# Mechanisms of Protein Search for Targets on DNA: Theoretical Insights

Alexey A. Shvets  [  Maria P. Kochugaeva  [  Anatoly B. Kolomeisky
<EMAIL_ADDRESS>[

###### Abstract

Protein-DNA interactions are critical for the successful functioning of all
natural systems. The key role in these interactions is played by processes of
protein search for specific sites on DNA. Although it has been studied for
many years, only recently microscopic aspects of these processes became more
clear. In this work, we present a review on current theoretical understanding
of the molecular mechanisms of the protein target search. A comprehensive
discrete-state stochastic method to explain the dynamics of the protein search
phenomena is introduced and explained. Our theoretical approach utilizes a
first-passage analysis and it takes into account the most relevant physical-
chemical processes. It is able to describe many fascinating features of the
protein search, including unusually high effective association rates, high
selectivity and specificity, and the robustness in the presence of crowders
and sequence heterogeneity.

MIT]Institute for Medical Engineering and Science, Massachusetts Institute of
Technology; Cambridge, MA 02142, USA Yale] Department of Biomedical
Engineering and System Biology Institute Yale University West Haven, CT,
06516, USA Rice] Department of Chemistry, Department of Chemical and
Biomolecular Engineering, and Center for Theoretical Biological Physics, Rice
University, Houston, Texas 77005, USA

##  1 Introduction

Dynamical nature of underlying processes is what distinguishes the living
systems from other processes. 1, 2. Biological processes constantly involve
time-dependent fluxes of energy and materials, which makes them strongly
deviating from equilibrium as long as organisms are alive. This implies that
the concepts of equilibrium thermodynamics have limited applications for
biological systems, while the role of methods that study the dynamical
transformations is much more important 3. In this review, we present our
theoretical views on dynamic aspects of the protein-DNA interactions, which
dominate in biological systems. Our approach is based on explicit calculations
of dynamic properties via a first-passage probabilities analysis. The first-
passage ideas have been already widely utilized in studies of various complex
processes in Chemistry, Physics and Biology 4, 5. We employ these ideas in
developing a discrete-state stochastic framework for analyzing the dynamics of
protein search for specific targets on DNA.

It is known that the beginning of most biological processes is associated with
specific protein molecules binding to specific target sequences on DNA because
these events initiate the cascades of corresponding biochemical and
biophysical processes 1, 2, 3. For example, to activate or to repress a gene
the corresponding transcription factor proteins must bind first to the gene
promoter’s region 1, 2. This fundamental aspect of protein-DNA interactions
has been studied extensively by various experimental and theoretical methods
6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 35, 24, 28, 25, 31, 30, 20, 21,
22, 23, 36, 27, 33, 34, 37, 26, 18, 19, 29, 32, 38. A special attention was
devoted to understanding the dynamics of the protein search for specific
targets on DNA. Many ideas have been proposed and critically discussed, but
only recently a clear molecular picture of the underlying processes started to
emerge 11, 12, 17.

Large amount of experimental observations on protein search phenomena, which
mostly come from the single-molecule measurements, suggests that it is a
complex dynamic phenomenon which combines three-dimensional (in the bulk
solution) and one-dimensional (on the DNA chain) motions 9, 10, 11, 12, 16.
But the most paradoxical observation is that, although the protein molecules
spend most of the search time ($\geq$90-99$\%$) on the DNA chain where they
diffuse very slowly, they still can find the targets very fast, in some cases
much faster than the bulk diffusion would allow 10, 11, 12. For example, the
measured association rate for lac-repressor was $\sim 10^{10}M^{-1}s^{-1}$
(two orders of magnitude faster than the diffusion limit!) 6, and many other
experimentally determined protein-DNA association rates were also
astonishingly high in comparison to typical biological binding rates. This is
known as a facilitated diffusion. Several theoretical ideas on the origin of
the facilitated diffusion, including lowering of dimensionality, electrostatic
effects, correlations between 3D and 1D motions, conformational transitions,
bending fluctuations, and hydrodynamics effects have been explored and
discussed 10, 11, 12. However, theoretical analysis shows that none of these
mechanisms can fully explain the facilitated diffusion in the protein search
17. To understand the dynamic aspects of protein-DNA interactions, we
developed a discrete-state stochastic framework to take into account the most
relevant physical-chemical processes in the system. The application of the
first-passage probabilities method allows us also to explicitly evaluate the
dynamic properties and to clarify dynamic aspects of the protein-DNA
interactions.

It is important to note that although there are still different opinions on
the theoretical foundations of the protein search phenomena, in this work we
mostly present our views on these problems, which, of course, are subjective.
In addition, there are many theoretical advances in our understanding of the
protein search dynamics, but we will concentrate only on few of them in order
to explain better the underlying molecular processes. Furthermore, there is a
huge number of investigations on the protein target search phenomena. Our goal
is not to cover all studies and all existing views but to present a clear
theoretical picture of these processes as we understand it now.

##  2 Simplest Discrete-State Stochastic Model of the Protein Target Search

Experiments clearly indicate that during the search the protein molecule is
alternating between freely diffusing behavior in the solution around the DNA
chain and non-specific associations to DNA, which also include scanning the
DNA chain 10, 11, 12. The process is completed when the protein molecule
reaches the specific target sequence on DNA for the first time. Stimulated by
this observations, we start with a simplest minimal model of the protein
search as presented in Figure 1. It is important to note that, in contrast to
other theoretical approaches 10, 11, 32, 15, this method is based on a
discrete-state stochastic description of the system. This is a more realistic
view of early stages of protein-DNA interactions because of intrinsically
discrete nature of molecular interactions in these systems.

Figure 1: A schematic view of a minimal discrete-state stochastic model of the
protein search for targets om DNA. The DNA chain has $L-1$ non-specific
binding sites and one specific target site. A protein molecule can diffuse
along the DNA segment with a rate $u$ in both directions. It can also
associate to DNA from the bulk solution (labeled as state $0$) with a rate
$k_{on}$ or it can dissociate back to the solution with a rate $k_{off}$. The
search is finished when the protein binds to the target site at the position
$m$ for the first time.

In this simple model, we consider a single protein molecule and a single DNA
molecule with a single target site: see Figure 1. The DNA chain is viewed as
having $L$ discrete binding sites, and one of them at the position $m$ is
considered to be the target for the protein molecule. Because the diffusion of
the proteins in the bulk is usually fast, all solutions states for the protein
are combined into one state that we label as a state $0$ (Figure 1). It is
assumed that from the bulk solution the protein molecule can bind with equal
probability to any site on DNA, and the total association rate to DNA is equal
to $k_{on}$, while the dissociation rate from DNA is $k_{off}$. The non-
specifically bound proteins can diffuse without bias along the DNA contour in
any direction with a rate $u$ (see Figure 1). Since the search process ends as
soon as the protein molecule arrives to the specific site for the first time,
we introduce a function $F_{n}(t)$, which is defined as a probability density
function of reaching the site $m$ (the target site) for the first time at time
$t$ if at $t=0$ the protein started in the state $n$ ($n=0$ is the bulk
solution, and $n=1,...,L$ are the protein-DNA bound states). This function is
also known as a first-passage probability density function 4, 5. To compute
these first-passage probabilities, we utilize backward master equations that
describe the temporal evolution of these quantities 4, 5, 17,

$\frac{dF_{n}(t)}{dt}=u\left[F_{n+1}(t)+F_{n-1}(t)\right]+k_{off}F_{0}(t)-(2u+k_{off})F_{n}(t),$
(1)

for $2\leq n\leq L-1$, while at the boundaries ($n=1$ or $n=L$) we have

$\frac{dF_{1}(t)}{dt}=uF_{2}(t)+k_{off}F_{0}(t)-(u+k_{off})F_{1}(t),$ (2)

and

$\frac{dF_{L}(t)}{dt}=uF_{L-1}(t)+k_{off}F_{0}(t)-(u+k_{off})F_{L}(t).$ (3)

For the state $n=0$, the backward master equation is different,

$\frac{dF_{0}(t)}{dt}=\frac{k_{on}}{L}\sum_{n=1}^{L}F_{n}(t)-k_{on}F_{n}(t).$
(4)

Here we used the fact that the rate to bind to any site on DNA is $k_{on}/L$,
so that the total association rate is equal to $k_{on}$. In addition, the
initial conditions require that $F_{m}(t)=\delta(t)$ and $F_{n\neq m}(t=0)=0$.
This means that if the protein molecule starts at the target site $m$ the
search is immediately accomplished.

It is important to explain the physical meaning of the backward master
equations because they differ from classical forward master equations widely
employed in Chemical Kinetics. It can be easily seen that all trajectories
that start at the state $n$ and finish at the target site $m$ can be divided
into several groups. For example, for $2\leq n\leq L-1$ all trajectories
starting at $n$ can be divided into three groups: 1) passing via the state
$n-1$, 2) passing via the state $n+1$ or 3) passing via the state $0$ in the
next time step. The fractions of those trajectories are given by
$u/(2u+k_{off})$, $u/(2u+k_{off})$ and $k_{off}/(2u+k_{off})$, respectively.
Equation (1) describes this partition of the trajectories in the time-
dependent manner because the first-passage probability flux to the target is
determined by these trajectories. Thus, the backward master equations reflect
the temporal evolution of the first-passage probabilities.

The most convenient way to analyze the dynamics in the system is to use
Laplace representations of the first-passage probability functions,
$\widetilde{F_{n}(s)}\equiv\int_{0}^{\infty}e^{-st}F_{n}(t)dt$. Then Equations
(1),(2), (3) and (4) can be written as simpler algebraic expressions:

$(s+2u+k_{off})\widetilde{F_{n}(s)}=u\left[\widetilde{F_{n+1}(s)}+\widetilde{F_{n-1}(s)}\right]+k_{off}\widetilde{F_{0}(s)};$
(5)
$(s+u+k_{off})\widetilde{F_{1}(s)}=u\widetilde{F_{2}(s)}+k_{off}\widetilde{F_{0}(s)};$
(6)
$(s+u+k_{off})\widetilde{F_{L}(s)}=u\widetilde{F_{L-1}(s)}+k_{off}\widetilde{F_{0}(s)};$
(7)
$(s+k_{on})\widetilde{F_{0}(s)}=\frac{k_{on}}{L}\sum_{n=1}^{L}\widetilde{F_{n}(s)}.$
(8)

In addition, from the initial conditions we have $\widetilde{F_{m}(s)}=1$.
These equations are solved assuming that the general form of the solution is
$\widetilde{F_{n}(s)}=Ay^{n}+B$, where the unknown coefficients $A$, $y$ and
$B$ are determined from the initial and boundary conditions 17. One could
argue that the target site $m$ divides the DNA molecule into two homogeneous
segments ($1\leq n\leq m$ and $m\leq n\leq L$), which can be considered
separately. It was shown 17 that this approach leads to explicit expressions
for the first-passage probability functions. Specifically, one obtains

$\widetilde{F_{0}(s)}=\frac{k_{on}(k_{off}+s)S_{1}(s)}{Ls(k_{off}+k_{on}+s)+k_{off}k_{on}S_{1}(s)},$
(9)

with an auxiliary function $S_{1}(s)$ defined as

$S_{1}(s)=\frac{y(1+y)(y^{-L}-y^{L})}{(1-y)(y^{1-m}+y^{m})(y^{m-L}+y^{1+L-m})};$
(10)

and with the parameters $y$ and $B$ given by

$y=\frac{s+2u+k_{off}-\sqrt{(s+2u+k_{off})^{2}-4u^{2}}}{2u};$ (11)
$B=\frac{k_{off}\widetilde{F_{0}(s)}}{(k_{off}+s)}.$ (12)

Explicit expressions for the first-passage probabilities provide a full
dynamic description of the protein search processes and any relevant
quantities can be easily computed. For example, the mean search time from the
bulk solution, which is inversely proportional to the chemical association
rate for the specific target site, can be found from17,

$T_{0}\equiv-\frac{\partial\widetilde{F_{0}(s)}}{\partial
s}\bigg{|}_{s=0}=\frac{1}{k_{on}}\frac{L}{S_{1}(0)}+\frac{1}{k_{off}}\frac{L-S_{1}(0)}{S_{1}(0)}.$
(13)

This result has a very clear physical meaning. Here the parameter $S_{1}(0)$
describes the average number of distinct sites that the protein molecule scans
during each visit to DNA while searching for the single specific site. Then,
on average, to find the target the protein must make $L/S_{1}(0)$ visits to
DNA because during every association $S_{1}(0)$ DNA sites are checked. Each
visit, on average, lasts $1/k_{on}$ while the protein scans for the target
diffusing along the DNA chain. The protein also makes $L/S_{1}(0)-1$
dissociations back into the solution. The number of dissociation events is
smaller by one than the number of association events because the last binding
to DNA leads to finding the specific site.

The results of our calculations for the mean search times are presented in
Figure 2. Our main finding here is that there are three dynamic search regimes
depending on the values of kinetic parameters. It is convenient to introduce
here a scanning length $\lambda=\sqrt{u/k_{off}}$, which gives the average
distance that the protein molecule travels on DNA during each search cycle.
This quantity is related to the parameter $S_{1}(0)$, but it is not the same
because the protein might visit the same sites several times. If the protein
molecule has a strong affinity to bind non-specifically to the DNA molecule
(small $k_{off}$, $\lambda>L$), then there will be only one searching cycle.
After binding to DNA the protein will not dissociate until it finds the
target. In this case, the mean search time scales as $\sim L^{2}$ because the
DNA-bound protein does a simple unbiased random walk. We call this dynamic
phase a random-walk regime. Because of the redundancy of the random walk the
search in this regime should be generally slow: many sites are repeatedly
visited. In the opposite limit of weak attractions between DNA and protein
molecules (large $k_{off}$, $\lambda<1$), the protein can bind to DNA but it
cannot slide because it quickly dissociates back into the solution. The
protein on average makes $L$ searching cycles ($T_{0}\sim L$). This dynamic
regime is called a jumping regime. The search in this regime is generally fast
as long as the associations are also fast. The most interesting behavior is
observed for the intermediate interactions, which we label as a sliding
regime. Here the scanning length $\lambda$ is larger than one but smaller than
the length of DNA $L$, and the number of searching cycles is also proportional
to $L$. But in this regime the system can reach the most optimal dynamic
behavior with the smallest search times. This search facilitation is achieved
due to the fact that the fluxes to the target are coming now from both the
bulk solution and from the DNA chain. This is one of the main mechanisms of
the facilitated diffusion of proteins during the target search, but other
processes like inter-segment transfer might also contribute significantly in
the facilitated diffusion 27.

Figure 2: Mean search times as a function of the scanning length parameter
$\lambda=\sqrt{u/k_{off}}$. The parameters utilized in calculations are:
$L=10^{3}$ bp, $u=k_{on}=10^{5}$ $s^{-1}$, and $m=L/2$. The transition rate
$k_{off}$ is varied to change $\lambda$.

##  3 The Effect of Multiple Targets and Traps

The advantage of the discrete-state stochastic framework with the first-
passage analysis presented above is that it can be extended and generalized to
more realistic biological situations. This allows us to investigate important
questions related to the mechanisms of the protein target search on DNA. Let
us present several specific examples, although many more results have been
obtained.17, 20, 21, 22, 23, 29, 27, 24, 28, 18, 19, 26, 25 We start with the
problem of how the presence of multiple target sites or multiple semi-specific
trap sites affect the dynamics of the protein search.

It is known that in eukaryotic cells multiple target sites are available on
the accessible DNA fragments 2, 1, 3, 40. The protein search is accomplished
in these systems when the protein molecules finds for the first time any of
the target sites. It has been argued that the mean search time in this system
might not decrease proportionally to the number of targets as one would
naively expect from simple-minded applications of chemical kinetics 18. This
is due to the complex mechanism of the protein search that involves both 3D
and 1D motions 18. Applying our discrete-state stochastic framework to this
problem, we consider a model with multiple targets at arbitrary locations as
presented in Figure 3. To describe the search dynamics in this system, we
again introduce the first-passage probability function $F_{n}(t)$ of finding
any of the targets at time $t$ if the process started at $t=0$ at the site
$n$. Targets are dividing the DNA chain into several homogeneous segments, and
this allows us to solve the corresponding backward master equations as
explained in Section 2. This leads to the following explicit expression for
the mean search time for any number of targets 18,

$T_{0}=\frac{1}{k_{on}}\frac{L}{S_{i}(0)}+\frac{1}{k_{off}}\frac{L-S_{i}(0)}{S{i}(0)},$
(14)

with a function $S_{i}(0)$ describing the average number of distinct sites
scanned by the protein on DNA with $i$ targets. This formula is a
generalization of Equation (13) when there is only one target ($i=1$).
Specific expressions for $S_{i}(0)$ for various numbers of randomly
distributed targets have been obtained 18. For example, for $i=2$ it was shown
that

$S_{2}(s)=\frac{(1+y)\left[2(1-y^{2L+m_{1}-m_{2}})+(1-y^{m_{2}-m_{1}})(y^{2m_{1}-1}+y^{1+2(L-m_{2})})\right]}{(1-y)(1+y^{2m_{1}-1})(1+y^{1+2(L-m_{2})})(1+y^{m_{2}-m_{1}})},$
(15)

where the parameter $y$ is given in Equation 11.

Figure 3: A schematic view of the discrete-state stochastic model of the
protein search with multiple specific sites. Targets are located at the sites
$m_{1}$ and $m_{2}$.

To understand the effect of multiple targets on the protein search dynamics,
we analyze the results of explicit calculations for mean search times as
presented in Figure 4. It is found that the presence of multiple targets does
not affect the overall dynamic phase diagram as compared with the single-
target case: three search regimes are again observed depending on the size of
the scanning length, the target size and the size of the DNA segment.
Generally, the search is faster in the multiple-target systems. However,
surprisingly, increasing the number of specific sites might not always
accelerate the search. To quantify this effect, we introduced an acceleration
parameter, $a_{n}=T_{0}(1)/T_{0}(n)$, where $T_{0}(n)$ is the mean search for
the system with $n$ targets. This ratio gives a numerical value of how faster
the search is in the presence of $n$ targets in comparison with the single-
target system. It is illustrated in Figure 5. One can see that there is a
range of parameters when the search dynamics in the system with two targets
can be slower than the dynamics in the system with one target. This happens in
the effectively 1D search regime (random-walk dynamic phase) when the single
target is located in the middle of the DNA chain, while two targets are close
to each other and located near one of the ends of the DNA segment. In this
case, for the protein molecule the two targets are viewed as effectively a
single target site (with the size equal to two target sites) because they are
so close to each other. But it is faster to find the target located in the
middle of the chain than the target positioned near the ends.17 This is the
main reason why having multiple targets does not always lead to decrease in
the search times. Thus, our theoretical analysis predicts that the degree of
acceleration due to the presence of multiple targets depends on the nature of
the dynamic search phase and on the location of the specific sites with
respect to each other and with respect to the middle point of DNA 18.

[a]
[b]
[c]

Figure 4: Dynamic phase diagrams for the protein search on DNA with one target
at the position $m$, with two targets at the positions $m_{1}$ and $m_{2}$ and
with the target and the trap at the positions $m_{1}$ and $m_{2}$,
respectively. Parameters used for calculations are: $k_{on}=u=10^{5}$ s-1 and
$L=10000$. a) $m=L/2$, $m_{1}=L/4$ and $m_{2}=3L/4$; b) $m=L/4$, $m_{1}=L/4$
and $m_{2}=L/2$; and c) $m=L/2$, $m_{1}=L/2$ and $m_{2}=L$. Adapted with
permission from Ref. 19.

Figure 5: Ratio of the mean search times as a function of the normalized
distance between the targets for single-target and two-target systems ($l$ is
the distance between between targets, $L$ is the DNA length). The single
target is in the middle of the chain. In the two-target system, one of the
specific sites is fixed at the end and the position of the second one is
varied. The parameters used in calculations are: $u=k_{on}=10^{6}$ s-1;
$k_{off}=10^{-4}$ s-1; and $L=10000$. Adapted with permission from Ref. 18.

Another important factor that might affect the protein search dynamics is the
existence of so-called semi-specific sites, or decoys, on DNA. These sites
have a chemical composition very similar to the specific targets with
differences in only one or few nucleotides. The protein molecule can be
trapped in these sites, and this should influence the search for real targets.
To analyze this effect, we can extend the simplest model to include the
possibility of traps, assuming that associations to these semi-specific sites
are effectively irreversible 19. This assumption is reasonable because the
search times in many systems are relatively short and the experimental
observations also limited in time. Thus the bindings to decoys can be viewed
as effectively irreversible. The first-passage analysis can be applied here,
but we have to notice that only a fraction of trajectories will reach the
correct target site. Then the main quantity of our calculations, the first-
passage probability function $F_{n}(t)$, is now a conditional probability for
the protein molecules not captured by the trap to find the target site.

Let us consider a system consisting of a single target at the site $m_{1}$ and
a single trap at the site $m_{2}$ on the DNA molecule with $L$ sites 19. The
scheme presented in Figure 3 is also a correct representation of this system
with the correction that instead of the second target there is a trap in the
site $m_{2}$, and the successful search corresponds to the protein molecule
finding the specific site $m_{1}$. Following our theoretical method, the
corresponding backward master equations can be solved and they yield the
Laplace transform of the first-passage probability function to find the target
if the protein starts from the bulk solution 19,

$\widetilde{F_{0}(s)}=\frac{k_{on}(k_{off}+s)S_{0}(s)}{Ls(k_{off}+k_{on}+s)+k_{off}k_{on}S_{2}(s)},$
(16)

with

$S_{0}(s)=\frac{(1+y)(1-y^{m_{1}+m_{2}-1})}{(1-y)(1+y^{2m_{1}-1})(1+y^{m_{1}-m_{2}})},$
(17)

and the parameters $y$ and $S_{2}$ given in Equations (11) and (15),
respectively. This allows us to evaluate all dynamic properties in the system
and to test the effect of traps.

The probability to reach the target (i.e., the fraction of the successful
trajectories) is now given by a so-called splitting probability function 4, 5,

$\Pi\equiv\widetilde{F_{0}(s=0)}=\frac{S_{0}(0)}{S_{2}(0)}.$ (18)

The mean search time, which is the conditional mean first-passage time to
reach the target, can be estimated by averaging over the successful
trajectories, producing

$T_{0}\equiv-\frac{\frac{\partial\widetilde{F_{0}(s)}}{\partial
s}\bigg{|}_{s=0}}{\Pi}=\frac{1}{k_{on}}\frac{L}{S_{2}(0)}+\frac{1}{k_{off}}\frac{L-S_{2}(0)}{S_{2}(0)}+\Pi\frac{\partial}{\partial
s}\left[\frac{S_{2}(s)}{S_{0}(s)}\right]\bigg{|}_{s=0}.$ (19)

Let us analyze this expression. On the left side, the division by the
splitting probability emphasizes the fact that this is the conditional mean
search time. It is also interesting to note that the first two terms on the
right side of the equation is exactly the mean search time for the system with
two targets and no traps (at the sites $m_{1}$ and $m_{2}$) as we discussed
above 18, while the third term is a correction which accounts for the fact
that the site at $m_{2}$ is actually the trap. The main reason for this is the
observation that the sites $m_{1}$ and $m_{2}$ are special locations where all
trajectories are end up in both systems, with two targets and with the target
and the trap. For the two-target case the mean search times are averaged over
all trajectories to both sites, while for the target and the trap system the
mean search times are obtained only by considering the trajectories finishing
at the target 19.

The results of calculations for the dynamic properties of the protein search
in the presence of traps are presented in Figures 4 and 6. Again, three
dynamic search phases are observed, but adding the trap generally facilitates
the search dynamics, which is a counter-intuitive result: see Figure 4.
However, this acceleration (in comparison with the single-target system) is
always associated with lowering of the probability of reaching the specific
target, as shown in Figure 6. This means that the protein molecules might
reach the target faster in the presence of the traps, but the fraction of such
events is decreasing. In addition, the search dynamics is sensitive to the
nature of the dynamic phase. The strongest effect due to the presence of the
trap is observed in the effective 1D random-walk regime (because it has only
one searching cycle) where the locations of the target and the trap strongly
influence the search. In other dynamic regimes, the effect is smaller.

Figure 6: Probability to reach the target as a function of the scanning length
for different distributions of the target and trap sites. Parameters used for
calculations are: $k_{on}=u=10^{5}$ s-1, $L=10000$ and $k_{off}$ is changing.
Symbols are from Monte Carlo computer simulations. Adapted with permission
from Ref. 19

##  4 Sequence heterogeneity

Real DNA molecules are heterogeneous polymers consisting of several types of
subunits. This means that the interactions between protein and DNA molecules
depend on the DNA sequence at the location where they meet. It is reasonable
to expect that this sequence dependence in the interaction strength should
affect the protein search dynamics because the diffusion rate for the non-
specifically bound proteins will be position-dependent 3, 11, 41. Similarly,
association and dissociation rates should also depend on the location of the
protein molecule on DNA. In addition, recent theoretical investigations
suggested that different DNA sequence symmetries might lead to additional
effective interactions between protein and DNA molecules 43, 44, 45, 46. The
discrete-state stochastic framework with the first-passage analysis is a
convenient tool to investigate the effect of DNA sequence heterogeneity and
symmetry on the protein search dynamics 20.

Our goal here is clarify the molecular origin of how the sequence
heterogeneity influences the protein target search. We assume here a
simplified picture of DNA, in which each monomer can be one of two chemical
species, $A$ or $B$, as presented in Figure 7 20. When the protein is bound to
the subunit $A$ ($B$), it interacts with energy $\varepsilon_{A}$
($\varepsilon_{B}$), and the difference between interaction energies is given
by a parameter $\varepsilon=\varepsilon_{A}-\varepsilon_{B}\geq 0$. This means
that the protein attracts stronger to the $B$ sites than to the $A$ sites. The
protein molecule can diffuse along DNA with a rate $u_{A}\equiv u$ or
$u_{B}=ue^{-\varepsilon}$, where $\varepsilon$ is measured in $k_{B}T$ units.
This reflects the assumption that if the protein interacts stronger with the
DNA at given location then it will move out of this site slower. In addition,
we assume that, independently of the chemical nature of the neighboring sites,
sliding out of the sites $A$ is characterized by the rate $u_{A}$, while the
diffusion out of the sites $B$ is given by $u_{B}$. From the bulk solution the
protein might associate to any site $A$ or $B$ on DNA with the corresponding
rates $k^{A}_{on}=k_{on}$ or $k^{B}_{on}=k_{on}e^{-\theta\varepsilon}$. Note
that for convenience the on-rates defined here as the rates per unit site, in
contrast to our definitions in the previous sections. Similarly, the
dissociations from the DNA chain are described by the rates
$k^{A}_{off}=k_{off}$ and $k^{B}_{off}=k_{off}e^{(\theta-1)\varepsilon}$.
Here, the parameter $0\leq\theta\leq 1$ specifies how the protein-DNA
interaction energy is distributed between the association and dissociation
transitions 20. The physical meaning of this parameter is that the protein
molecule tends to bind faster and to dissociate slower from the stronger
attracting sites $B$, as compared with the weaker attracting $A$ sites. The
parameter $\theta$ accounts for these effects.

Figure 7: A simplified view on the protein search on DNA with two different
types of subunits, $A$ and $B$. a) A general scheme; b) DNA is viewed as a
symmetric block copolymer with the target in the middle of the chain; c) DNA
is viewed as alternating copolymer with different compositions of the subunits
flanking the target in the middle of the chain. Adapted with permission from
Ref. 20

To quantify the role of sequence heterogeneity, we consider the DNA molecule
with a fixed chemical composition (the fractions of $A$ and $B$ monomers are
the same), but with different arrangements of subunits. Two limiting cases are
specifically analyzed. One of them views the DNA molecule as two homogeneous
segments of only $A$ and only $B$ subunits separated by the target in the
middle of the chain (Figure 7). Another one is the DNA chain with the
alternating $A$ and $B$ sites. The block copolymer has two homogeneous
sequence segments, while the alternating polymers are more heterogeneous. It
is important to note that in both cases, the overall interaction between the
protein and DNA is the same (because the overall chemical composition in both
cases is identical), and thus our analysis probes only the effect of the
heterogeneity and symmetry in the subunit positions, in contrast to other
theoretical treatments 42.

Figure 8: The ratio of the mean search times for the alternating DNA sequences
and for the block copolymer DNA sequences as a function of the scanning length
$\lambda=\sqrt{u/k_{off}}$. Three different chemical compositions near the
target ($T$) are distinguished, namely, $ATA$, $ATB$, $BTB$. The transition
rates are $u=10^{5}\,s^{-1}$ and $k_{on}=0.1\,s^{-1}$. The DNA length is
$L=1000$, the loading parameter is $\theta=0.5$, and the energy difference of
interactions for the protein with $A$ and $B$ sites is $\varepsilon=5$
$k_{B}T$. Adapted with permission from Ref. 20.

Applying again the first-passage approach and solving the corresponding
equations leads to the explicit expressions for mean search times for all
situations shown in Figure 7 20. For example, for the block copolymer DNA
sequences, we obtain

$T_{0}=\frac{k_{off}+k_{on}\left[(L/2-P_{A})+e^{\varepsilon}(L/2-P_{B})\right]}{k_{on}k_{off}(1+P_{A}+e^{\theta\varepsilon}P_{B})},$
(20)

where

$P_{i}=\frac{x_{i}^{1-L/2}-x_{i}^{1+L/2}}{(1-x_{i})(x_{i}^{1+L/2}+x_{i}^{{}_{L}/2})},$
(21)
$x_{i}=\frac{2u_{i}+k_{off}^{(i)}-\sqrt{(2u_{i}+k_{off}^{(i)})^{2}-4u_{i}^{2}}}{2u_{i}},$
(22)

for $i=A$ or $B$. The expressions for the mean search time for alternating
sequences are quite bulky and can be found in Ref. 20.

The results of our calculations are presented in Figure 8, where the ratio of
the mean search times for the block copolymer and alternating sequences are
plotted. The analysis of this figure produces several interesting
observations. First, we see that three dynamic search regimes are also found
in this system and the effect of sequence heterogeneity on protein search
dynamics depends on the nature of the dynamic phase. In the jumping regime
when the protein does not slide along the DNA contour ($\lambda<1$), the
symmetry of the sequence does not play any role. This is because in this case
the process is taking place only via associations and dissociations (3D
search), and the structure of the DNA chain is not important. The situation is
different for the intermediate sliding regime (3D+1D search, $1<\lambda<L$)
where in most cases, the search on alternating sequences is faster. This can
be explained by noticing that the search time in this dynamic phase is
proportional to $L/\lambda$, which gives the average number of cycles before
the protein can find the target. In the block copolymer sequence, the protein
mostly comes to the target from the $B$ segment because of stronger
interactions with these sites, i.e, it comes from one side of the DNA
molecule. In the alternating sequences, the protein can reach the target from
both sides of DNA, and this lowers the overall search time. It can be shown
analytically that the scanning length on the alternating segment is larger
than the scanning length for the $B$ segment, i.e., $\lambda_{AB}>\lambda_{B}$
20. Then the search is faster for the alternating sequences because
$L/\lambda_{AB}<L/\lambda_{B}$, i.e., the number of searching cycles is lower
for the alternating sequences, which helps to find the target faster. The only
deviation from this picture is found for $ATA$ sequences, which corresponds to
having two $A$ sites around the target site, where for the small range of
parameters the search is slower than in the block copolymer sequence. This
effect can be explained by the fact that the protein does not sit at $A$ sites
for the long time and it moves quickly away, effectively increasing the
barrier to enter the target via DNA 20. Thus, our theory predicts that the
composition of the DNA flanking sites around the target sequences might affect
the dynamics of reaching them. It is interesting to note that recent
experiments are consistent with our theoretical predictions 47.

In the random-walk regime (1D search, $\lambda>L$), the effect of the sequence
heterogeneity is even stronger. The protein molecule finds the specific
binding site up to 2 times faster for more heterogeneous alternating DNA
sequences. To understand this behavior, we note that in this case the mean
first-passage time to reach the target is a sum of residence times on the DNA
sites since the protein will not dissociate until the target is located so
that all trajectories to the target are one-dimensional. Because the target is
in the middle of the chain, the mean time to reach the target from the block
copolymer sequence can be approximated as $T_{0}\simeq(L/4)\tau_{B}$, where
$\tau_{B}$ is the average residence time on any site $B$. The protein prefers
to start the search at any position on the $B$ segment with equal probability,
i.e., the distance to the target varies from 0 to $L/2$. Then, the average
starting position of the protein is $L/4$ sites away from the target. For the
alternating sequences, the average distance to the target is approximately the
same ($L/4$), but the chemical composition of intermediate sites on the path
to the target is different, yielding, $T_{0}\simeq(L/8)\tau_{A}+(L/8)\tau_{B}$
($\tau_{A}$ is the residence time on $A$ sites). The protein spends much less
time on $A$ subunits, and this leads to faster search for the alternating DNA
sequences. For $\tau_{A}\ll\tau_{B}$, this also explains the factor of 2 in
the search speed. In this case, the $B$ subunits can be viewed as effective
traps that slow down the search dynamics. Thus, our theoretical calculations
make surprising predictions that the sequence heterogeneity almost always lead
to faster protein search for targets on DNA despite the fact that it lowers
the effective protein-DNA binding affinity 43, 44, 45, 46. And the stronger
the contribution of the 1D search modes, the more relevant will be the effect
of sequence heterogeneity.

##  5 The Effect of Crowding on DNA in the Protein Target Search

Living cells are typically crowded with a large number of molecules, and many
of them are attached to the DNA chains 1, 2. This should prevent the fast
protein search for targets on DNA, and earlier theoretical studies supported
this prediction 49. However, surprisingly, experiments show that crowding on
DNA does not affect much the effectiveness of the protein target search 33,
34, and this was also found in MD simulations 48. By applying the discrete-
state stochastic approach, we were able to clarify the role of the crowding on
DNA in the protein target search.

To analyze this problem, the model illustrated in Figure 9 is considered.
There is a single DNA molecule with $L+1$ binding sites, and one of them is
the target (at the site $m$). On the DNA chain there is also a crowding
particle that can diffuse with a rate $u_{ob}$, but it cannot leave DNA. A
single protein molecule starts from the solution (state 0) and it can bind to
any site on DNA that is not occupied by the crowder with a rate $k_{on}$ (rate
per site). The bound protein molecule can diffuse with a rate $u$, and there
is an exclusion interaction between the protein and the crowder. Finally, the
protein molecule can dissociate from DNA to the bulk solution with a rate
$k_{off}$: see Figure 9.

Figure 9: A schematic view of the protein target search in the presence of a
moving obstacle on DNA. The crowding particle cannot dissociate from DNA,
while the protein molecule can dissociate into the solution, labeled as state
$0$, and return back to the DNA chain.

Investigating the model with the mobile crowding particle on DNA first using
Monte Carlo computer simulations, it is found that there are three search
regimes depending on the main length scales in the system. This is shown in
Figure 10 for the mean search times to find the target as a function the
scanning length $\lambda$. We can understand the complex dynamics in this
system using the following arguments. If the diffusion rate of the crowder is
much smaller than other rates ($u_{ob}\ll u$, $k_{on}$ and $k_{off}$), then
the protein molecule will find the target before the crowding particle can
move away from its original location. But we already explicitly solved the
problem of the protein target search with static obstacles using the same
discrete-state stochastic approach with the first-passage analysis 23. Then
the mean search time in the system with movable crowder can be approximated as
the average over all possible static locations of the crowding particle 21,
yielding

$\langle
T_{0}\rangle\simeq\frac{1}{L}\left(\sum_{l_{ob}=1}^{m-1}T_{ob}(l_{ob})+\sum_{l_{ob}=1}^{L-m}T_{ob}(l_{ob})\right),$
(23)

where

$T_{ob}=\frac{k_{off}+k_{on}(L-S_{ob}(0))}{k_{on}k_{off}S_{ob}(0)},$ (24)

is the mean search time with the static obstacle located at a distance
$l_{ob}$ from the target. An auxiliary function $S_{ob}$ is given by 23

$S_{ob}(s)=\frac{y(y^{-m}-y^{m})}{(1-y)(y^{m}+y^{1+m})}+\frac{y(1-y^{2l_{ob}-2})}{(1-y)(1+y^{2l_{ob}-1})}$
(25)

with the parameter $y$ specified in Equation 11.

This simple approximate theory works quite well in the dynamic regimes where
3D pathways are important for the search ($\lambda<L$). However, theoretical
arguments fail in the random-walk regime where 1D dynamics dominate the
search. These results are expected. The protein molecule that collides with
the crowding particle on DNA in dynamic regimes with 3D pathways will have the
opportunity to dissociate into the bulk solution and to avoid the blocking
effect. But in the random-walk regime (1D search) there is no such
opportunity, and the search times will definitely increase. Computer
simulations also indicate that the search times in this regime depend on the
diffusivity of the crowding particle. The search is faster for more mobile
crowders: see Figure 10.

Figure 10:  Mean search times to find the target in the system with a mobile
crowder on DNA. The DNA chain has $L=1000$ sites, and the target is in the
middle of the chain, $m=L/2$. Parameters used for calculations are
$k_{on}=0.1$ s-1, $u=10^{5}$ s-1 and variable $u_{ob}$. Solid curves
correspond to analytical results for DNA without obstacles and for DNA with a
static obstacle, which are averaged over all initial positions of the crowder.
Symbols correspond to Monte Carlo computer simulations. Dashed lines describe
the approximate theory, as explained in the text. Adapted with permission from
Ref. 21.

The dynamics in the random-walk regime can be explained using the following
arguments. The overall search can be viewed as consisting of two terms,

$\langle T_{0}^{ob}\rangle\simeq T_{0}+\langle T_{bl}\rangle,$ (26)

where $T_{0}$ is the search in the random-walk regime without any crowders,
and it is given in Equation 13. The second term is the average time it takes
for the crowder to diffuse away and clear the path for the protein to reach
the target without interference 21. It was shown that this blocking time
$T_{bl}$ depends on the location of the target and the diffusion rate of the
crowding particle $u_{ob}$ 21,

$\langle T_{bl}\rangle=\frac{m^{4}+(L-m)^{4}}{16u_{ob}(L^{2}+m^{2}-mL)}.$ (27)

This simple theoretical arguments show excellent agreement with Monte Carlo
computer simulations: see dashed lines in Figure 10. But more importantly,
they provide a clear molecular picture on the role of the crowding on DNA in
the protein target search. If the protein search is dominated by 1D pathways
and the mobility of the crowder is low the search dynamics will be
significantly slowed down. But if the search involves mostly 3D pathways and
the crowder is mobile the mean search times will not be affected much. It
seems that real biological systems operate in 3D+1D regime, and crowding
particles diffuse with the rates comparable to the searching proteins ($u\sim
u_{ob}$) 3. Then one might conclude that the effect of the crowders on DNA
should be minimal. This fully agrees with experimental observations and with
results from MD simulations 34, 48.

##  6 Conclusions and Future Directions

Although protein search for targets on DNA is a very complex phenomenon that
involves multiple biochemical and biophysical processes, significant advances
in our understanding of the underlying molecular mechanisms have been achieved
in recent years. A major role in this success is due to analysis of the
systems using the discrete-state stochastic framework supplemented by explicit
calculations via the first-passage probabilities method. In this review, we
presented and explained this theoretical approach by considering the protein
target search in various systems. It is important to emphasize that the main
advantage of our theoretical approach is the ability to obtain analytical
results that clarify the physics of the underlying processes. In addition, the
method can be easily extended in many directions, as shown in this work, as
well as in other cases which we did not discuss in this work, such as the role
of conformational transitions 24 and the effect of DNA loop formation in the
protein target search 23. Furthermore, our theoretical calculations using this
theoretical framework were successful in explaining the experimental
observations on homology search by RecA protein filaments 25 and the dynamics
of CRISPR genome interrogation 29.

Several important dynamic features of the protein search for targets on DNA
have been identified from theoretical analysis. It is found that the dynamic
phase diagram of the protein target search always shows thee dynamic regimes,
which are determined by the three relevant length scales in the system: the
size of DNA, the average scanning length of the non-specifically bound
proteins, and the size of the target sequence. Depending on the dynamic phase,
the search is dominated by the 3D motions (jumping regime), 1D motions
(random-walk regime) or a combination of 3D and 1D motions in the sliding
regime. The analysis shows that the most optimal search dynamics can be
achieved in the dynamic regime when the protein molecules explore both 1D and
3D pathways during the search. In this case, the protein can reach the target
by sliding from the DNA chain or by directly binding from the solutions.
Theoretical calculations also indicate that the presence of several target
sites influences the search dynamics differently depending on the locations of
the targets on DNA and distances between them. Surprising observations are
found in the system with semi-specific sites, which are viewed as effective
traps. It is shown that the search dynamics can be faster in this case, but it
comes with the price of lowering the yield of the protein molecules reaching
the target. We also investigated the effect of sequence heterogeneity and
symmetry in the protein search dynamics. Our calculations indicate that the
search is faster for more heterogeneous sequences, and the chemical
composition around the target is also an important factor in this process.
Furthermore, our method allowed us to probe the effect of crowding on DNA in
the protein target search. It is shown that it depends on the dynamic phase
and on the mobility of the crowding particles. The crowders influence the
protein search stronger when 1D pathways dominate and when the diffusivity of
the crowding particle is small enough so that the protein will be frequently
blocked during the process. Increasing the mobility of the crowders and/or
increasing the contribution of 3D search pathways lowers the effect of the
crowding. These theoretical arguments fully agree with experimental
observations and MD computer simulations.

Despite tremendous progress in theoretical understanding of the protein target
search phenomena, there are many questions remain on the molecular mechanisms
of these processes. It is still unclear what is the nature of protein-DNA
interactions in the regions surrounding the target sequences. Is the effective
potential created by these interactions drives the protein molecule to the
target like a funnel or is it completely random? How large is the size of the
flanking segments that affect the finding of the target? What is the role of
DNA topology in the protein target search? This is especially important for
proteins that have several binding sites for DNA which can form DNA loops and
other complex structures. Another interesting question is the role of various
DNA and protein conformations in these processes. It is clear that further
progress in understanding protein target search phenomena depends on combining
theoretical, computational and experimental methods.

A.B.K. acknowledges the support from the Welch Foundation (C-1559), from the
NSF (CHE-1664218) and from the Center for Theoretical Biological Physics
sponsored by the NSF (PHY-1427654).

## References

  * 1 Alberts, B. et al., Molecular Biology of Cell, 6th ed., Garland Science, New York, 2014.
  * 2 Lodish, H. et al., Molecular Cell Biology, 6th ed., W. H. Freeman, New York, 2007.
  * 3 Phillips, R.; Kondev, J.; Theriot, J. Physical Biology of the Cell, 2nd ed., Garland Science, New York, 2012.
  * 4 Van Kampen, N.G. Stochastic Processes in Physics and Chemistry, 3rd ed., North Holland, Amsterdam, 2007.
  * 5 Redner, S. A Guide to First-Passage Processes, Cambridge University Press, Cambridge, 2001.
  * 6 Riggs, A.D.; Bourgeois, S.; Cohn, M. The lac-represser-operator interaction: IIIKinetic studies. J.Mol. Biol. 1970, 53, 401-417.
  * 7 Berg, O.G.; Winter, R.B.; von Hippel, P.H. Diffusion-driven mechanisms of protein translocation on nucleic acids: I. Models and theory. Biochemistry 1981, 20, 6929-6948.
  * 8 Berg, O.G.; von Hippel, P.H. Diffusion-controlled macromolecular interactions. Annu. Rev. Biophys. Biophys. Chem. 1985, 14, 131-160.
  * 9 Gowers, D.M.; Wilson, G.G.; Halford, S.E. Measurements of the contributions of 1D and 3D pathways to the translocation of protein along DNA. Proc. Natl. Acad. Sci. USA 2005, 102, 15883-15888.
  * 10 Halford, S.E.; Marko, J.F. How do site-specific DNA-binding proteins find their targets? Nucl. Acids Res. 2004, 32, 3040-3052.
  * 11 Mirny, L.; Slutsky, M.; Wunderlich, Z.; Tafvizi, A.; Leith, J.; Kosmrlj, A. How a protein searches for its site on DNA: the mechanism of facilitated diffusion. J. Phys. A: Math. Theor. 2009, 42, 434019.
  * 12 Kolomeisky, A.B. Physics of protein-DNA interactions: Mechanisms of facilitated target search. Phys. Chem. Chem. Phys. 2011, 13, 2088-2095.
  * 13 Hu, T.; Grosberg, A.Y.; Shklovskii, B.I. How proteins search for their specific sites on DNA: the Role of DNA conformations. Biophys. J. 2006, 90, 2731-2744.
  * 14 Hu, L.; Grosberg, A.Y.; Bruinsma, R. Are DNA transcription factor proteins maxwellian demons? Biophys. J. 2008, 95, 1151-1156.
  * 15 Bauer, M.; Metzler, R. Generalized facilitated diffusion model for DNA-binding proteins with search and recognition states. Biophys. J. 2012, 102, 2321-2330.
  * 16 Sheinman, M.; Benichou, O.; Kafri, Y.; Voituriez, R. Classes of fast and specific search mechanisms for proteins on DNA. Rep. Progr. Phys. 2012, 75, 026601.
  * 17 Veksler, A.; Kolomeisky, A.B. Speed-selectivity paradox in the protein search for targets on DNA: Is it real or not? J. Phys. Chem. B 2013, 117, 12695-12701.
  * 18 Lange, M.; Kochugaeva, M.; Kolomeisky, A.B. Protein search for multiple targets on DNA. J. Chem. Phys. 2015, 143, 105102.
  * 19 Lange, M.; Kochugaeva, M.; Kolomeisky, A.B. Dynamics of the protein search for targets on DNA in the presence of traps. J. Phys. Chem. B 2015, 119, 12410-12416.
  * 20 Shvets, A.A.; Kolomeisky, A.B. Sequence heterogeneity accelerates protein search for targets on DNA. J. Chem. Phys. 2015 143, 245101.
  * 21 Shvets, A.A.; Kolomeisky, A.B. Crowding on DNA in protein search for targets. J. Phys. Chem. Lett. 2016, 7, 2502-2506.
  * 22 Shvets, A.A.; Kochugaeva, M.; Kolomeisky, A.B. The role of static and dynamic obstacles in the protein search for targets on DNA. J. Phys. Chem. B 2015, 120, 5802-5809.
  * 23 Shvets, A.A.; Kolomeisky, A.B. The role of DNA looping in the search for specific targtes on DNA by multisite proteins. J. Phys. Chem. Lett. 2016, 7, 5022-5027.
  * 24 Kochugaeva, M.P.; Shvets, A.A.; Kolomeisky, A.B. How conformational dynamics influences the protein search for targets on DNA. J. Phys. A: Math. Theor. 2016, 49, 444004.
  * 25 Kochugaeva, M.P.; Berezhkovskii, A.A.; Kolomeisky, A.B. Optimal length of conformational transitions region in the protein search for targets on DNA. J. Phys. Chem. Lett. 2017, 8, 4049-4054.
  * 26 Shin, J.; Kolomeisky, A.B. Surface-assisted dynamic search processes. J. Phys. Chem. B 2018, 122, 2243-2250.
  * 27 Esadze, A.; Kemme, C.A.; Kolomeisky, A.B.; Iwahara, J. Positive and negative impacts of nonspecific sites during targte location by a sequence-specific DNA-binding protein: Origin of the optimal search at physiological ionic strength. Nucl. Acids Res. 2014, 42, 7039-7046.
  * 28 Kochugaeva, M.P.; Shvets, A.A.; Kolomeisky, A.B. On the mechanism of homology search by ReacA protein filaments. Biophys. J. 2017 112, 859-867.
  * 29 Shvets, A.A.; Kolomeisky, A.B. Mechanism of genome interrogation: How CRISPR RNA-guided Cas9 proteins locate specific targets on DNA. Biophys. J. 2017, 112, 1416-1424.
  * 30 Tafvizi, A.; Huang, F.; Fersht, A.R.; Mirny, L.A.; van Oijen, A.M. A single-molecule characterization of p53 search on DNA. Proc. Natl. Acad. Sci. USA 2011, 108, 563-568.
  * 31 Slutsky, M.; Mirny, L.A. Kinetics of protein-DNA interaction: facilitated target location in sequence-dependent potential. Biophys. J. 2004, 87, 4021-4035.
  * 32 Benichou, O.; Kafri, Y., Sheinman, M.; Voituriez, R. Searching fast for a target on DNA without falling to traps. Phys. Rev. Lett. 2009, 103, 138102-138104.
  * 33 Hammar, P.; Leroy, P.; Mahmutovic, A.; Marklund, E.G.; Berg, O.G.; Elf, J. The lac Repressor displays facilitated Diffusion in Living Cells. Science 2012, 336, 1595-1598.
  * 34 Mahmutovic, A;, Berg, O.G.; Elf, J. What matters for Lac repressor search in vivo - sliding, hopping, intersegment transfer, crowding on DNA or recognition? Nucl. Acids Res. 2015, 43, 3454-3464.
  * 35 Cuculis, L.; Abil, Z.; Zhao, H.; Schroeder, C.M. Direct observation of TALE protein dynamics reveals a two-state search mechanism. Nat. Commun. 2015, 6.
  * 36 Zandarashvili, L.; Esadze, A.; Vuzman, D.; Kemme, C.A.; Levy, Y.; Iwahara, J. Balancing between affinity and speed in target DNA search by zinc-finger proteins via modulation of dynamic conformational ensemble. Nucl. Acid Res. 2015, 112, E5142-E5149.
  * 37 Reingruber, J.; Holcman, D. Transcription factor search for a DNA promoter in a three-state model. Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 2011, 84, 020901-020904.
  * 38 Koslover, E. F.; de la Rosa, M. A. D.; Spakowitz, A. J. Theoretical and computational modeling of target-site search kinetics in vitro and in vivo. Biophys. J. 2011, 101, 856-865.
  * 39 Chu, X.; Liu, F.; and Maxwell, B.A.; Wang, Y.; Suo, Z.; Wang, H.; Han, W.; Wang, J. Dynamic conformational change regulates the protein-DNA recognition: an investigation on binding of a Y-family polymerase to its target DNA. PLoS Comput. Biol. 2014, 10, e1003804.
  * 40 Townson, S.A.; Samuelson, J.C.; Bao, Y.; Xu, S.-Y.; Aggarwal, A.K. BstYI Bound to Noncognate DNA Reveals a ”Hemispecific” Complex: Implications for DNA Scanning. Structure 2007, 15, 449-459.
  * 41 Bauer, M.; Rasmussen, E.S.; Lomholt, M.A.; Metzler, R. Real sequence effects on the search dynamics of transcription factors on DNA. Sci. Rep. 2015, 5, 10072.
  * 42 Brackley, A.A.; Cates, M.A.; Marenduzzo, D. Facilitated diffusion on mobile DNA: Configurational traps and sequence heterogeneity. Phys. Rev. Lett. 2012, 109, 168103.
  * 43 Afek, A.; Sela, I., Musa-Lempel, N.; Lukatsky, D.B. Nonspecific transcription-factor-DNA binding influences nucleosome occupancy in yeast. Biophys. J. 2011, 101, 2465-2475.
  * 44 Afek, A.; Lukatsky, D.B. Nonspecific protein-DNA binding is widespread in the yeast genome. Biophys. J. 2012, 102, 1881-1888.
  * 45 Afek, A.; Lukatsky, D.B. Positive and negative design for nonconsensus protein-DNA binding affinity in the vicinity of functional binding sites. Biophys. J. 2013, 105, 1653-1660.
  * 46 Afek, A.; Schipper, J.L.; Horton, J.; Gordan, R.; Lukatsky, D.B. Protein-DNA binding in the absence of specific base-pair recognition. Proc. Natl. Acad. Sci. USA 2014, 111, 17140-17145.
  * 47 Le, D.D.; Shimko, T.C.; Aditham, A.K.; Keys, A.M.; Longwell, S.A.; Orenstein, Y.; Fordyce, P.M. Comprehensive, high-resolution binding energy landscapes reveal context dependencies of transcription factor binding. Proc. Natl. Acad. Sci. USA 2018, 115, E3702-E3711.
  * 48 Marcovitz, A.; Levy, Y. Obstacles may facilitate and direct DNA search by proteins. Biophys. J. 2013, 104, 2042-20152.
  * 49 Gomez, D.; Klumpp, S. Facilitated diffusion in the presence of obstacles on the DNA. Phys. Chem. Chem. Phys. 2016, 18, 11184-11192.