11institutetext: Michael A. Lomholt 22institutetext: PhyLife - Center for
Physical Life Science, University of Southern Denmark, Campusvej 55, 5230
Odense M, Denmark 22email<EMAIL_ADDRESS>33institutetext: Ralf Metzler
44institutetext: Institute of Physics & Astronomy, University of Potsdam,
Karl-Liebknecht-Str 24/25, Haus 28, D-14476 Potsdam-Golm, Germany 44email:
<EMAIL_ADDRESS>
# Target search on DNA - effect of coiling

Michael A. Lomholt and Ralf Metzler

###### Abstract

Some proteins can find their targets on DNA faster than by pure diffusion in
the 3-dimensional cytoplasm, through the process of facilitated diffusion:
They can loosely bind to DNA and temporarily slide along it, thus being guided
by the DNA molecule itself to the target. This chapter examines this process
in mathematical detail with a focus on including the effect of DNA coiling on
the search process.

##  0.1 Introduction

Already in 1968 Adam and Delbrück adam68  proposed that diffusion limited
biological processes could be aided by the mechanism of dimensional reduction.
A prime example of this are proteins searching for a specific location on a
long DNA molecule, typically megabases long. In 1970 Riggs et al. riggs70
observed the very fast association of the lac repressor protein to its
specific operator site on DNA, and Richter and Eigen suggested in 1974
richter74  that it was exactly by exploiting the possibility of one
dimensional diffusion along the DNA that the lac repressor could be so fast.
Detailed mathematical modelling of this facilitated diffusion process, during
which the protein can switch between three dimensional diffusion in water and
one dimensional diffusion along the DNA has been refined in a number of
articles by Berg and various coauthors berg76 ; berg77 ; berg81 ; berg82 ;
hippel89 .

The present chapter examines the effect of DNA coiling on the target search
process of proteins searching for a target on DNA. The sliding of DNA-binding
proteins was demonstrated in single DNA experiments for both single-strand DNA
binders sokolov05  and double-strand binders wang06 ; hammar12 . Berg and
Blomberg suggested already in 1977 berg77  that coiling would speed up the
target localisation by removing some of the correlations that would be present
for a straight DNA molecule. The speed-up by DNA coiling has been confirmed
experimentally by Gowers and Halford exploiting supercoiling of DNA in
gowers03 , and later by optical tweezers experiments by Broek et al. broek08 .
The theory of the effect of DNA coiling that is presented in this chapter is
focused on modelling the latter experiment. It builds on the facilitated
diffusion model by Berg and Ehrenberg berg82  for straight DNA and it
constitutes a significantly more detailed version of the mathematical theory
presented in lomholt09 .

The account begins with a re-derivation of Smoluchowski’s 1916 result for
diffusion limited reactions with spherical targets smoluchowski16  in Section
0.2. This is the target localisation rate that dimensional reduction helps in
surpassing. Section 0.2 also introduces, for this simple case, some of the
mathematical apparatus that the remaining sections use. Next the facilitated
diffusion process is presented from the point of view of diffusion along the
DNA in Section 0.3. The specific theory for straight and coiled DNA then
follows in Sections 0.4 and 0.5. In Section 0.6 the mechanism of
intersegmental transfers is briefly discussed before ending with conclusions
in Section 0.7. Additionally, the Matlab scripts used for calculating target
localisation rates and creating the figures in this chapter can be found at
coiledDNA .

##  0.2 Smoluchowski result

We will first revisit the case of diffusing proteins reacting with a spherical
target at the origin as originally studied by Smoluchowski in 1916
smoluchowski16 . To find out how fast a protein can find the target we will
study the diffusion limited binding to the target. Our dynamical quantity is
the three dimensional volume density $n=n(r,t)$. We assume spherical symmetry
so the density only depends on the radial distance $r$ to the origin and time
$t$. The dynamics follows the diffusion equation

$\frac{\partial}{\partial t}n(r,t)=D_{\rm
3d}\frac{1}{r^{2}}\frac{\partial}{\partial
r}\left(r^{2}\frac{\partial}{\partial r}n(r,t)\right),$ (1)

where $D_{\rm 3d}$ is the protein diffusion constant. The proteins react with
the target placed at the origin whenever they are within a distance $b$ of it
(measured from the centre of the protein to the centre of the target). This
means that we have an absorbing boundary condition at $r=b$,

$n(r=b,t)=0.$ (2)

Far away from the target we will assume that the density approaches a constant
value $n_{\rm bulk}$, meaning that we have the second boundary condition

$\lim_{r\to\infty}n(r,t)=n_{\rm bulk}.$ (3)

To solve the diffusion equation we Laplace transform the density $n(r,t)$,

$n(r,u)=\int_{0}^{\infty}dt\;e^{-ut}n(r,t).$ (4)

Note that we indicate a Laplace transformed quantity by a change of the
variable to Laplace time $u$. When Laplace transforming the diffusion equation
it is converted into the ordinary differential equation

$un(r,u)-n_{\rm bulk}=D_{\rm 3d}\frac{1}{r^{2}}\frac{\partial}{\partial
r}\left(r^{2}\frac{\partial}{\partial r}n(r,u)\right),$ (5)

where we assumed that initially the concentration is constant $n(r,t=0)=n_{\rm
bulk}$. The general solution to this equation is

$n(r,u)=\frac{n_{\rm bulk}}{u}+\frac{A(u)}{r}e^{\sqrt{u/D_{\rm
3d}}r}+\frac{B(u)}{r}e^{-\sqrt{u/D_{\rm 3d}}r},$ (6)

where $A$ and $B$ are integration constants that can depend on $u$. These can
be identified via the two boundary conditions, leading to the unique selection

$n(r,u)=\frac{n_{\rm bulk}}{u}\left(1-\frac{b}{r}e^{-\sqrt{u/D_{\rm
3d}}(r-b)}\right).$ (7)

The current density along the radial direction is $-D_{\rm 3d}\partial
n/\partial r$. The current $j$ of proteins into the target is minus this
current times the surface area. In Laplace space we find

$\displaystyle j(u)$ $\displaystyle=4\pi b^{2}D_{\rm 3d}\left.\frac{\partial
n(r,u)}{\partial r}\right|_{r=b}$ $\displaystyle=4\pi D_{\rm 3d}bn_{\rm
bulk}\left(\frac{1}{u}+\frac{b}{\sqrt{D_{\rm 3d}u}}\right).$ (8)

Transforming back to real time this gives

$j(t)=4\pi D_{\rm 3d}bn_{\rm bulk}\left(1+\frac{b}{\sqrt{\pi D_{\rm
3d}t}}\right).$ (9)

From this solution for the current into the target we see that it reaches the
stationary value $j_{\rm stat}=4\pi D_{\rm 3d}bn_{\rm bulk}$ at times $t\gg
b^{2}/D_{\rm 3d}$. With inverse mobility $1/D_{\rm 3d}$ and with the square of
the target radius $b$, this characteristic time increases, as expected.

Let us for the moment assume that we are in the eventual stationary situation
with a constant flux of proteins into the target. Then at any given moment
there is a probability density $j_{\rm stat}$ of a protein arriving at the
target. If we look at the waiting time $\tau$ before the next protein arrives,
then this will be exponentially distributed with rate parameter $j_{\rm
stat}$, i.e., the average waiting time reads

$\langle\tau\rangle=j_{\rm stat}^{-1}.$ (10)

This equation converts between the macroscopic view of the equations for the
density of proteins and the microscopic view of the arrival of the first
protein to the target. When $\langle\tau\rangle\gg b^{2}/D_{\rm 3d}$, i.e.,
when $n_{\rm bulk}\ll b^{-3}$, then the transient contribution (the second
term) in Eq. (9) is unimportant for the statistics of the protein binding. In
the following we will assume that we are at such dilute concentrations of
proteins such that we only need to study the stationary regime to know the
binding statistics. Furthermore, we will remove the dependence on
concentration by defining the rate constant for the binding reaction as

$k_{\rm on}\equiv\frac{j_{\rm stat}}{n_{\rm bulk}}=4\pi D_{\rm 3d}b.$ (11)

This is indeed Smoluchowski’s result smoluchowski16 .

The experimental motivation for developing another model than the above for
DNA binding proteins comes from experiments on the Lac repressor riggs70
where a value $k_{\mathrm{on}}\approx
10^{10}/[(\mathrm{mol}/\mathrm{l})\cdot\mathrm{sec}]$ was measured. We can
compare this with Smoluchowski’s theory for spherical targets by estimating
its prediction for $k_{\mathrm{on}}$. For a typical diameter of a
transcription factor of $2R=5\,{\rm nm}$ by Stokes formula we obtain
$D_{\mathrm{3d}}=6\pi\eta R\approx 10^{2}\mu\mbox{m}^{2}/\mbox{s}$ where we
used the viscosity of water $\eta\approx 10^{-3}\,{\rm Pa}\cdot{\rm s}$. If we
say the protein has to situate itself on the DNA within a precision matching
the length of one base-pair it corresponds to a target of size $2b\approx
0.3\,$nm. Consequently we find $k_{\mathrm{on}}\approx 0.2\,\mu{\rm
m}^{3}/{\rm s}\approx 10^{8}/[(\mathrm{mol}/\mathrm{l})\cdot\mathrm{sec}]$.
That some proteins can operate faster than this upper diffusion limit for
spherical targets motivates the facilitated diffusion model in the next
section.

##  0.3 Facilitated Diffusion along DNA

The faster search can be explained by the fact that proteins can bind loosely
(”non-specifically” ptashne02 ; stormo98 ; buchler03 ) to the DNA, and then
slide along the DNA diffusively, searching for the target. To model this
process mathematically we will formulate it as a diffusion process along the
DNA. Thus our central dynamic quantity $n(x,t)$ is now the density of loosely
bound proteins per length of the DNA at position $x$, where $x$ is the
distance along the DNA contour with the origin defined to be at the target. We
label the diffusion constant along the DNA $D_{\rm 1d}$ to distinguish it from
the diffusion constant $D_{\rm 3d}$ for unbound proteins. The diffusion
equation governing $n(x,t)$ then has several extra terms as compared with Eq.
(1),

$\displaystyle\frac{\partial n(x,t)}{\partial t}=\left(D_{\rm
1d}\frac{\partial^{2}}{\partial x^{2}}-k^{\rm ns}_{\rm
off}\right)n(x,t)-j(t)\delta(x)+G(x,t)$ $\displaystyle+k^{\rm ns}_{\rm
off}\int_{-\infty}^{\infty}dx^{\prime}\int_{0}^{t}dt^{\prime}\,W_{\rm
bulk}(x-x^{\prime},t-t^{\prime})n(x^{\prime},t^{\prime}).$ (12)

Here $k^{\rm ns}_{\mathrm{off}}$ is the dissociation rate from the DNA of the
non-specifically bound proteins. $j(t)$ is, as in the previous section, the
number of proteins per time binding to the target site located at $x=0$ and
they are thus removed. $W_{\rm bulk}(x-x^{\prime},t-t^{\prime})$ is the joint
probability that a protein returns to the point $x$ at time $t$ after falling
of the DNA at $x^{\prime}$ at time $t^{\prime}$. The remaining term, $G(x,t)$,
accounts for the proteins that bind to the DNA at time $t$ and position $x$
without previously having been bound between time zero and time $t$. We will
see later that details of $G(x,t)$ (such as if it is uniform in $x$) will not
matter for dilute protein concentrations. We are assuming that any protein
diffusing across the target will be absorbed by it, imposing the boundary
condition $n(x=0,t)=0$. This boundary condition determines the current $j(t)$.

In the above we made a number of assumptions to simplify the interplay between
one dimensional diffusion along the DNA and diffusion in the bulk around the
DNA: We assumed that the DNA is long enough that we can take the end points to
infinity when considering the diffusion to the target, i.e., we can integrate
to infinity in the convolution integral containing $W_{\rm bulk}$. This
assumption should hold, if the absorption at the target only depletes proteins
at a distance from the target that is small compared with the distance to the
end points. In the next section we will introduce an effective sliding length
that provides a length scale for this depletion. Another assumption is that
there is no dependence between excursion events to the bulk. A static coiled
configuration of the DNA could lead to such dependence, since two excursions
following each other could bring the protein back and forth between two parts
of the DNA that are far away from each measured along the DNA with high
probability. The assumption should hold for straight or quickly fluctuating
DNA though. Assuming we are in either of these situations the kernel $W_{\rm
bulk}(x-x^{\prime},t-t^{\prime})$ is assumed to be homogeneous in space and
time too, i.e., the distance and duration of excursions in the bulk do not
depend on the initial location or time. Additionally, we assume that the
diffusion along the DNA is homogeneous, i.e., it does not depend on the local
sequence of nucleotides. See for instance slutsky04 ; benichou09  for
discussions of how sequence heterogeneity could be a barrier for fast search,
but also for a mechanism with which proteins can circumvent this barrier.

As in the previous section, we want to find the stationary flux into the
target $j_{\rm stat}$, and we will use the same technique of Laplace
transforming, replacing time $t$ with the Laplace variable $u$. Furthermore,
we will Fourier transform along the DNA:

$n(q,u)=\int_{-\infty}^{\infty}dx\;e^{-iqx}n(x,u),$ (13)

similarly indicating the transform by replacing the position coordinate $x$
with wavenumber $q$. Applying the two transforms Eq. (12) becomes

$\frac{1}{W(q,u)}n(q,u)-n(q,t=0)=G(q,u)-j(u),$ (14)

where

$\frac{1}{W(q,u)}=u+D_{\rm 1d}q^{2}+k^{\rm ns}_{\rm off}[1-W_{\rm
bulk}(q,u)].$ (15)

To extract the long time behaviour we rewrite Eq. (14) to arrive at

$n(q,u)=n^{\rm ns}(q,u)-j(u)W(q,u),$ (16)

where we have introduced

$n^{\rm ns}(q,u)=W(q,u)[G(q,u)+n_{0}(q)].$ (17)

$n^{\rm ns}(x,t)$ is the density of proteins on the DNA in the absence of the
sink at the target. For this situation we would expect equilibration to occur,
such that eventually we will have a constant density $n^{\rm ns}_{\rm eq}$. In
the following we will neglect all dependence on the initial distribution of
proteins by assuming we are in this equilibrium limit of loose binding to the
DNA. Due to the linearity of our equations we expect this assumption to hold
in the dilute limit, since the equilibration time should be independent of the
protein concentration, while the search time will increase as concentration is
lowered sokolov05 . In the Laplace domain this means that $n^{\rm ns}(x,u)\sim
n^{\rm ns}_{\rm eq}/u$, where $\sim$ indicates that we are looking at the
behaviour at long times, which by Tauberian theorems hughes95  is equivalent
to studying the limit of $u$ going to zero. Equations (8) and (9) provide an
example of this, where it can be found that the term that dominates at small
$u$ converts to a term that dominates at large $t$, and vice versa for the
other term.

We can find the current $j(u)$ by inverse Fourier transforming Eq. (16) at
$x=0$ using the boundary condition at the target, $n(x=0,t)=0$, to arrive at

$j(u)=\frac{n^{\rm ns}(x=0,u)}{W(x=0,u)}.$ (18)

To proceed we will assume that the value $W(x=0,u=0)$ is finite and non-zero,
which will be the case for the three dimensional diffusion problems we study
here with the bulk space stretching far in all directions, leading to long
tailed distributions for the bulk excursions. With this assumption we find the
limiting behaviour at small $u$: $j(u)\sim k_{\rm 1d}n^{\rm ns}_{\rm
eq}/[uW(x=0,u=0)]$, i.e., that the current becomes stationary at long times
with value

$j_{\rm stat}\sim k_{\rm 1d}n^{\rm ns}_{\rm eq}$ (19)

where $k_{\rm 1d}=1/W(x=0,u=0)$, or explicitly

$k_{\rm 1d}^{-1}=\int_{-\infty}^{\infty}\frac{dq}{2\pi}\frac{1}{D_{\rm
1d}q^{2}+k_{\rm off}^{\rm ns}(1-\lambda_{\rm bulk}(q))},$ (20)

with the distribution $\lambda_{\rm bulk}(x)=W_{\rm bulk}(x,u=0)$ of the
relocation lengths along the DNA contour of the three-dimensional bulk
excursions.

To proceed and calculate the integral in Eq. (20) we need to know
$\lambda_{\rm bulk}(q)$. There is, however, a situation we can discuss already
without a detailed form of $\lambda_{\rm bulk}(q)$: If $\lambda_{\rm bulk}(x)$
is sufficiently long-tailed for a protein not to rebind to a point where it
was previously bound to the DNA, then for the purpose of calculating the
integral in Eq. (20) we can effectively consider $\lambda_{\rm bulk}(q)$ to
vanish for $q\neq 0$ (it will always be unity for $q=0$ by normalisation).
Thus $\lambda_{\rm bulk}(q)$ will not contribute significantly to the integral
and we can discard it and then integrate analytically to find

$k_{\rm 1d}=2\sqrt{D_{\rm 1d}k^{\rm ns}_{\rm off}}.$ (21)

If we multiply this result on both sides with $n^{\rm ns}_{\rm eq}$ it can be
rewritten as

$j_{\rm stat}=2l_{\rm sl}k^{\rm ns}_{\rm off}n^{\rm ns}_{\rm eq}.$ (22)

To understand this expression, first note that $k^{\rm ns}_{\rm off}n^{\rm
ns}_{\rm eq}$ is the rate of unbinding events per length of the DNA. In the
equilibrated situation this is equivalent to the rate of rebinding events per
length. If such a rebinding happens within a distance of $l_{\rm sl}$ on
either side of the target, then the protein will be likely to find the target,
thus resulting in Eq. (22) for the rate of target localisation.

In the previous section we preferred to express results in terms of $k_{\rm
on}=j_{\rm stat}/n_{\rm bulk}$, where $n_{\rm bulk}$ was the density per
volume in the bulk, i.e., the density of non-bound proteins in the water
around the DNA. At equilibrium of non-specific binding the two concentrations
are related by the non-specific binding constant per length of DNA, which is
$K_{\rm ns}=n^{\rm ns}_{\rm eq}/n_{\rm bulk}$. Using this we can write

$k_{\rm on}=k_{\rm 1d}K_{\rm ns}.$ (23)

Note that if a significant fraction of the proteins are bound to the DNA it
becomes necessary to distinguish the bulk concentration of proteins $n_{\rm
bulk}$ from the total amount of proteins per volume $n_{\rm total}$. If
$l_{\rm DNA}$ is the length of the DNA divided by the volume of the
surrounding water, then the concentrations are related by

${n}_{\rm total}={n}_{\rm bulk}+l^{\rm total}_{\rm DNA}n^{\rm ns}_{\rm
eq}=(1+l^{\rm total}_{\rm DNA}K_{\rm ns}){n}_{\rm bulk}.$ (24)

In terms of $n_{\rm total}$ the target localisation rate can therefore be
written as

$j_{\rm stat}=\frac{k_{\rm on}}{1+K_{\rm ns}l_{\rm DNA}^{\rm total}}n_{\rm
total}.$ (25)

In the following two sections we will aim at calculating $k_{\rm on}$ first
for straight and then coiled DNA.

##  0.4 Straight DNA

Figure 1: Schematic of the bulk excursion of a protein being released at the
cylindrical DNA-interaction surface. This process is modelled by the diffusion
equation (26) and the boundary condition (27).

In this section we will obtain $W_{\rm bulk}$ for the case when the DNA is
completely straight. This can be achieved, e.g., by stretching the DNA in an
optical tweezers setup pant04 ; broek08  or in microfluidic setups krog18b .
We can then consider the problem as a diffusion problem around a cylinder. We
first determine the probability density $P$ for a single protein to be at a
position in space at a certain time $t$. In this cylindrical geometry, the DNA
is situated along the $x$-axis, and the protein will interact with it when it
are at a distance $r=r_{\rm int}$ from the $x$-axis (see Fig. 1). For the
directions orthogonal to the $x$-axis we use polar coordinates with $r$ the
radial coordinate. Since we are only interested in the value of $x$ at which
the protein rebinds to the DNA, we will drop the dependence on the polar angle
and $P$ is then rotationally symmetric around the $x$-axis. Thus we have
$P=P(x,r,t)$, and the three dimensional diffusion equation can be written as

$\frac{\partial{P}}{\partial t}=D_{\rm 3d}\left(\frac{\partial^{2}}{\partial
x^{2}}+\frac{1}{r}\frac{\partial}{\partial r}r\frac{\partial}{\partial
r}\right){P},$ (26)

with the diffusion constant $D_{\rm 3d}$. The boundary condition we use at
$r=r_{\rm int}$ can be expressed as the probability flux density in the radial
direction,

$-\left.D_{\rm 3d}\frac{\partial{P}}{\partial r}\right|_{r=r_{\rm
int}}=-\left.\frac{k^{\rm ns}_{\rm on}}{2\pi r_{\rm int}}{P}\right|_{r=r_{\rm
int}}+\frac{\delta(x)}{2\pi r_{\rm int}}\delta(t).$ (27)

On the right hand side the first term represents the probability rate of the
protein loosely binding to the DNA per surface area of the cylinder. $k^{\rm
ns}_{\rm on}$ is a rate constant with units of area per time. The second term
represents the protein release from the boundary at $x=0$ at time $t=0$. Note
that consistency with non-specific equilibration, $n^{\rm ns}_{\rm eq}k^{\rm
ns}_{\rm off}={n}_{\rm bulk}k^{\rm ns}_{\rm on}$, means that $k^{\rm ns}_{\rm
on}=K_{\rm ns}k^{\rm ns}_{\rm off}$. That the protein enters the bulk via the
boundary condition means that we will take the bulk to be empty initially:
$P(x,r,t=0)=0$. Additionally we assume that $P$ vanishes infinitely far away
from the spatial origin at all times.

To solve Eq. (26) we Laplace transform the time $t$ as usual and also Fourier
transform along the $x$-axis again to convert the derivatives with respect to
$x$ to factors of $iq$, i.e.,

$uP(q,r,u)=D_{\rm 3d}\left(-q^{2}+\frac{1}{r}\frac{\partial}{\partial
r}r\frac{\partial}{\partial r}\right){P}(q,r,u).$ (28)

The general solution of this equation is given in terms of modified Bessel
functions $I_{n}$ and $K_{n}$ as

$P(q,r,u)=AI_{0}({\bar{q}}r)+BK_{0}({\bar{q}}r),$ (29)

where ${\bar{q}}=\sqrt{q^{2}+u/D_{\rm 3d}}$. The boundary condition that $P$
vanishes infinitely far from the origin tells us that $A=0$. Inserting the
term with $B$ into the remaining boundary condition (27) gives

$B=\frac{1}{k^{\rm ns}_{\rm on}K_{0}({\bar{q}}r_{\rm int})+2\pi r_{\rm
int}D_{\rm 3d}{\bar{q}}K_{1}({\bar{q}}r_{\rm int})}.$ (30)

Now we have arrived at a result that we can use to calculate $W_{\rm bulk}$,
since it is simply given by $W_{\rm bulk}(x,t)=k^{\rm ns}_{\rm on}P(x,r=r_{\rm
int},t)$, i.e., the probability that the protein binds at the boundary at
position $x$ and time $t$. This produces the result

$W_{\mathrm{bulk}}^{\rm cyl}(q,u)=\left(1+\frac{2\pi D_{\rm 3d}{\bar{q}}r_{\rm
int}K_{1}({\bar{q}}r_{\rm int})}{k^{\rm ns}_{\rm on}K_{0}({\bar{q}}r_{\rm
int})}\right)^{-1}.$ (31)

The superscript cyl indicates that this is the solution when the DNA is
forming a straight cylinder.

Figure 2: Plot of $k_{\rm on}$ as a function of $k_{\rm on}^{\rm ns}$ for
$D_{\rm 3d}=0.1\,\mu{\rm m}^{2}/{\rm ms}$, $D_{\rm 1d}=10^{-4}\,\mu{\rm
m}^{2}/{\rm ms}$, $r_{\rm int}=3\,{\rm nm}$ and $k_{\rm off}^{\rm ns}=1\,{\rm
ms}^{-1}$. The black solid line is the full numerical result based on Eqs.
(20), (23) and (31), while the red dashed line matching at low $k_{\rm
on}^{\rm ns}$ is the result in Eqs. (21) & (23) and the blue dash-dotted line
is the result in Eq. (33).

We are now in a position where we can evaluate the integral in Eq. (20)
numerically, since we can obtain $\lambda_{\rm bulk}$ from
$W_{\mathrm{bulk}}^{\rm cyl}$ simply by setting $u=0$, i.e., ${\bar{q}}=|q|$.
The numerical result for $k_{\rm on}$ is shown as a function of $k^{\rm
ns}_{\rm on}$ in Fig. 2, together with two asymptotic approximations. The
asymptotic expression for small $k^{\rm ns}_{\rm on}$ is based on Eqs. (21)
and (23) and is limited by the non-specific binding to DNA. This limit holds
when $k^{\rm ns}_{\rm on}\ll D_{\rm 3d}$. In the opposite limit, $k^{\rm
ns}_{\rm on}\gg D_{\rm 3d}$, the detailed form of $W_{\mathrm{bulk}}^{\rm
cyl}(q,u=0)$ at small values of $|q|$ becomes important. To obtain an
expression for $k_{\rm on}$ in this limit, we use that the asymptotic
behaviour of the modified Bessel functions at small arguments are:
$K_{1}(x)\approx 1/x$ and $K_{0}(x)\approx-\ln(x)$. Thus at small $|q|$,
$W_{\mathrm{bulk}}^{\rm cyl}(q,u=0)$ will deviate from unity only by a small
term that decays slowly as $-1/\ln(|q|r_{\rm int})$. If we ignore this
logarithmic factor and examine the behaviour of the denominator of the
integrand in Eq. (20), we see that the slow logarithmic behaviour becomes
dominant at $|q|$ below $1/l_{\rm sl}^{\rm eff}$ where

$l_{\rm sl}^{\rm eff}=\sqrt{\frac{k_{\rm on}^{\rm ns}}{(2\pi D_{\rm
3d})}}l_{\rm sl}.$ (32)

If we simply replace the last term in the denominator by its asymptotic value
at $q=1/l_{\rm sl}^{\rm eff}$, i.e., we set $1-\lambda_{\rm bulk}(q)=2\pi
D_{\rm 3d}/[k_{\rm on}^{\rm ns}\ln(l_{\rm sl}^{\rm eff}/r_{\rm int})]$ then we
can perform the integral and obtain

$k_{\rm on}\sim\frac{4\pi D_{\rm 3d}l_{\rm sl}^{\rm eff}}{[\ln(l_{\rm sl}^{\rm
eff}/r_{\rm int})]^{1/2}}$ (33)

This is the approximate asymptotic expression plotted for large $k_{\rm
on}^{\rm ns}$ in Fig. (2). The crossover to this asymptotic behaviour occurs
around $k_{\rm on}^{\rm ns}\approx 2\pi D_{\rm 3d}$, but the final convergence
only sets in very slowly as $k_{\rm on}^{\rm ns}$ increases due to the
involved logarithmic behaviours.

In the limit when $k_{\rm on}^{\rm ns}\gg D_{\rm 3d}$ the proteins that fall
of the DNA will have a tendency to rebind immediately, thus performing what is
in effect just a tiny hop on the spot. The tiny probability that the protein
avoids such a hop must be proportional to $D_{\rm 3d}/k_{\rm on}^{\rm ns}$ by
dimensional arguments. Therefore the number of hops and thus the time spent
loosely bound on the DNA scales with $k_{\rm on}^{\rm ns}/D_{\rm 3d}$. The
effective distance covered by one-dimensional diffusion along the DNA while
sliding and hopping on the spot will therefore increase by a factor
proportional to $\sqrt{k_{\rm on}^{\rm ns}/D_{\rm 3d}}$ since the displacement
in Brownian motion increase by the square root of time. With this reasoning an
interpretation becomes evident of $l_{\rm sl}^{\rm eff}$ as the effective
sliding length resulting from combining sliding events connected by quick
rebindings. By comparing Eq. (33) with the Smoluchowski result (11) this
reasoning also provides an interpretation of $l_{\rm sl}^{\rm eff}$ as an
increased target size (sometimes referred to as an antenna hu06 ) with
$1/[\ln(l^{\rm eff}_{\rm sl}/r_{\rm int})]^{1/2}$ entering as a geometrical
factor due to the cylindrical shape of the target.

Finally, we note that the results of Eq. (20) and (31) match the result
obtained by Berg and Ehrenberg in 1982 berg82  for infinitely long straight
DNA.

##  0.5 Coiled DNA

Figure 3: Left: The DNA is considered made up of locally straight rods with
the protein diffusing among them. Right: The diffusion problem is solved for
each rod individually and then combined assuming independent interactions.

We now turn to the calculation of $k_{\rm on}$ when the DNA is coiled. To
reach an analytical expression, we assume that the DNA has a sufficiently high
persistence length, such that locally around the target the DNA can
effectively be thought of as a number of cylindrical straight rods passing
close to each other (see left part of Fig. 3). The idea behind this is that we
can solve the diffusion problem for a single straight rod (see right part of
Fig. 3), and then we can combine the solutions by assuming independent
interactions. In this independent interactions approach, the individual
solutions are combined by letting the actual binding event be the binding
event that happens first among the multiple rods. This means that we can
calculate the “survival” probability of the protein not having bound anywhere
as simply the product of all the probabilities for each rod that it has not
yet bound to that individual rod.

We call $P_{\rm surv}^{\rm foreign}(t)$ the survival probability that the
protein has not yet bound to any foreign straight rod ignoring the existence
of the (non-foreign) straight rod that it unbound from. With the assumption of
independent interactions we can write

$P_{\rm surv}^{\rm foreign}(t)=\left[1-J_{\rm single}(t)\right]^{N}.$ (34)

where $N$ is the, for the moment finite, number of foreign straight rods and
$J_{\rm single}(t)$ is the probability that it has reacted with a particular
foreign rod ignoring the existence of all the others. Assuming we have found
$J_{\rm single}(t)$ and thereby $P_{\rm surv}^{\rm foreign}(t)$ we will then
calculate $W_{\rm bulk}$ as

$W_{\rm bulk}(q,t)=W_{\rm bulk}^{\rm cyl}(q,t)P_{\rm surv}^{\rm
foreign}(t)\quad(q\neq 0)$ (35)

where $W_{\rm bulk}^{\rm cyl}$ is the solution found in the last section when
the DNA was a single straight rod. Here we have again used the assumption of
independent interactions to calculate the probability that it returns to the
straight rod it unbound from, i.e., we multiply the solution without foreign
rods with the probability that it has not yet bound to any foreign rod. Also,
we have assumed that binding to a foreign rod leads the protein so far away
from where it unbound that it effectively bound someplace uniformly-random
along the contour, and that these excursions therefore only contribute at
$q=0$ to $W_{\rm bulk}(q,t)$. This is the same kind of approximation that was
used previously when arriving at the reaction limited result in Eq. (21).

The single foreign rod binding probability $J_{\rm single}$ will be found by
solving the usual three dimensional diffusion equation, Eq. (26), but now with
the initial condition that the protein can start anywhere in the bulk relative
to the rod. This means that the initial condition is $P(x,r,t=0)=1/V$ where
$V$ is the volume of the bulk water. After a Laplace transformation of time
$t$ we therefore get

$uP(r,u)-\frac{1}{V}=D_{\rm 3d}\frac{1}{r}\frac{\partial}{\partial
r}r\frac{\partial}{\partial r}{P}(r,u),$ (36)

where the $x$ dependence has been left out since the problem is uniform along
the $x$ axis. The general solution to this equation is

$P(r,u)=\frac{1}{uV}+AI_{0}(\sqrt{u/D_{\rm 3d}}r)+BK_{0}(\sqrt{u/D_{\rm
3d}}r)$ (37)

with the integration constants $A$ and $B$ that may depend on $u$. As one of
the boundary conditions we take that the probability should approach the
constant $1/V$ far from the DNA. Since we anyway intend to take the limit of
the volume expanding to infinity in all directions we can write this boundary
condition in Laplace space as $\lim_{r\to\infty}P(r,u)=1/(uV)$. Thus we have
that $A=0$. For the boundary condition on the DNA surface we employ Eq. (27)
without the release from the surface, i.e.,

$-\left.D_{\rm 3d}\frac{\partial{P}}{\partial r}\right|_{r=r_{\rm
int}}=-\left.\frac{k^{\rm ns}_{\rm on}}{2\pi r_{\rm int}}{P}\right|_{r=r_{\rm
int}}.$ (38)

We then obtain

$P(r,u)=\frac{1}{uV}-\frac{k_{\rm on}^{\rm ns}K_{0}(\sqrt{u/D_{\rm
3d}}r)/(uV)}{k^{\rm ns}_{\rm on}K_{0}(\sqrt{u/D_{\rm 3d}}r_{\rm int})+2\pi
r_{\rm int}\sqrt{D_{\rm 3d}u}K_{1}({\bar{q}}r_{\rm int})}$ (39)

We can now find the probability of having reacted with a single rod, $J_{\rm
single}(t)=\int_{0}^{t}Lk_{\rm on}^{\rm ns}P(r_{\rm
int},t^{\prime})dt^{\prime}$, which in Laplace space becomes

$J_{\rm single}(u)=\frac{Lk_{\rm on}^{\rm ns}}{u^{2}V}\left(1+\frac{k_{\rm
on}^{\rm ns}K_{0}(\sqrt{u/D_{\rm 3d}}r_{\rm int})}{2\pi\sqrt{uD_{\rm
3d}}r_{\rm int}K_{1}(\sqrt{u/D_{\rm 3d}}r_{\rm int})}\right)^{-1}.$ (40)

This puts us in a position to combine the results for each rod via Eq. (34),
and take the limit of $N$, $L$ and $V$ growing to infinity with $L/V=l_{\rm
DNA}/N$ for constant $l_{\rm DNA}$. This results in

$P_{\rm surv}^{\rm foreign}(t)=\exp\left[-J_{\rm cap}(t)\right],$ (41)

where

$J_{\rm cap}(u)=\frac{k_{\rm on}^{\rm ns}l_{\rm
DNA}}{u^{2}}\left(1+\frac{k_{\rm on}^{\rm ns}K_{0}(\sqrt{u/D_{\rm 3d}}r_{\rm
int})}{2\pi\sqrt{uD_{\rm 3d}}r_{\rm int}K_{1}(\sqrt{u/D_{\rm 3d}}r_{\rm
int})}\right)^{-1}.$ (42)

We can evaluate the inverse Laplace transform of $J_{\rm cap}(u)$ numerically
and thereby obtain $P_{\rm surv}^{\rm foreign}(t)$. However, to obtain $W_{\rm
bulk}(q,u)$ we would have to perform a Laplace transform on top of that. To
reduce the number of transforms, i.e., integrals, that need to be performed we
will use a convenient approximation for $J_{\rm cap}(t)$. To argue for this
approximation we first note that for $k^{\rm ns}_{\rm on}\gg D_{\rm 3d}$ we
can divide the $u$-dependence of $J_{\rm cap}(u)$ into the three regimes

$J_{\rm cap}(u)=\left\\{\begin{array}[]{ll}k^{\rm ns}_{\rm on}l_{\rm
DNA}{u^{-2}}&,\;u^{-1}\ll\frac{D_{\rm 3d}r_{\rm int}^{2}}{(k^{\rm ns}_{\rm
on})^{2}}\\\ 2\pi l_{\rm DNA}r_{\rm int}\sqrt{D_{\rm
3d}}u^{-3/2}&,\;\frac{D_{\rm 3d}r_{\rm int}^{2}}{(k^{\rm ns}_{\rm on})^{2}}\ll
u^{-1}\ll\frac{r_{\rm int}^{2}}{D_{\rm 3d}}\;.\\\ \frac{4\pi l_{\rm DNA}D_{\rm
3d}}{u^{2}\ln(D_{\rm 3d}/(ur_{\rm int}^{2}))}&,\;u^{-1}\gg\frac{r_{\rm
int}^{2}}{D_{\rm 3d}}\end{array}\right.$ (43)

When the regimes are well separated in this way, with decaying $u$ dependence,
they can each be Laplace inverted independently to find

$J_{\rm cap}(t)=\left\\{\begin{array}[]{ll}k^{\rm ns}_{\rm on}l_{\rm
DNA}t&,\;t\ll\frac{D_{\rm 3d}r_{\rm int}^{2}}{(k^{\rm ns}_{\rm on})^{2}}\\\
4l_{\rm DNA}r_{\rm int}\sqrt{\pi D_{\rm 3d}t}&,\;\frac{D_{\rm 3d}r_{\rm
int}^{2}}{(k^{\rm ns}_{\rm on})^{2}}\ll t\ll\frac{r_{\rm int}^{2}}{D_{\rm
3d}}\;,\\\ \frac{4\pi l_{\rm DNA}D_{\rm 3d}t}{\ln(D_{\rm 3d}t/r_{\rm
int}^{2})}&,\;t\gg\frac{r_{\rm int}^{2}}{D_{\rm 3d}}\end{array}\right.$ (44)

where Tauberian theorems hughes95  were applied in the last regime. When
$k^{\rm ns}_{\rm on}\gg D_{\rm 3d}$ the first regime will be very brief and
the probability of being captured by a foreign rod will only accumulate to
$l_{\rm DNA}r_{\rm int}^{2}D_{\rm 3d}/k^{\rm ns}_{\rm on}$. This will be a
very small number, and therefore we will ignore this regime. In the last
regime we will approximate the slowly varying logarithm with a constant. Thus
we will assume $J_{\rm cap}(t)\approx k_{\rm cap}t$ in this regime where the
precise value of $k_{\rm cap}$ will be fixed later, but it will be somewhere
around $l_{\rm DNA}D_{\rm 3d}$. Combining the last two regimes we see that a
function that gives the correct approximate behaviour in each is

$P_{\rm app}(t)=\exp[-J_{\rm app}(t)]\;\;\mathrm{with}\;\;J_{\rm
app}(t)=4l_{\rm DNA}r_{\rm int}\sqrt{\pi D_{\rm 3d}t}+k_{\rm cap}t,$ (45)

and this is the function we will replace
$P_{\mathrm{surv}}^{\mathrm{foreign}}$ with. To fix $k_{\rm cap}$ we choose
the condition that we want the same probability that the protein returns to
the non-foreign rod for both of $P_{\mathrm{surv}}^{\mathrm{foreign}}$ and
$P_{\mathrm{app}}$. This gives the following equation for $k_{\rm cap}$

$\displaystyle\int_{0}^{\infty}dt\;W_{\rm bulk}^{\rm cyl}(q=0,t)\left[P_{\rm
surv}^{\rm foreign}(t)-P_{\rm app}(t)\right]=0,$ (46)

which can be handled numerically. This equation involves two Laplace
inversions and an integral over these for each numerical guess at a solution,
but we have a good guess by using $l_{\rm DNA}D_{\rm 3d}$ as a starting value
for $k_{\rm cap}$. See Fig. 4 for a comparison of $J_{\rm cap}(t)$ and $J_{\rm
app}(t)$. The numerical Laplace inversions where handled using the Talbot and
Euler algorithms presented in abate06 .

Figure 4: Plot of $J_{\rm cap}(t)$ as a function of time $t$ for $D_{\rm
3d}=0.1\,\mu{\rm m}^{2}/{\rm ms}$, $r_{\rm int}=3\,{\rm nm}$ and $k_{\rm
on}^{\rm ns}=10^{4}\,\mu{\rm m}^{2}/{\rm ms}$. The black solid line is the
result of numerically Laplace inverting Eq. (42), while the red dashed line
matching at low $t$ is the result for the first regime in Eq. (44) and the
blue dash-dotted line is the approximation $J_{\rm app}(t)$ given in Eq. (45).

The $\sqrt{t}$ part of $P_{\rm app}$ can be rewritten as a Laplace transform
in the following way

$\displaystyle P_{\rm app}(t)$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}ds\;e^{-(s+k_{\rm cap})t}\frac{a}{2\sqrt{\pi
s^{3}}}e^{-a^{2}/(4s)},$ (47)

where $a=4\sqrt{\pi}r_{\rm int}l_{\rm DNA}\sqrt{D_{\rm 3d}}$. This is very
convenient when it is inserted in Eq. (35) instead of
$P_{\mathrm{surv}}^{\mathrm{foreign}}$ and Laplace transformed to obtain
$W_{\rm bulk}(q,u)$. Then one obtains

$W_{\rm bulk}(q,u)=\int_{0}^{\infty}ds\;\frac{a\,e^{-a^{2}/(4s)}}{2\sqrt{\pi
s^{3}}}W_{\rm bulk}^{\rm cyl}(q,u+s+k_{\rm cap}).$ (48)

With this expression the number of involved integrals have been reduced to a
single one, and the expression is now ready to be used inside the integral of
Eq. (20). Along the way in the derivation of the expression we assumed $D_{\rm
3d}\ll k_{\rm on}^{\rm ns}$. But note that in the opposite limit $D_{\rm
3d}\gg k_{\rm on}^{\rm ns}$ the expression is consistent with the vanishing of
$\lambda_{\rm bulk}(q)$ for $q\neq 0$ and thus it extrapolates to the result
in Eq. (21) in that limit.

To get some analytic insight into the effects of the coiling or presence of
foreign DNA we can apply approximations similar to the ones taken in the
$D_{\rm 3d}\ll k_{\rm on}^{\rm ns}$ regime of the straight rod. Thus we
approximate $W_{\rm bulk}^{\rm cyl}$ by setting $q=1/l_{\rm sl}^{\rm eff}$,
and furthermore we approximate $k_{\rm cap}=D_{\rm 3d}l_{\rm DNA}$. Using this
we can write

$\displaystyle 1-\lambda_{\rm bulk}(q)$
$\displaystyle\approx\int_{0}^{\infty}ds\;\frac{a\,e^{-a^{2}/(4s)}}{2\sqrt{\pi
s^{3}}}[1-W_{\rm bulk}^{\rm cyl}(q=1/l_{\rm sl}^{\rm eff},u=s+k_{\rm cap})]$
$\displaystyle=C_{1}+C_{2}.$ (49)

In the second line we split the integral over $s$ in two parts, where $C_{1}$
represents the integral from $s=0$ to $s=D_{\rm 3d}/r_{\rm int}^{2}$ and
$C_{2}$ the remaining integral to infinity.

To obtain an approximation for $C_{1}$ we assume that the dominant
contribution of the integral occurs for $s\ll k_{\rm cap}+D_{\rm 3d}/(l_{\rm
sl}^{\rm eff})^{2}$ and thus we simply ignore the $s$ dependence in $W_{\rm
bulk}^{\rm cyl}$. This gives

$\displaystyle C_{1}$ $\displaystyle\approx\int_{0}^{D_{\rm 3d}/r_{\rm
int}^{2}}ds\;\frac{a\,e^{-a^{2}/(4s)}}{2\sqrt{\pi s^{3}}}[1-W_{\rm bulk}^{\rm
cyl}(q=1/l_{\rm sl}^{\rm eff},u=k_{\rm cap})]$ $\displaystyle\approx 1-W_{\rm
bulk}^{\rm cyl}(q=1/l_{\rm sl}^{\rm eff},u=k_{\rm cap})$
$\displaystyle\approx-\frac{2\pi D_{\rm 3d}}{k_{\rm on}^{\rm
ns}\ln\left(\sqrt{(l_{\rm sl}^{\rm eff})^{-2}+l_{\rm DNA}}r_{\rm
int}\right)}.$ (50)

For the approximation in the second line we used $a^{2}\ll D_{\rm 3d}/r_{\rm
int}^{2}$ to extend the integral back to infinity again, and normalisation of
the factor in front of the square brackets to perform the integration. In the
third line we used the asymptotic expansion of the modified Bessel functions
at small arguments, as we did when deriving Eq. (33).

To obtain an approximation for $C_{2}$ we do the opposite as for $C_{1}$ and
ignore instead $1/l_{\rm sl}^{\rm eff}$ and $k_{\rm cap}$ for the remainder of
the integral. This gives

$\displaystyle C_{2}$ $\displaystyle\approx\int_{D_{\rm 3d}/r_{\rm
int}^{2}}^{\infty}ds\;\frac{a\,e^{-a^{2}/(4s)}}{2\sqrt{\pi s^{3}}}[1-W_{\rm
bulk}^{\rm cyl}(q=0,u=s)]$ (51) $\displaystyle\approx\int_{D_{\rm 3d}/r_{\rm
int}^{2}}^{\infty}ds\,\frac{a}{2\sqrt{\pi s^{3}}}\frac{1}{1+k_{\rm on}^{\rm
ns}/(2\pi r_{\rm int}\sqrt{D_{\rm 3d}s})}$ (52) $\displaystyle=4r_{\rm
int}^{2}l_{\rm DNA}\frac{2\pi D_{\rm 3d}}{k_{\rm on}^{\rm
ns}}\ln\left(1+\frac{k_{\rm on}^{\rm ns}}{2\pi D_{\rm 3d}}\right).$ (53)

For the approximation in the second line we used $a^{2}\ll D_{\rm 3d}/r_{\rm
int}^{2}$ to discard the exponential and we used that the asymptotic expansion
of the two modified Bessel functions are identical for ${\bar{q}}r_{\rm
int}\gg 1$ to discard these functions too.

Finally, setting $1-\lambda_{\rm bulk}(q)=C_{1}+C_{2}$ we can perform the $q$
integral in Eq. (20) and reach the approximate expression

$k_{\rm on}\sim 4\pi D_{\rm 3d}l_{\rm sl}^{\rm eff}\sqrt{\frac{k_{\rm on}^{\rm
ns}}{2\pi D_{\rm 3d}}(C_{1}+C_{2})}$ (54)

Figure 5: Plot of $k_{\rm on}/k_{\rm on}^{\rm cyl}$ as a function of
$\sqrt{l_{\rm DNA}}l_{\rm sl}^{\rm eff}$ for $D_{\rm 3d}=0.1\,\mu{\rm
m}^{2}/{\rm ms}$, $D_{\rm 1d}=10^{-4}\,\mu{\rm m}^{2}/{\rm ms}$, $r_{\rm
int}=3\,{\rm nm}$, $k_{\rm on}^{\rm ns}=10^{4}\,\mu{\rm m}^{2}/{\rm ms}$ and
$k_{\rm off}^{\rm ns}=1\,{\rm ms}^{-1}$ (giving $l_{\rm sl}^{\rm
eff}=1.26\,\mu{\rm m}$). $k_{\rm on}^{\rm cyl}$ represents the value for
straight DNA, i.e., for $l_{\rm DNA}=0$. The black solid line is the numerical
result obtained using Eq. (48) in Eq. (20), while the red dashed line is the
result in Eq. (54) divided by the result in Eq. (33). The value of $k_{\rm
on}^{\rm ns}$ has been chosen to be far inside the regime $k_{\rm on}^{\rm
ns}\gg D_{\rm 3d}$.

In Fig. 5 the ratio of $k_{\rm on}$ for the coiled and straight DNA is plotted
as a function of $l_{\rm DNA}^{1/2}$ for both using Eq. (48) in Eq. (20) and
for the approximate result of Eq. (54) divided by the asymptotic straight
approximation in Eq. (33). The approximate result shows clear deviations, but
it qualitatively captures the steep increase in $k_{\rm on}$ when $l_{\rm
DNA}$ becomes denser than $1/(l_{\rm sl}^{\rm eff})^{2}$.

##  0.6 Intersegmental transfers

Some proteins, such as the lac repressor hippel89 , are able to bind to more
than one DNA strand simultaneously. As a consequence these proteins can move
directly from one segment of DNA to another, without having to unbind from the
DNA and diffuse temporarily in the bulk water. This mechanism is called an
intersegmental transfer hippel89 , and it can be incorporated in the diffusion
equation along the DNA, Eq. (12), by introducing the extra term

$\displaystyle\frac{\partial n(x,t)}{\partial t}=\left(D_{\rm
1d}\frac{\partial^{2}}{\partial x^{2}}-k^{\rm ns}_{\rm
off}\right)n(x,t)-j(t)\delta(x)+G(x,t)$ $\displaystyle+k^{\rm ns}_{\rm
off}\int_{-\infty}^{\infty}dx^{\prime}\int_{0}^{t}dt^{\prime}\,W_{\rm
bulk}(x-x^{\prime},t-t^{\prime})n(x^{\prime},t^{\prime})$ $\displaystyle-
k_{\rm intra}\left(n(x,t)-\int_{-\infty}^{\infty}dx^{\prime}\,\lambda_{\rm
intra}(x-x^{\prime})n(x^{\prime},t)\right).$ (55)

Here $k_{\rm intra}$ is the rate at which intersegmental transfers are
occurring and $\lambda_{\rm intra}(x)$ is the distribution of distances the
protein transfers measured along the DNA. We here again assume independence
between transfers, i.e., that the the dynamics of the conformation of the coil
is fast compared with the rate of the transfers $k_{\rm intra}$, to simplify
the modelling considerably. By applying the same mathematical steps as in
Section 0.3 we find that the rate $k_{\rm 1d}$ can now be expressed as

$k_{\rm 1d}^{-1}=\int_{-\infty}^{\infty}\frac{dq}{2\pi}\frac{1}{D_{\rm
1d}q^{2}+k_{\rm off}^{\rm ns}(1-\lambda_{\rm bulk}(q))+k_{\rm
intra}(1-\lambda_{\rm intra}(q))}.$ (56)

We conclude from this, that the intersegmental transfers will increase $k_{\rm
1d}$ and result in faster target search, since the extra term contributes
positively in the denominator of the integrand. Thus increasing $k_{\rm
intra}$ will always result in faster target search. This is opposed to
increasing $k_{\rm off}^{\rm ns}$, since here there will be the opposing
effects of increasing $k_{\rm 1d}$ but lowering the density of loosely bound
proteins $n_{\rm ns}^{\rm eq}$ on the DNA.

There are two limits in which we can simplify the form of the extra term in
Eq. (56). One occurs when the transfers are typically very long, so that they
do not lead the protein to a place it has already visited. In this case we can
make the approximation $\lambda_{\rm intra}(q)\approx 0$ that we have
previously used for $\lambda_{\rm bulk}$ also. This limit has been studied in
sheinman09 , and it was found there that the intersegmental transfers can have
a significant impact with respect to lowering search times. Another relatively
simple limit, studied in lomholt05search , can emerge if the polymer that the
protein diffuses along is a freely fluctuating coil with low persistence
length (i.e., for DNA with a persistence length of around 150 base pairs
manning06 , this would require a coarse-graining in a sufficiently long
chain). In this case the distribution $\lambda_{\rm intra}(x)$ can have a
power law tail at large $x$, i.e., $\lambda_{\rm
intra}(x)\propto|x|^{-1-\alpha}$ ($x$ large), where for instance $\alpha=0.5$
for Gaussian chains. Motions with power law tails for jumping distances with
$0<\alpha<2$ are called Lévy flights. If the power law tail sets in at
distances short enough such that the motion is dominated by one dimensional
diffusion along the chain at these distances, then it will only be the power
law tail of the intersegmental transfer term that contributes to the integral
in Eq. (56), and we can approximate $k_{\rm intra}(1-\lambda_{\rm
intra}(q))\approx D_{\rm L}|q|^{\alpha}$ with $D_{\rm L}$ being a diffusion
constant for the Lévy flight. This $q$-scaling means that the corresponding
convolution term in Eq. (55) can be written as a fractional derivative. See
lomholt05search  for more on this including a study of optimal search
strategies, i.e., values of $k_{\rm off}^{\rm ns}$ that minimises search time
in this limit.

##  0.7 Conclusion

The mathematical theory presented in this chapter provides a way to make
predictions about the effect of coiling of DNA for target search by proteins.
For proteins with only one DNA binding site the effect of coiling is
quantified through the concentration $l_{\rm DNA}$ of DNA around the target.
However, the theory relies on many assumptions that limit the applications.
These include that: the DNA molecule is long and has a large persistence
length, correlations with previous bulk excursions vanish (possibly through
dynamics of the DNA), and dilute protein concentration. Thus there are many
relevant alternatives of modelling the effect of DNA coiling on target search
that has not been covered in this chapter. Just a few examples are: scaling
regimes when the persistence length cannot be assumed to be long hu06 , effect
of large scale heterogeneous network structure nyberg21 ; hedstrom23 , or
effect of the fractal nature of chromatin benichou11 .

Sequence heterogeneity was mentioned in Section 0.3 as a potential barrier for
fast search. However, combined with looping, sequence heterogeneity in the
form of auxiliary sites within looping distance of the target, can also help
in accelerating search bauer15 . Without looping such sites will act as traps,
and this introduces additional long time scales in the search process.
Modelling such traps will mean that the assumption of non-specific
equilibration and exponentially distributed search time employed in this
chapter will have to be examined carefully benichou09 .

We finish by noting that more recent studies have shown that at low chemical
concentrations chemical rate constants (i.e., inverse mean search times) may
become insufficient. Instead, even in quite generic geometries, repeated
first-passage times may be significantly different from each other
monasterio11 ; mattos12 , with characteristic times spanning several orders of
magnitude godec16 ; grebenkov18 . Given that transcription factors in living
cells may reach such low numbers (ten or few tens in the entire cell), it will
be of interest to explore the full span of relevant target localisation times,
in supercomputing studies ma20  or experimentally. In such scenarios of low
copy numbers the bursty protein production of a DNA-binding proteins encoded
in one gene leads to strong concentration fluctuations at the target gene, in
situations when the gene-gene distance is short pulkkinen13 .

Facilitated diffusion was demonstrated to occur both in vitro and in living
cells. Single-DNA experiments unveiled contributions of DNA coiling to the
localisation dynamics of DNA-binding proteins to their specific target site on
the DNA. In the future it will be of interest to follow individual proteins
from their production yu06  across the cell rienzo14 , to their eventual
binding to the DNA.

## References

  * [1] G. Adam and M. Delbrück.  Structural Chemistry and Molecular Biology, chapter Reduction of Dimensionality in Biological Diffusion Processes.  Freeman, San Francisco, 1968.
  * [2] A. D. Riggs, S. Bourgeois, and M. Cohn.  The lac repressor-operator interaction.  J. Mol. Biol., 53:401, 1970.
  * [3] P. H. Richter and M. Eigen.  Diffusion controlled reaction rates in spheroidal geometry: Application to repressor-operator association and membrane bound enzymes.  Biophys. Chem., 2(3):255–263, 1974.
  * [4] O. G. Berg and C. Blomberg.  Association kinetics with coupled diffusional flows. special application to the lac repressor–operator system.  Biophys Chem., 4:367–381, 1976.
  * [5] O. G. Berg and C. Blomberg.  Association kinetics with coupled diffusion. an extension to coiled-chain macromolecules applied to the lac repressor-operator system.  Biophys Chem., 7:33–39, 1977.
  * [6] O. G. Berg, R. B. Winter, and P. H. von Hippel.  Diffusion-driven mechanisms of protein translocation on nucleic acids. 1. models and theory.  Biochemistry, 20(24):6929–6948, 1981.
  * [7] O. G. Berg and M. Ehrenberg.  Association kinetics with coupled three- and one-dimensional diffusion. chain-length dependence of the association rate of specific DNA sites.  Biophys Chem., 15:41–51, 1982.
  * [8] P. H. von Hippel and O. G. Berg.  Facilitated target location in biological systems.  J. Biol. Chem., 264:675, 1989.
  * [9] I. Sokolov, R. Metzler, K. Pant, and M. Williams.  Target search of $n$ sliding proteins on a DNA.  Biophys. J., 89:895, 2005.
  * [10] Y. M. Wang, R. H. Austin, and E. C. Cox.  Single molecule measurements of repressor protein 1d diffusion on DNA.  Phys. Rev. Lett., 97:048302, 2006.
  * [11] P. Hammar, P. Leroy, A. Mahmutovic, E. G. Marklund, O. G. Berg, and J. Elf.  The lac repressor displays facilitated diffusion in living cells.  Science, 336:1595, 2012.
  * [12] D. M. Gowers and S. E. Halford.  Protein motion from non-specific to specific DNA by three-dimensional routes aided by supercoiling.  EMBO J., 22:1410±1418, 2003.
  * [13] B. van den Broek, M. A. Lomholt, S.-M. J. Kalisch, R. Metzler, and G. J. L. Wuite.  How DNA coiling enhances target localization by proteins.  Proc. Natl. Acad. Sci. USA, 105:15738, 2008.
  * [14] M. A. Lomholt, B. van den Broek, S.-M. Kalisch, G. J. L. Wuite, and R. Metzler.  Facilitated diffusion with DNA coiling.  Proc. Natl. Acad. Sci. USA, 106:8204, 2009.
  * [15] M. von Smoluchowski.  Drei Vorträge über Diffusion, Brownsche Molekularbewegung und Koagulation von Kolloidteilchen.  Phys. Z., 17:557, 1916.
  * [16] `https://github.com/mlomholt/coiledDNA`.
  * [17] M. Ptashne and A. Gann.  Genes and Signals.  Cold Spring Harbor Lab. Press, Plainview, NY, 2002.
  * [18] G. D. Stormo and D. S. Fields.  Specificity, free energy and information content in protein-DNA interactions.  Trends Biochem. Sci., 23:109, 1998.
  * [19] N. E. Buchler, U. Gerland, and T. Hwa.  On schemes of combinatorial transcription logic.  Proc. Natl. Acad. Sci. USA, 100:5136, 2003.
  * [20] M. Slutsky and L. A. Mirny.  Kinetics of protein-DNA interaction: Facilitated target location in sequence-dependent potential.  Biophysical Journal, 87:4021–4035, 2004.
  * [21] O. Bénichou, Y. Kafri, M. Sheinman, and R. Voituriez.  Searching fast for a target on DNA without falling to traps.  Phys. Rev. Lett., 103:138102, 2009.
  * [22] Barry D. Hughes.  Random Walks and Random Environments, volume 1.  Oxford University Press, Oxford, 1995.
  * [23] K. Pant, R. L. Karpel, I. Rouzina, and M. C. Williams.  Mechanical measurement of single-molecule binding rates: Kinetics of DNA helix-destabilization by T4 gene 32 protein.  J. Mol. Biol., 336:851–870, 2004.
  * [24] J. Krog, M. Alizadehheidari, E. Werner, S. K. Bikkarolla, J. O. Tegenfeldt, B. Mehlig, M. A. Lomholt, F. Westerlund, and T. Ambjörnsson.  Stochastic unfolding of nanoconfined DNA: experiments, model and Bayesian analysis.  J. Chem. Phys., 149:215101, 2018.
  * [25] T. Hu, A. Y. Grosberg, and B. I. Shklovskii.  How proteins search for their specific sites on DNA: the role of DNA conformation.  Biophys. J., 90:2731–2744, 2006.
  * [26] J. Abate and W. Whitt.  A unified framework for numerically inverting Laplace transforms.  INFORMS J. Comput., 18(4):408–421, 2006.
  * [27] M. Sheinman and Y. Kafri.  The effects of intersegmental transfers on target location by proteins.  Phys. Biol., 6:016003, 2009.
  * [28] M. A. Lomholt, T. Ambjörnsson, and R. Metzler.  Optimal target search on a fast-folding polymer chain with volume exchange.  Phys. Rev. Lett., 95:260603, 2005.
  * [29] G. S. Manning.  The persistence length of DNA is reached from the persistence length of its null isomer through an internal electrostatic stretching force.  Biophys. J., 91:3607, 2006.
  * [30] M. Nyberg, T. Ambjörnsson, P. Stenberg, and L. Lizana.  Modeling protein target search in human chromosomes.  Phys. Rev. Res., 3:013055, 2021.
  * [31] L. Hedström and L. Lizana.  Modelling chromosome-wide target search.  New J. Phys., 25:033024, 2023.
  * [32] O. Bénichou, C. Chevalier, B. Meyer, and R. Voituriez.  Facilitated diffusion of proteins on chromatin.  Phys. Rev. Lett., 106:038102, 2011.
  * [33] M. Bauer, E. S. Rasmussen, M. A. Lomholt, and R. Metzler.  Real sequence effects on the search dynamics of transcription factors on DNA.  Sci. Rep., 5, 2015.
  * [34] C. Mejía-Monasterio, G. Oshanin, and G. Schehr.  First passages for a search by a swarm of independent random searchers.  J. Stat. Mech., 2011:P06022, 2011.
  * [35] T. Mattos, C. Mejía-Monasterio, R. Metzler, and G. Oshanin.  First passages in bounded domains: When is the mean first passage time meaningful?  Phys. Rev. E, 86:031143, 2012.
  * [36] A. Godec and R. Metzler.  Universal proximity effect in target search kinetics in the few encounter limit.  Phys. Rev. X, 6:041037, 2016.
  * [37] D. Grebenkov, R. Metzler, and G. Oshanin.  Strong defocusing of molecular reaction times: geometry and reaction control.  Commun. Chem., 1:96, 2018.
  * [38] J. Ma, M. Do, M. A. Le Gros, C. S. Peskin, C. A. Larabell, Y. Mori, and S. A. Isaacson.  Strong intracellular signal inactivation produces sharper and more robust signaling from cell membrane to nucleus.  PLoS Comp. Biol., 16:e1008356, 2020.
  * [39] O. Pulkkinen and R. Metzler.  Distance matters: The impact of gene proximity in bacterial gene regulation.  Phys. Rev. Lett., 110:198101, 2013.
  * [40] J. Yu, J. Xiao, X. Ren, K. Lao, and X. S. Xie.  Probing gene expression in live cells, one protein molecule at a time.  Science, 311:1600, 2006.
  * [41] C. Di Rienzo, V. Piazza, E. Gratton, F. Beltram, and F. Cardarelli.  Probing short-range protein brownian motion in the cytoplasm of living cells.  Nat. Commun., 5:5891, 2014.