of the effective rate of 1D-jumps:
weff ¼
L2
sd
¼ w1
¼ w1
wa 1þq wd þ w2 2 w þw wd þ wa a d ð1 qÞ
s1D 1þq s3D þ w2 s1D þ s3D ð1 qÞ2 s1D þ s3D
ð41Þ
As compared to Eq. (25) the w2 rate has increased by a factor > 1; this results in a further increase of the weff rate and, consequently, in shortening of the searching time.
1þq ð1qÞ2
4. Concluding remarks An analytical solution for the time of searching for a target site has been obtained for the case of a combination of two types of stochastic motion. The first motion was always treated as one-dimensional diffusion (only jumps to the adjacent sites were allowed); for the second motion an arbitrary probability distribution of the jump length has been assumed. For the simplest situation of a 1D-like second motion with a faster jumping rate we have obtained an analytical expression for the Laplace transform of the searching kinetics and an approximate expression for the kinetics
19
K.L. Ivanov, N.N. Lukzen / Chemical Physics 435 (2014) 14–20
g~ðR þ x0 jx0 ; sÞ ; g~ðR þ x0 jR þ x0 ; sÞ
in the time domain. Effective time of searching has been derived; we argue that the second motional process with a faster rate facilitates the search. It is shown that the kinetics strongly deviates from exponential and follows a t 3=2 power-law, which is a consequence of the diffusion type of motion. We have shown that when the jumps are allowed not only to the adjacent sites the search is facilitated even further. In general, our treatment can be useful for describing kinetics of diffusional search for targets by motions of two types; an important example of such process is searching for the specific binding site on the DNA molecule.
~j1 ðsÞ
Acknowledgements
Here the delta-like initial condition for the Green function is taken into account by the term dðx x0 Þ. To get rid of the second derivatives let us perform the Fourier transformation in the coordinate domain. These yields the following two equations for the Fou~F ðkjx0 ; sÞ: rier-transformed quantities, g~F ðkjx0 ; sÞ and h
This work has been supported by the Ministry of Education and Science of the Russian Federation, the Russian Foundation for Basic Research, Russia (Grants Nos. 11-03-00356 and 12-03-33082), SB RAS (project No. 71) and grant of the President of RF MD3279.2014.2. We are thankful to Dr. V.V. Koval (Novosibirsk) for stimulating discussions and to Prof. V.A. Avetisov (Moscow) and Dr. D.A. Grebenkov (Paris) for providing useful literature references.
C 1 ðxÞ
C 2 ðxÞ
(
ðA7Þ
2 ~F ðkjx0 ; sÞ þ 1 expðikx0 Þ ðs þ D1 k þ wd Þg~F ðkjx0 ; sÞ ¼ wa h 2p 2 ~ ðkjx ; sÞ ¼ w g~ ðkjx ; sÞ ðs þ D k þ w Þh 2
a
F
0
d F
ðA8Þ
0
These are two linear algebraic equations, which are easy to ~F ðkjx0 ; sÞ: solve and to find g
expðikx0 Þ s þ D2 k þ wa 2 2 2p ðs þ D1 k þ wd Þðs þ D2 k þ wa Þ wd wa ðA9Þ
To evaluate the flux one should perform the inverse Fourier transformation, in our case, it is sufficient to perform only the cosine-transformation. To do so, let us expand both the numerator 2 and denominator of g~F ðkjx0 ; sÞ as polynomials of k and then evaluate the Fourier integrals by using the formula:
Z
1
dk
cosðkxÞ
1
2
k þa
pffiffiffi p ¼ pffiffiffi expðx aÞ a
ðA10Þ
The Fourier original is then as follows:
g~ðx þ x0 jx0 ; sÞ ¼
ðA2Þ
where
b ¼ D1 0 ; U b ¼ wd wa ; V b ¼ v ðxÞ 0 ; C0 ¼ 1 D 0 D2 0 0 0 wd wa ðA3Þ
These matrices describe diffusion (along the two chains), dissociation-association, specific binding occurring with a positiondependent rate, v ðxÞ, and the initial condition given by the term containing the delta-function. As previously, we assume that specific binding can only occur in a narrow zone, i.e., we will take the following v ðxÞ rate
v ðxÞ ¼ ksb dðx R x0 Þ
8 2 ~ > < s D1 d 2 þ wd g~ðxjx0 ; sÞ ¼ wa hðxjx 0 ; sÞ þ dðx x0 Þ dx 2 > d ~ : ~ s D2 dx 2 þ wa hðxjx0 ; sÞ ¼ wd g ðxjx0 ; sÞ
ðA1Þ
with the two components being the probability densities to find the enzyme with the coordinate x on the two chains. Equation for CðxÞ in the Laplace domain can be written as
! 2 d ~ sÞ þ dðx x0 ÞC0 ~ sÞ ¼ V b b b ðxÞCðx; s D 2 U Cðx; dx
Here we have introduced the Green function for the continuous diffusion case. To calculate its [1, 1]-element one should solve two ^ g ¼ ðGÞ ^ coupled equations for two matrix elements of G, 11 and ^ . These two equations are h ¼ ðGÞ 21
g~F ðkjx0 ; sÞ ¼
One can also assume that the enzyme motion is continuous, with the diffusion coefficients being equal to D1 and D2 for the first and second chain, respectively. The enzyme starts searching from the point with the coordinate x0 at the first chain; when it reaches the coordinate ðR þ x0 Þ (on the first chain as well) specific binding occurs immediately. The distance R corresponds to L links of the DNA chain, each being of the length l0 : R ¼ L l0 . Dissociationassociation can be introduced in the same way as in the model of discrete jumps. To write down the kinetic equations let us introduce a vector
ðA6Þ
2
Appendix A. Continuous diffusion model
CðxÞ ¼
^ where g ¼ ðGÞ 11
ðA4Þ
pffiffiffiffiffi pffiffiffiffiffi 1 a1 a0 expðx a1 Þ a2 a0 expðx a2 Þ pffiffiffiffiffi pffiffiffiffiffi þ a1 a2 2D1 a1 a2 a2 a1 ðA11Þ
Here
a0 ¼ s2c þ uac s1c þ s2c þ udc þ uac a1;2 ¼ 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðs1c þ s2c þ udc þ uac Þ2 4ðs1c s2c þ s1c uac þ s2c udc Þ ðA12Þ 2 For brevity we used the following notations:
s1c ¼
s ; D1
s2c ¼
s ; D2
udc ¼
wd ; D1
uac ¼
wa D2
ðA13Þ
To proceed further it is convenient to use a close analogy between the quantities ai used here and ki in the discrete case, see Eq. (19). When at small s one of the two terms in Eq. (A11), for clarity, the second term, has a singularity much simpler approximate expressions for ~j1 ðsÞ can be used. Detailed analysis shows that this is indeed the case therefore, as previously, assuming that s1c ; s2c udc ; uac we obtain:
This approximation (contact approximation) for the reactivity dramatically simplifies the derivation. The flux of specific binding takes the form
~j1 ðsÞ expðpffiffiffiffiffiffiffi ssd Þ
~jðsÞ ¼
The effective searching time can be thus found by redefining the kinetic parameters:
Z
~ sÞdx ¼ ksb C ~ 1 ðR þ x0 ; sÞ 1 ^ Cðx; VðxÞ 0
ðA5Þ
When the rate constant, ksb , is taken infinitely large (the process is diffusion-limited) one can obtain the following expression for j-component of interest (the result is an exact analogue of the previously derived expression, see Eq. (10)):
sd ¼ R2 =Deff ; Deff ¼ ¼ D1
ðA14Þ
D1 wa þ D2 wd wd þ wa
s1D s3D þ D2 s1D þ s3D s1D þ s3D
ðA15Þ
20
K.L. Ivanov, N.N. Lukzen / Chemical Physics 435 (2014) 14–20
Here the definition of the times s1D ; s3D remains the same as before as well as interpretation of Deff . Hence, the analysis of the searching time given in Section 3.2 is valid in the continuous case as well. The coincidence of the results 2
l2
2
l2
in the limit s1c l0 ; sc2 l0 1 is fully consistent: at times t D01 ; D02 discrete jumps are physically equivalent to continuous diffusion. Therefore the inverse Laplace transformation of j1 gives the following results at long times:
j1 ðtÞ
1 2
rffiffiffiffiffiffiffiffi
s sd d exp 3 4t pt
ðA16Þ
The effective searching time is as follows
s d ¼ R2
s1D þ s3D D1 s1D þ D2 s3D
ðA17Þ
As compared to the purely 1D case (the second chain is absent) where sd0 ¼ R2 =D1 the searching time changes by a factor:
sd ¼ sd0 1 þ
1 s3D s1D þs3D
D2 D1
1
References [1] M. von Smoluchowski, Z. Phys. Chem. 92 (1917) 129. [2] A.D. Riggs, S. Bourgeois, M. Cohn, J. Mol. Biol. 52 (1970) 401.
ðA18Þ
[3] P. Hammar, P. Leroy, A. Mahmutovic, E.G. Marklund, O.G. Berg, J. Elf, Science 336 (2012) 1595. [4] O.G. Berg, R.B. Winter, P.H. von Hippel, Biochemistry 20 (1981) 6929. [5] P.H. von Hippel, O.G. Berg, J. Biol. Chem. 264 (1989) 675. [6] P.H. Richter, M. Eigen, Biophys. Chem. 2 (1974) 255. [7] O.G. Berg, C. Blomberg, Biophys. Chem. 4 (1976) 367. [8] G. Adam, M. Delbrück, Structural Chemistry and Molecular Biology, W.H. Freeman and Co., New York, 1968. [9] Y.M. Wang, R.H. Austin, E.C. Cox, Phys. Rev. Lett. 97 (2006) 048302. [10] J. Gorman, E.C. Greene, Nat. Struct. Mol. Biol. 15 (2008) 768. [11] J. Elf, G.-W. Li, X.S. Xie, Science 316 (2007) 1191. [12] K. Loth, M. Gnida, J. Romanuka, R. Kaptein, R. Boelens, J. Biomol. NMR 56 (2013) 41. [13] L. Mirny, M. Slutsky, Z. Wunderlich, A. Tafvizi, J. Leith, A. Kosmrlj, J. Phys. A: Math. Theor. 42 (2009) 434013. [14] G. Kolesov, Z. Wunderlich, O.N. Laikova, M.S. Gelfand, L.A. Mirny, Proc. Nat. Acad. Sci. U.S.A. 104 (2007) 13948. [15] H.-X. Zhou, Biophys. J. 88 (2005) 1608. [16] A.G. Cherstvy, A.B. Kolomeisky, A.A. Kornyshev, J. Phys. Chem. B 112 (2008) 4741. [17] A.G. Cherstvy, J. Phys. Chem. B 113 (2009) 4242. [18] R.K. Das, A.B. Kolomeisky, Phys. Chem. Chem. Phys. 12 (2010) 2999. [19] T. Hu, A. Grosberg, B. Shklovskii, Biophys. J. 90 (2006) 2731. [20] A.B. Kolomeisky, Phys. Chem. Chem. Phys. 13 (2011) 2088. [21] J.C. Genereux, A.K. Boal, J.K. Barton, J. Am. Chem. Soc. 132 (2010) 891. [22] P.-W. Fok, C.-L. Guo, T. Chou, J. Chem. Phys. 129 (2008) 235101. [23] V.A. Avetsov, A.K. Bikulov, A.P. Zubarev, J. Phys. A: Math. Theor. 42 (2009) 085003. [24] A.M. Berezhkovskii, A.V. Barzykin, J. Chem. Phys. 136 (2012) 054115. [25] J.-F. Rupprecht, O. Bénichou, D.S. Grebenkov, R. Voituriez, J. Stat. Phys. 147 (2012) 891. [26] J.-F. Rupprecht, O. Bénichou, D.S. Grebenkov, R. Voituriez, Phys. Rev. E 86 (2012) 041135. [27] S. Halford, J. Marko, Nucleic Acids Res. 32 (2004) 3040.
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Chemical Physics journal homepage: www.elsevier.com/locate/chemphys
Kinetics of target searching by means of two diffusion-like motions Konstantin L. Ivanov ⇑, Nikita N. Lukzen a b
International Tomography Center SB RAS, Institutskaya Str. 3a, Novosibirsk 630090, Russia Novosibirsk State University, Pirogova Str. 2, Novosibirsk 630090, Russia
a r t i c l e
i n f o
Article history: Received 9 December 2013 In final form 4 March 2014 Available online 12 March 2014 Keywords: Diffusion-controlled processes Diffusional search DNA repair
a b s t r a c t A theoretical approach to stochastic searching of a small target is developed, which can be applied, for instance, to searching for a damaged site on the DNA molecule by a DNA repair enzyme. It is assumed that the searching molecule moves along a chain of sites by means of stochastic jumps to the adjacent positions (one-dimensional ‘sliding’); the second motion models diffusion in three dimensions. A general expression is obtained for the flux to the damaged site. A special case is analyzed where the second motion is treated as effective 1D diffusion; the effective searching time is estimated. It is shown that the faster second motion shortens the searching time. A more realistic case where the second kind of motion leads to jumps not only to the adjacent site is also treated: it is shown that the diffusional search is facilitated even further. Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction Biological processes are frequently dealing with protein–nucleic acid interactions, notably, with specific binding of various enzymes to nucleic acids. Among them is DNA repair (i.e., removal of molecular lesions caused by metabolic activities and environmental factors), which is an important process for keeping the integrity and stability of the genome. Understanding the mechanism of searching for the particular binding site on the large DNA molecule is an intriguing problem. It is known that the search is much more efficient as compared to the simple prediction coming from the Smoluchowski theory [1]: there is experimental evidence (both in in vitro [2] and in vivo [3] studies) that the rate constant, ks , exceeds the Smoluchowski-like one, kSm ¼ 4pRD, by a few orders of magnitude. (Here R is the sum of the linear sizes of the damaged site and the enzyme and D is the mutual diffusion coefficient of the enzyme and DNA). To rationalize this unexpected result several theoretical models have been proposed, with the most well-known being the Berg–Winter–von Hippel (BWH) model [4,5], which suggests that first the enzyme binds to an arbitrary site on the DNA (non-specific binding) and searches for the damaged site in the course of subsequent stages of 1D and 3D diffusion. 1D diffusion proceeds by means of ‘sliding’ [2,6,7] of the enzyme along the DNA strand, the sliding increases the effective size of the target ⇑ Corresponding author at: International Tomography Center SB RAS, Institutskaya Str. 3a, Novosibirsk 630090, Russia. Tel.: +7 (383) 330 8868; fax: +7 (383) 333 1399. E-mail address: [email protected] (K.L. Ivanov). http://dx.doi.org/10.1016/j.chemphys.2014.03.001 0301-0104/Ó 2014 Elsevier B.V. All rights reserved.
[8]. After a sliding stage the enzyme dissociates from the DNA and diffuses in three dimensions in the solution, then it binds again to the DNA molecule and the sequence is repeated until the enzyme finds its target and binds to it (specific binding) so that the DNA repair process starts. It is often argued that the combination of 1D and 3D diffusion facilitates the process and makes the time required for searching for the damaged site considerably shorter. This is because ‘sliding’ of a large enzyme molecule along the DNA is relatively slow [3,9–12]; thus, purely 1D diffusional motion makes the search inefficient. On the other hand, 1D diffusion increases the effective target size: in this way the combination of two diffusional motions reduces the searching time. Previously, several models have been considered to describe the complex process of searching by 1D and 3D diffusional walks. These are the co-localization mechanism [13,14] (assuming that that proteins are produced near their binding sites on the DNA thereby reducing the number of searching cycles) and the correlation mechanism [15–18] (taking account of the correlations between 1D sliding and 3D diffusion). Hu et al. [19] have developed a theoretical approach, which take into account the role of DNA conformation by using an electrostatic analogy. Description of these theoretical results can be found, for instance, in a recent review by Kolomeisky [20]. Analytical solutions of the problem are usually difficult to obtain even assuming simplified geometry of DNA and protein. Solutions for realistic cases are probably impossible because the enzyme is not a spherical molecule, moreover, it can bind to DNA only in specific orientations; DNA is also a highly complex molecule having different conformations. One should note that fast searching for DNA damages can be sometimes
K.L. Ivanov, N.N. Lukzen / Chemical Physics 435 (2014) 14–20
accounted for by an alternative mechanism (charge transport through the DNA for carrying out redox chemistry at a distance) [21,22]; however, here we will not consider this mechanism and focus on subsequent diffusional motions of two different types. The aim of this work is developing a physically reasonable model of searching, which enables analytical solution for the searching time after non-specific binding of the enzyme to DNA. Although a lot has already been done in this field we will propose a model, which allows one obtaining relatively simple analytical expressions for the searching time and for the kinetics of searching for the damaged site. On one hand, we will try to make the model simple; on the other hand, we will need to take account of two types of stochastic motions. To tackle the problem analytically we will exploit the formalism of generating functions. We will find out whether the combination of 1D and 3D diffusion facilitates the search or does not and also discuss peculiarities of diffusion, which make the kinetics different from frequently assumed monoexponential function. Last but not least, our general formulas will not be limited to the specific problem of the DNA repair kinetics but will also be applicable to other cases of searching for different targets by two types of stochastic motions. 2. Methods 2.1. Theoretical model Describing subsequent stages of 1D and 3D diffusion of the repair enzyme is generally a complex problem, which can be solved analytically only assuming specific models for the enzyme motion. Here we propose a new model, which enables analytical treatment of the problem and obtaining both the effective DNA repair time, i.e., the time required to find the damaged nucleotide, and the repair kinetics. The situation treated here is presented in Fig. 1. Here we assume that the enzyme can move along the DNA strand by stochastic jumps to the nearest nucleotides, the jump rate is equal to w1 . Taking 3D diffusion into account is generally problematic; to minimize this problem we assume that 3D diffusion is a process of jumping along a parallel chain with the jump rate equal to w2 . Of course, such assumption does not consider (i) the peculiarities of 3D diffusion (which is non-redundant in contrast to redundant 1D diffusion) and (ii) the conformation of the DNA. Nonetheless, as will be shown, it can account for the effect of facilitating the 1D search, which becomes inefficient when the enzyme spends too long time on the DNA fragment as the number of sites (nuclepffiffi otides) visited by the repair enzyme grows with time only as t. In this case dissociation from the DNA with subsequent diffusion in space (being faster than ‘sliding’) and association to a new site facilitates the search, since more sites on the DNA can be visited in the unit time. Here we will thus assume that w2 w1 (free diffusion in space is much faster than the sliding process);
2.2. General results for repair flux To proceed further let us define the notations. Here we will introduce a set of values C i , which are equal to the probability to find the enzyme on the i-th nucleotide at the first chain; C 0i is then the probability to find the enzyme on the i-th site at the second chain. It is more convenient to derive equations in the matrix-form, for this reason we will work with a set of vectors
Ci ¼
wd wa k
dissociation and association will be modeled as transitions between the two chains with rates wd and wa , respectively. We will assume that both processes are slow even as compared to 1D ‘sliding’, in other words, the enzyme makes many stochastic jumps before it dissociates or re-associates. Thus, the following hierarchy of rates is assumed: w2 w1 wa ; wb . This assumption is consistent with the current view on the DNA repair processes [3]. Motion along the two chains thus models 1D and 3D diffusion, dissociation and re-association events are taken into account by the transitions between the chains. We also assume that each chain is infinite to ignore boundary effects. At t ¼ 0 the enzyme is located on the k-th nucleotide, while the n-th nucleotide is damaged. When the enzyme reaches the damaged site, specific binding occurs immediately and irreversibly; thus the search is accomplished. In this model it is therefore sufficient to calculate the time of first arrival from the initially occupied site to the damaged site. Our model allows reasonable estimates for the searching time in the case where two different stochastic motions are present although the 3D motion is accounted for in a greatly simplified way. It is worth noting that replacing 3D diffusion by effectively 1D diffusion is also done in the correlation mechanism [16,20]. In principle, one can omit the second chain and introduce 3D diffusion by assuming that the enzyme can jump not only to the nearest site but to other sites as well (see Ref. [4] and Appendix therein). However, to do so it is required to calculate the specific distribution of rates of jumping to other sites. In our treatment the second stochastic process is incorporated in the model from the very beginning and is treated as effective 1D diffusion. As will be shown in the end of the paper, our method also allows one to consider jumps between arbitrary positions, i.e., to extend the treatment to a more realistic type of motion, which captures some properties of the 3D diffusion. We will obtain analytical results for one example of a specific probability distribution of the jump length. It is worth noting that our model is bearing some similarity to the one proposed by Avetisov and coworkers [23] who treated conformational changes in frozen proteins with a hierarchy of the transition rates to describe dynamics of biomolecules on very different time scales. Cycles of 3D diffusion and 2D diffusion have also been treated in the problem of diffusional search for a small hole in a spherical cavity by Berezhkovskii and Barzykin (who developed a coarse-grained model for the search process) [24] and later by Rupprecht et al. (who obtained exact solution of the problem) [25,26].
w2 w2
n
w1 w1 Fig. 1. Model of searching for a damaged site on the DNA molecule. The repair enzyme jumps along the DNA chain (1D sliding), which is an infinite chains of sites (nucleotides); the rate of jumps to the adjacent sites is equal to w1 . 3D diffusional motion is modeled by jumps along a parallel chain; the rate of such jumps to the adjacent sites is taken equal to w2 . Transitions between the two chains (dissociation and re-association) are occurring at rates wd and wa . The searching process starts at the k-th site of the first chain; the target site is the n-th site on the first chain: upon reaching this site irreversible specific binding immediately occurs.
15
Ci
C 0i
ð1Þ
Kinetic behavior of the system is then described by the following set of equations for Ci written in the Laplace domain:
~i ¼ W ~ i1 þ C ~ iþ1 Þ din K ~ n þ dik f k ; c UÞ b C c ðC bC ðs þ 2 W c ¼ w1 0 ; U b ¼ wd wa where W wd wa 0 w2
ð2Þ
Hereafter s is the Laplace variable, tilde stands for the Laplace. .. In Eq. (2) transformed quantities; the matrices are defined by .c c -matrix stands for the stochastic jumps between the the W
16
K.L. Ivanov, N.N. Lukzen / Chemical Physics 435 (2014) 14–20
b adjacent positions in each chain; the U-matrix describes the transitions between the chains, i.e., dissociation-association. For the 1 coming from k-th equation there is also a source term f 0 ¼ 0 the initial condition (at t ¼ 0 the enzyme is located on the k-th site), in the n-th equation there is an additional term in the b Cn and describes irreversible right-hand side, which is equal to K b ¼ ksb 0 with specific binding at the corresponding site. Here K 0 0 ksb being the rate constant for the specific binding process. To solve the kinetics equations it is convenient to work with the generating function (GF), FðzÞ, for the sequence of Ci s. It is defined as follows:
FðzÞ ¼
1 X
~ i zi C
ð3Þ
physical meaning of g m is as follows: it is the probability to find the enzyme at certain time m nucleotides away from the starting site at the first chain assuming that the starting site was at the first chain as well. To obtain the analytical result for the searching time and searching kinetics it becomes necessary to write down explicit ^ expression for the [1, 1]-element of the matrix GðzÞ. The expression for this element and j1 ðtÞ will be given in the following section. 3. Results and discussion 3.1. Case of jumps only to the adjacent sites In the simplest case described above where only jumps between the adjacent sites are possible we obtain the following result ^ for the [1, 1]-element of the G-matrix:
i¼1
Here z is a complex variable. In this definition we include all powers of z but not only positive ones (as is usually done). When z ¼ eih , the series from Eq. (3) is, in fact, the Fourier series expan~ i s are the Fourier harmonics sion of the GF with respect to h, i.e., C of F. Thus, the knowledge of the GF provides full information about the quantities of interest. The equation for the GF can be obtained from the original set of equations (2) by multiplying the equation for Ci by zi and summing all the equations:
1 c b ^ b Cn W U FðzÞ ¼ AFðzÞ s z2þ ¼ zk f 0 zn K z
ð4Þ
^ 11 ðzÞ ¼ G
s þ wa w2 ðz 2 þ 1=zÞ ðs þ wd w1 ðz 2 þ 1=zÞÞðs þ wa w2 ðz 2 þ 1=zÞÞ wd wa ð11Þ
Now let us calculate the quantities g m as the Fourier harmonics ^ 11 ðzÞ: of G
gm ¼
1 2p
Z p
^ 11 ðeih Þdh eimh G
ð12Þ
p
To evaluate these integrals it is convenient to substitute z as eih ^ 11 ðzÞ: in the expression for G
It is then easy to obtain the general solution of this equation:
^ 1 zk f 0 zn K b Cn FðzÞ ¼ ðAÞ
ð5Þ
In the following consideration we will be interested in the rate of specific binding, which is given by the flux from the n-th site at the first chain to the new state (specific binding state), where the repair process becomes operative. This flux can be defined in the following way:
b Cn ðtÞ jðtÞ ¼ K
ð6Þ
Thus, our final goal is evaluating Cn . Let us express this quantity ^ 1 ^ ¼ ðAÞ from the GF. To do this, let us assume that the matrix G (having the meaning of the free propagator of the system in the absence of specific binding) has already been expanded in a series with different powers of z:
^ GðzÞ ¼
1 X
s þ wa þ 2w2 ð1 cos hÞ ðs þ wd þ 2w1 ð1 cos hÞÞðs þ wa þ 2w2 ð1 cos hÞÞ wd wa 1 s2 þ ua þ 2ð1 cos hÞ ¼ w1 ðs1 þ ud þ 2ð1 cos hÞÞðs2 þ ua þ 2ð1 cos hÞÞ ud ua
^ 11 ðhÞ ¼ G
ð13Þ Here we introduced: s1;2 ¼ s=w1;2 ; ud ¼ wd =w1 ; ua ¼ wa =w2 . We ^ 11 ðhÞ even further by noting that the denominator can simplify G is a second-order polynomial of cos h with two roots, k1;2 :
k0 cos h ^ 11 ðhÞ ¼ 1 G 2w1 ðk1 cos hÞðk2 cos hÞ 1 k1 k0 1 k2 k0 1 þ ¼ 2w1 k1 k2 k1 cos h k2 k1 k2 cos h
ð14Þ
By using the following expressions for the integrals
^ i zi G
ð7Þ
i¼1
By setting equal the coefficients at zn in the left-hand side and right-hand side of expression (5) for the GF we obtain:
1 2p
pffiffiffiffiffiffiffiffiffiffiffiffiffi m Z p Z p cosðmhÞ ða a2 1Þ 1 sinðmhÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi ; dh ¼ dh ¼ 0 2p p a cos h a2 1 p a cos h
ð15Þ
we obtain that
~n ¼ G ^ nk f 0 G ^0 K ~n bC C
ð8Þ
From the vector ^jðsÞ we will need only its first component, j1 , which is equal to
~j1 ðsÞ fnk ; where f0 qffiffiffiffiffiffiffiffiffiffiffiffiffink qffiffiffiffiffiffiffiffiffiffiffiffiffink k1 k21 1 k2 k22 1 k1 k0 k2 k0 qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi þ f nk ¼ k1 k2 k2 k1 2 k1 1 k22 1
^j1 ðsÞ ¼ ksb g nk g nk ; 1 þ ksb g 0 g0
To reduce the otherwise long notations let us introduce three new quantities:
The quantity of interest then takes the form:
~jðsÞ ¼ K ~n ¼ K ^0 K ^ nk f 0 bC b ðE ^þG b Þ1 G
^ mÞ where g m ¼ ðG 11
ð9Þ
ð10Þ
Here the approximate equality is obtained assuming infinitely fast specific binding upon arrival to the n-th site, that is ksb g 0 1, which we will always use in the consideration below. Thus, for the diffusion-limited searching process we have obtained a simple general result: to evaluate all kinetic quantities of interest it is sufficient to compute only two quantities, g nk and g 0 . The
s1 s2 þ ud ua ; 4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 X¼ ðs1 þ s2 þ ud þ ua Þ2 4ðs1 s2 þ s1 ua þ s2 ud Þ; 4 s1 þ s2 þ ud þ ua : Y¼ 4
ð16Þ
D¼
ð17Þ
17
K.L. Ivanov, N.N. Lukzen / Chemical Physics 435 (2014) 14–20
Then
fnk ¼
DþX 2X
qffiffiffiffiffiffiffiffiffiffiffiffiffiffink qffiffiffiffiffiffiffiffiffiffiffiffiffiffink k1 k21 1 k2 k22 1 DX qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 2X 2 k1 1 k22 1
ð18Þ
In the notation given by Eq. (17) explicit expressions for ki ’s are:
k0 ¼ 1 þ
s2 þ u a ; 2
k1;2 ¼ 1 þ Y X:
ð19Þ
Combining all these formulas one can obtain the expression for the flux. However, expressions (16), (18) are still quite cumbersome; moreover, it is necessary to perform the inverse Laplace transformation to obtain j1 ðtÞ. As a consequence, when the rates defined in Section 2.1 are related arbitrarily, the result for j1 ðtÞ is very complex. However, in our model we assume specific relations between the rates, which are also reasonable from the experimental point of view. These are sufficient to arrive at relatively simple analytical expressions for the flux and the searching time. To proceed further let us expand the expressions for fm assuming s1 ; s2 ud ; ua 1. This corresponds to a situation where (i) diffusion along the chains is much faster than dissociationassociation and (ii) relatively long times, t > T, are treated, at which all the kinetics processes have been established (jumps along the chains and dissociation-association have occurred many times); here T w1a ; w1 . d Let us note that in expression (18)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k21;2 1 ¼ ðk1;2 1Þðk1;2 þ 1Þ 2ðY XÞ
ð20Þ
Since at s1 ; s2 ! 0 we obtain Y ! X, one of the two terms in expression (18) for fnk is dominating (in our notations this is the second term) because the denominator tends to zero. Neglecting the other term we obtain a much simpler expression for the flux
~j1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffink pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffink k2 k22 1 1 2ðY XÞ
ud þ ua 1 ðs1 s2 Þðud ua Þ þ 4 ud þ ua 4
ð22Þ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffink qffiffiffiffiffiffiffiffiffiffiffiffiffink s1 ua þ s2 ud 1 ¼ 1 s=weff ud þ ua pffiffiffiffiffiffiffi expð ssd Þ
w1 wa þ w2 wd ; wd þ wa
sd ¼
ðn kÞ weff
2
ð24Þ
3.2. Effective searching time and kinetics In Eqs. (23) and (24) we have introduced the effective time of the process, sd , which can be written as follows:
sd ¼ L2 =weff ; weff ¼ w1
wa wd þ w2 wd þ wa wd þ wa
s1D þ s3D ; w1 s1D þ w2 s3D
ð25Þ
Here L ¼ ðn kÞ is the initial distance to the damaged site (in the dimensionless units it is equal to the number of sites); it depends on the parameter c ¼ wd =wa given by the ratio of the dissociation and association rates. The physical meaning of sd is the characteristic time, at which the flux is changing; thus, it is the
sd , and the effective rate, weff , are
weff ¼ w1
s1D s3D þ w2 s1D þ s3D s1D þ s3D
ð27Þ
As compared to the purely 1D diffusion case where the searching time is s0 ¼ L2 =w1 , the time changes by the following factor:
sd 1 ¼ s0 s1Dsþ1Ds3D þ s1Dsþ3Ds3D
w2 w1
¼
1 1 þ s1Dsþ3Ds3D
w2 w1
1
ð28Þ
2 Thus, when w1 < w2 (3D diffusion is faster) so that w 1 > 0, w1 the effective searching time becomes shorter. Of course the size of this effect depends on the dissociation-association rates, which is described by the factor s1Dsþ3Ds3D in the denominator. The inverse Laplace transformation of Eq. (23) gives the following result for the flux, j1 ðtÞ, at long times:
1 2
rffiffiffiffiffiffiffiffi
s sd d exp 3 4t pt
ð29Þ
This is a function, which is equal to zero at t ¼ 0 (as initially there are no enzymes at the damaged site) going through a maximum at t max ¼ sd =6 and decaying as / t 3=2 at t ! 1. Thus, sd gives the characteristic time, at which the target site is reached. Interestingly, the mean searching time, t, is infinite for this power-law time dependence:
Z 0
This formula describes the kinetics of the searching process with weff being the effective jumping rate and sd being the searching time:
weff
sd ¼ L2
hti ¼ ð23Þ
ð26Þ
Then the searching time,
ð21Þ
which gives us that
~j1 ðsÞ
s1D ¼ 1=wd ; s3D ¼ 1=wa
j1 ðtÞ
When s1 ; s2 ud ; ua we obtain
X
effective searching time. It is worth noting that the integral of the flux over time is equal to 1, i.e., the particles do not escape by going to infinite distances and never coming back; thus, the flux indeed changes solely due to the specific binding. According to Eq. (25) at long times the motion is qualitatively 1D diffusion-like and a the effective jumping rate is given by the sum of two rates: w1 w wþw a d wd and w2 w þwa , which correspond to the effective jumping rates for d the first and second chain, respectively. Each rate has a very clear meaning: is given by the original jumping rate for the corresponding chain multiplied by the fraction of time that the enzyme resides on this chain. As it is often done [13,19,27], we can rewrite the results in terms of the durations of 1D and 3D diffusion stages, which we introduce as:
1
tj1 ðtÞdt ! 1
ð30Þ
This is a clear indication that due to the diffusion-like motion the time-behavior is considerably more complex than one would expect from the formal chemical kinetics. Diffusional ‘tails’ of the kinetics make the behavior different from exponential with a consequence that the moments of the process, e.g., hti, are not defined. The time dependence of the flux is illustrated in Fig. 2. When the time, sd , becomes shorter the flux, j1 ðtÞ, reaches its maximum earlier and the maximum is higher. This indicates that the enzyme finds its target faster. Here we do not show results for very large w2 (the highest w2 is 10w1 ) because it is difficult to show the corresponding kinetics on the same scale with that for the purely 1D-searching. For very large w2 rates the maximum shifts to even shorter times (equal to sd =6) and j1 ðtmax Þ becomes even higher. Interestingly, one can obtain the same results assuming continuous diffusion model and redefining some of the kinetic parameters. This is demonstrated in Appendix A. Thus, in this simplest case there is no principal difference between the discrete and continuous models. However, we would like to focus on the jumping model because it provides a rather straightforward extension of the results to the situation, where 3D diffusion is described by a distribution of jumping lengths.
18
K.L. Ivanov, N.N. Lukzen / Chemical Physics 435 (2014) 14–20
Let us now rewrite the expression for ½1 AðhÞ:
5
1 AðhÞ ¼
3
The full analytical treatment is still very cumbersome. However, ^ 11 ðzÞ as if one can write G
0
j1(t)
ð1 þ qÞð1 cos hÞ 1 þ q2 2q cos h
4
2
1 k0 cos h þ const 2w1 ðk1 cos hÞðk2 cos hÞ
1 0 0.0
0.5
1.0
t/
1.5
2.0
0
ð36Þ
ð37Þ
and obtain that one of the roots ki (for clarity, let it be k2 ) tends to one faster than the other, one can immediately find an approximate expression for the flux (similar to how it has been done to derive Eq. (21))
Fig. 2. Kinetics of diffusional search. Time dependence of the flux, j1 ðtÞ, shown for different values of w2 =w1 equal to 10 (solid line), 5 (dashed line) and 3 (dotted line). Results for the purely 1D-search are shown by a thin solid line. Here s1D ¼ s3D , the results are shown for a dimensionless quantity j1 ðtÞ s0 .
qffiffiffiffiffiffiffiffiffiffiffiffiffiffink pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nk ~j1 k2 k2 1 ð1 2ðY XÞÞ 2
3.3. Case of jumps to remote sites
The idea behind this derivation is the same as before; the quantities X and Y are also introduced in the same way. This procedure ^ 11 ðzÞ is indeed is valid, since for AðhÞ is given by Eq. (36) function G of the form given by Eq. (37). If we assume that s1 ; s2 ud ; ua 1 for the difference of X and Y we arrive at the expresion:
Let us assume that in the 3D-case the jumps are occurring not only between the adjacent sites but also over one site, over two sites, etc. To take this effect into account, we will assume that there is a distribution, PðkÞ, of probabilities for jumping from the i-th site to the ði kÞ-th sites. Hereafter we will always take a symmetric distribution, i.e., PðkÞ ¼ PðkÞ, the distribution will also be normalP ized 1 k¼1 PðkÞ ¼ 1. Let us modify the original set of equations (2) ~ i taking such distribution into account: for C ~i ¼ W ~ i1 þ C ~ iþ1 c UÞ b C c1 C ðs þ 2 W ~ i1 þ P1 C ~ iþ1 þ P 2 C ~ i2 þ P 2 C ~ iþ2 þ P 3 C ~ i3 þ P 3 C ~ iþ3 þ ... c 2 P1 C þW ~ n þ dik f k bC din K
ð31Þ
Here
c1 ¼ W
w1
0
0
0
;
c2 ¼ W
0
0
0 w2
ð32Þ
By applying the same mathematical procedure as before we obtain the equation for the GF:
1 c ^ c2 U b FðzÞ ¼ AFðzÞ W 1 2½AðzÞ 1 W s z2þ z ~n bC ¼ zk f 0 zn K
ð33Þ
P1 1
where AðzÞ ¼ 2 k¼1 PðkÞðzk þ 1=zk Þ. In the following analysis one can use expressions obtained above and substitute there w2 z 2 þ 1z by 2w2 ½AðzÞ 1. When z is substituted as eih , then P ^ AðzÞ becomes AðhÞ ¼ 1 k¼1 PðkÞ cosðkhÞ. The result for G11 ðhÞ then takes the form:
^ 11 ðhÞ ¼ G
s þ wa þ 2w2 ð1 cos hÞ ðs þ wd þ 2w1 ð1 cos hÞÞðs þ wa þ 2w2 ½1 AðhÞÞ wd wa ð34Þ
For an arbitrary AðhÞ function further analysis becomes problematic, at least, in the analytical form. Previously we have considered a special case of Pð1Þ ¼ 1, PðkÞ ¼ 0 ðk > 1Þ, which was relatively simple. To demonstrate effects of non-local jumps let us consider AðhÞ corresponding to PðkÞ – 0 for k > 1, in which analytical solution is still feasible. Such a case is given, for instance, by a situation where PðkÞ decays with the index k as the geometrical progression with q < 1, that is PðkÞ ¼ ð1 qÞqk1 . Then AðzÞ takes the form:
AðzÞ ¼
1q z 1=z þ 2 1 qz 1 q=z
ð35Þ
2ðY XÞ ¼
s1 ua þ s2 ud 1þq ua þ ð1qÞ 2 ud
ð38Þ
ð39Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi Thus, indeed k2 1 and 1= k22 1 has a singularity. Then one can use the same strategy as before and just redefine the searching time as
sd ¼ L2
wa þ wd s1D þ s3D ¼ L2 1þq 1þq w1 wa þ ð1qÞ w1 s1D þ ð1qÞ 2 w2 wd 2 w2 s3D
ð40Þ
For q ¼ 0 (jumps only between the adjacent sites) the result the same as previously. However, when q ! 1 (jumps to remote sites are possible) and w2 is non-zero the searching time formally tends to 0. Thus, when the jumps between remote sites are possible the searching time can be dramatically shortened. This situation corresponds, for instance, to long-range hopping along the DNA fragment or inter-segment transfer of the enzyme [4]. Of course the result sd ! 0 at q ! 1 is misinterpretation as the approximations made in order to obtain the flux, jðtÞ, are limited to relatively long times, i.e., t > T; the searching n times o cannot be shorter than T. Thus, in this case sd ¼ max w1a ; w1 . Nevertheless, increase of q d clearly leads to the reduction of the searching time. One can interpret Eq. (40) in terms of the effective rate of 1D-jumps:
weff ¼
L2
sd
¼ w1
¼ w1
wa 1þq wd þ w2 2 w þw wd þ wa a d ð1 qÞ
s1D 1þq s3D þ w2 s1D þ s3D ð1 qÞ2 s1D þ s3D
ð41Þ
As compared to Eq. (25) the w2 rate has increased by a factor > 1; this results in a further increase of the weff rate and, consequently, in shortening of the searching time.
1þq ð1qÞ2
4. Concluding remarks An analytical solution for the time of searching for a target site has been obtained for the case of a combination of two types of stochastic motion. The first motion was always treated as one-dimensional diffusion (only jumps to the adjacent sites were allowed); for the second motion an arbitrary probability distribution of the jump length has been assumed. For the simplest situation of a 1D-like second motion with a faster jumping rate we have obtained an analytical expression for the Laplace transform of the searching kinetics and an approximate expression for the kinetics
19
K.L. Ivanov, N.N. Lukzen / Chemical Physics 435 (2014) 14–20
g~ðR þ x0 jx0 ; sÞ ; g~ðR þ x0 jR þ x0 ; sÞ
in the time domain. Effective time of searching has been derived; we argue that the second motional process with a faster rate facilitates the search. It is shown that the kinetics strongly deviates from exponential and follows a t 3=2 power-law, which is a consequence of the diffusion type of motion. We have shown that when the jumps are allowed not only to the adjacent sites the search is facilitated even further. In general, our treatment can be useful for describing kinetics of diffusional search for targets by motions of two types; an important example of such process is searching for the specific binding site on the DNA molecule.
~j1 ðsÞ
Acknowledgements
Here the delta-like initial condition for the Green function is taken into account by the term dðx x0 Þ. To get rid of the second derivatives let us perform the Fourier transformation in the coordinate domain. These yields the following two equations for the Fou~F ðkjx0 ; sÞ: rier-transformed quantities, g~F ðkjx0 ; sÞ and h
This work has been supported by the Ministry of Education and Science of the Russian Federation, the Russian Foundation for Basic Research, Russia (Grants Nos. 11-03-00356 and 12-03-33082), SB RAS (project No. 71) and grant of the President of RF MD3279.2014.2. We are thankful to Dr. V.V. Koval (Novosibirsk) for stimulating discussions and to Prof. V.A. Avetisov (Moscow) and Dr. D.A. Grebenkov (Paris) for providing useful literature references.
C 1 ðxÞ
C 2 ðxÞ
(
ðA7Þ
2 ~F ðkjx0 ; sÞ þ 1 expðikx0 Þ ðs þ D1 k þ wd Þg~F ðkjx0 ; sÞ ¼ wa h 2p 2 ~ ðkjx ; sÞ ¼ w g~ ðkjx ; sÞ ðs þ D k þ w Þh 2
a
F
0
d F
ðA8Þ
0
These are two linear algebraic equations, which are easy to ~F ðkjx0 ; sÞ: solve and to find g
expðikx0 Þ s þ D2 k þ wa 2 2 2p ðs þ D1 k þ wd Þðs þ D2 k þ wa Þ wd wa ðA9Þ
To evaluate the flux one should perform the inverse Fourier transformation, in our case, it is sufficient to perform only the cosine-transformation. To do so, let us expand both the numerator 2 and denominator of g~F ðkjx0 ; sÞ as polynomials of k and then evaluate the Fourier integrals by using the formula:
Z
1
dk
cosðkxÞ
1
2
k þa
pffiffiffi p ¼ pffiffiffi expðx aÞ a
ðA10Þ
The Fourier original is then as follows:
g~ðx þ x0 jx0 ; sÞ ¼
ðA2Þ
where
b ¼ D1 0 ; U b ¼ wd wa ; V b ¼ v ðxÞ 0 ; C0 ¼ 1 D 0 D2 0 0 0 wd wa ðA3Þ
These matrices describe diffusion (along the two chains), dissociation-association, specific binding occurring with a positiondependent rate, v ðxÞ, and the initial condition given by the term containing the delta-function. As previously, we assume that specific binding can only occur in a narrow zone, i.e., we will take the following v ðxÞ rate
v ðxÞ ¼ ksb dðx R x0 Þ
8 2 ~ > < s D1 d 2 þ wd g~ðxjx0 ; sÞ ¼ wa hðxjx 0 ; sÞ þ dðx x0 Þ dx 2 > d ~ : ~ s D2 dx 2 þ wa hðxjx0 ; sÞ ¼ wd g ðxjx0 ; sÞ
ðA1Þ
with the two components being the probability densities to find the enzyme with the coordinate x on the two chains. Equation for CðxÞ in the Laplace domain can be written as
! 2 d ~ sÞ þ dðx x0 ÞC0 ~ sÞ ¼ V b b b ðxÞCðx; s D 2 U Cðx; dx
Here we have introduced the Green function for the continuous diffusion case. To calculate its [1, 1]-element one should solve two ^ g ¼ ðGÞ ^ coupled equations for two matrix elements of G, 11 and ^ . These two equations are h ¼ ðGÞ 21
g~F ðkjx0 ; sÞ ¼
One can also assume that the enzyme motion is continuous, with the diffusion coefficients being equal to D1 and D2 for the first and second chain, respectively. The enzyme starts searching from the point with the coordinate x0 at the first chain; when it reaches the coordinate ðR þ x0 Þ (on the first chain as well) specific binding occurs immediately. The distance R corresponds to L links of the DNA chain, each being of the length l0 : R ¼ L l0 . Dissociationassociation can be introduced in the same way as in the model of discrete jumps. To write down the kinetic equations let us introduce a vector
ðA6Þ
2
Appendix A. Continuous diffusion model
CðxÞ ¼
^ where g ¼ ðGÞ 11
ðA4Þ
pffiffiffiffiffi pffiffiffiffiffi 1 a1 a0 expðx a1 Þ a2 a0 expðx a2 Þ pffiffiffiffiffi pffiffiffiffiffi þ a1 a2 2D1 a1 a2 a2 a1 ðA11Þ
Here
a0 ¼ s2c þ uac s1c þ s2c þ udc þ uac a1;2 ¼ 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðs1c þ s2c þ udc þ uac Þ2 4ðs1c s2c þ s1c uac þ s2c udc Þ ðA12Þ 2 For brevity we used the following notations:
s1c ¼
s ; D1
s2c ¼
s ; D2
udc ¼
wd ; D1
uac ¼
wa D2
ðA13Þ
To proceed further it is convenient to use a close analogy between the quantities ai used here and ki in the discrete case, see Eq. (19). When at small s one of the two terms in Eq. (A11), for clarity, the second term, has a singularity much simpler approximate expressions for ~j1 ðsÞ can be used. Detailed analysis shows that this is indeed the case therefore, as previously, assuming that s1c ; s2c udc ; uac we obtain:
This approximation (contact approximation) for the reactivity dramatically simplifies the derivation. The flux of specific binding takes the form
~j1 ðsÞ expðpffiffiffiffiffiffiffi ssd Þ
~jðsÞ ¼
The effective searching time can be thus found by redefining the kinetic parameters:
Z
~ sÞdx ¼ ksb C ~ 1 ðR þ x0 ; sÞ 1 ^ Cðx; VðxÞ 0
ðA5Þ
When the rate constant, ksb , is taken infinitely large (the process is diffusion-limited) one can obtain the following expression for j-component of interest (the result is an exact analogue of the previously derived expression, see Eq. (10)):
sd ¼ R2 =Deff ; Deff ¼ ¼ D1
ðA14Þ
D1 wa þ D2 wd wd þ wa
s1D s3D þ D2 s1D þ s3D s1D þ s3D
ðA15Þ
20
K.L. Ivanov, N.N. Lukzen / Chemical Physics 435 (2014) 14–20
Here the definition of the times s1D ; s3D remains the same as before as well as interpretation of Deff . Hence, the analysis of the searching time given in Section 3.2 is valid in the continuous case as well. The coincidence of the results 2
l2
2
l2
in the limit s1c l0 ; sc2 l0 1 is fully consistent: at times t D01 ; D02 discrete jumps are physically equivalent to continuous diffusion. Therefore the inverse Laplace transformation of j1 gives the following results at long times:
j1 ðtÞ
1 2
rffiffiffiffiffiffiffiffi
s sd d exp 3 4t pt
ðA16Þ
The effective searching time is as follows
s d ¼ R2
s1D þ s3D D1 s1D þ D2 s3D
ðA17Þ
As compared to the purely 1D case (the second chain is absent) where sd0 ¼ R2 =D1 the searching time changes by a factor:
sd ¼ sd0 1 þ
1 s3D s1D þs3D
D2 D1
1
References [1] M. von Smoluchowski, Z. Phys. Chem. 92 (1917) 129. [2] A.D. Riggs, S. Bourgeois, M. Cohn, J. Mol. Biol. 52 (1970) 401.
ðA18Þ
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