# Coarse-grained dynamics of transiently-bound fast linkers

Sophie Marbach  CNRS, Sorbonne Université, Physicochimie des Electrolytes et
Nanosystèmes Interfaciaux, F-75005 Paris, France  Courant Institute of
Mathematical Sciences, New York University, NY, 10012, U.S.A.  Christopher E.
Miles<EMAIL_ADDRESS><EMAIL_ADDRESS>Both authors contributed
equally to this work. Department of Mathematics, University of California,
Irvine, Irvine, USA, 92697.

###### Abstract

Transient bonds between fast linkers and slower particles are widespread in
physical and biological systems. In spite of their diverse structure and
function, a commonality is that the linkers diffuse on timescales much faster
compared to the overall motion of the particles they bind to. This limits
numerical and theoretical approaches that need to resolve these diverse
timescales with high accuracy. Many models, therefore, resort to effective,
yet ad-hoc, dynamics, where linker motion is only accounted for when bound.
This paper provides a mathematical justification for such coarse-grained
dynamics that preserves detailed balance at equilibrium. Our derivation is
based on multiscale averaging techniques and is broadly applicable. We verify
our results with simulations on a minimal model of fast linker binding to a
slow particle. We show how our framework can be applied to various systems,
including those with multiple linkers, stiffening linkers upon binding, or
slip bonds with force-dependent unbinding. Importantly, the preservation of
detailed balance only sets the ratio of the binding to the unbinding rates,
but it does not constrain the detailed expression of binding kinetics. We
conclude by discussing how various choices of binding kinetics may affect
macroscopic dynamics.

††preprint: AIP/123-QED

Transient bonds between fast molecules and other slower molecules are found
ubiquitously throughout physical systems. Such bonds enable momentum transfer
at microscopic scales and are at the root of diverse phenomena, including
linkers that tune the material properties of polymers [1, 2, 3], shape spatial
organization and function of biological systems[4, 5, 6], and determine the
rate of tethered chemical reactions [7]. Stochastic modeling of transient
binding of linkers via numerical or theoretical approaches is, therefore, of
widespread interest.

The diversity of phenomena goes hand in hand with a variety of systems and
hence of lexical terms: examples include cross-links in polymer meshes [1, 2],
metal-ligand bonds in self-assembled porous materials [8, 9], complementary
DNA pairs hybridizing between colloids [10, 11, 12, 13, 14, 15], and fast
myosin motors binding to slender actin fibers.[4, 5, 16] Henceforth, the word
linker will refer to any molecule with (a) a binding end that can form a bond
with another molecule and (b) that relaxes rapidly to a finite length. In the
previous examples, the linker is the polymer linker, ligand, DNA, or myosin
molecule. The bond will refer to the chemical bond formed between a linker and
a slower molecule (or another linker). In the previous examples, the bond is
the weak polymer-polymer adhesion, the metal-ligand chemical bond, the
hybridized DNA section, or the high-affinity myosin head after ATP hydrolysis.

Figure 1: Coarse-graining principle for a fast linker forming a transient
bond. (a) Cartoon of a common coarse-graining procedure where fast linkers
(pink) between slow fibers (blue) are modeled only when they are bound. In
this work, we study a minimal system where (b) a fast jiggling linker (pink)
binds and unbinds rapidly to a binding site on a slow particle with rates
$q_{\rm on}$ and $q_{\rm off}$ that depend on the relative distance. The 1D
position of the linker is $x_{\ell}$ and that of the slow particle $x_{s}$. In
this paper, we use multiscale averaging techniques to coarse-grain the
dynamics of the fast linker while preserving detailed balance. In the coarse-
grained model, only the slow particle dynamics are specified, with effective
kinetic rates $q_{\rm on}^{\rm eff}(x_{s})$ and $q_{\rm off}^{\rm eff}(x_{s})$
that only depend on the position of the slow particle $x_{s}$.

Despite enormous variations in mechanochemical properties of linking
molecules, one unifying feature is their diffusion on characteristic
timescales much faster than the overall motion of the fibers or objects they
link [17, 18]. This rapid diffusion of (often many) linking molecules creates
disparate time and length scales that must be resolved, often creating a
bottleneck for numerical or theoretical investigations [19, 8, 9]. To
alleviate this, it is common to rely on coarse-grained descriptions of the
linkers, either for their dynamics, binding kinetics, or both. In these
coarse-grained scenarios, crosslinkers are replaced by effective laws so that
their detailed dynamics need not be considered directly [20, 21, 22, 23, 16,
24, 25, 26, 27, 28]. For example, if a fast linker binds transiently to a slow
particle, a coarse-grained simulation or model would only specify the dynamics
of the slow particle; see illustration in Fig. 1. While the proposed effective
dynamics may be relatively intuitive in some scenarios, there does not seem to
be a systematic procedure for justifying and comparing this coarse-graining
across different systems [22, 16, 24]. This paper provides a mathematical
justification for such dynamics and, more importantly, a framework to derive
effective dynamics in various settings.

In establishing the justification for the effective dynamics with fast
linkers, several questions arise. Importantly, when linker dynamics are in
thermodynamic equilibrium, with binding and unbinding rates that depend on
mutual distance, detailed balance has to be enforced [18, 29]. An immediate
consequence is that the equilibrium distribution of the system is determined
by the binding and unbinding rates. As these detailed rates are often
unmeasured, a variety of choices are taken in the literature [20, 21, 22, 23,
16, 24, 25, 26, 27]. These choices of specific forms of binding and unbinding
spatial dependence introduce ambiguity in interpreting the resulting dynamics.
We, therefore, briefly discuss choices of kinetic rates and their consequences
in the coarse-graining procedure.

The paper is organized as follows. We first consider a minimal system (Fig.
1-b) made of a single fast linker binding to a slow particle (Sec. I) and
catalog the variety of modeling choices at this microscopic level. Next, we
rigorously coarse grain fast linker dynamics to obtain an effective model for
the slow particle (Sec. II) that we validate with numerical simulations (Sec.
III). We then show, with 3 examples, how we can apply our formalism to more
complex yet common setups (Sec. IV): we investigate (i) 2 fast reactive
linkers connecting to each other, (ii) a linker that stiffens upon binding,
and (iii) a slip bond with force-dependent unbinding. Finally, we discuss how
the choice of microscopic kinetic rates can affect the coarse-grained dynamics
both at short and long timescales in nontrivial ways (Sec. V). We hope our
framework will help to investigate systems with fast transient crosslinkers
more systematically.

##  I General system with transient crosslinkers

###  I.1 Microscopic dynamics

We consider the motion of a relatively slow particle representing, for
example, a slender actin filament, a cell, or a colloid [16, 18]. The slow
particle diffuses, for simplicity, in one spatial dimension (see Fig. 1-b,
blue particle) and we will later discuss how to extend the results to 3D. The
position of the slow particle at time $t$ is $x_{s}(t)$. The diffusion
coefficient of the particle is $D_{s}=k_{B}T/\gamma_{s}$, where $k_{B}$ is
Boltzmann’s constant, $T$ is temperature, and $\gamma_{s}$ is the friction
coefficient of the particle. The particle evolves in an external potential
$\mathcal{U}_{s}(x_{s})$, that could represent connections with other
particles. The forces in the unbound state on the particle are thus simply
$\mathcal{F}_{\rm u}=-\partial_{s}\mathcal{U}_{s}(x_{s})$.

We also track the motion of a relatively fast linker, for example, a myosin
head [16] or the sticky ends of a single-stranded DNA filament [18, 22]. The
fast linker’s position is $x_{\ell}$, and the linker diffuses with diffusion
coefficient $D_{\ell}=k_{B}T/\gamma_{\ell}$, where $\gamma_{\ell}$ is the
friction coefficient of the fast particle (see Fig. 1-b, pink linker). Here,
the linker is attached to another slow object or an immobile surface; for now,
we will consider it connected to a fixed point. This assumption ensures that
the binding is localized in space. The linker usually resists extension, as it
is made of a polymer or a protein that resists uncoiling [17, 18]. It is hence
reasonable to assume the linker is submitted to a recoil force,
$\mathcal{F}_{\ell}=-k_{\ell}(x_{\ell}-x_{\ell,0})$ where $k$ is a spring
constant [30], and $x_{\ell,0}$ is the rest length of the linker. Note that,
as long as the force is conservative, extending our approach to other
expressions is straightforward.

The unbound dynamics are

$\begin{cases}\frac{dx_{s}}{dt}&=\frac{\mathcal{F}_{\rm
u}(x_{s},x_{\ell})}{\gamma_{s}}+\sqrt{\frac{2k_{B}T}{\gamma_{s}}}\eta_{s}(t)\\\
&=-\frac{\partial_{s}\mathcal{U}_{s}(x_{s})}{\gamma_{s}}+\sqrt{\frac{2k_{B}T}{\gamma_{s}}}\eta_{s}(t)\\\
\frac{dx_{\ell}}{dt}&=-\frac{k_{\ell}}{\gamma_{\ell}}(x_{\ell}-x_{\ell,0})+\sqrt{\frac{2k_{B}T}{\gamma_{\ell}}}\eta_{\ell}(t)\end{cases}$
(1)

where the $\eta_{i}(t)$ are uncorrelated Gaussian white noises, where
$\langle\eta_{i}(t)\rangle=0$ and
$\langle\eta_{i}(t)\eta_{j}(t^{\prime})\rangle=\delta_{ij}\delta(t-t^{\prime})$
where $\delta_{ij}$ is the Kronecker symbol and $\langle\cdot\rangle$ is an
average over realizations of the noise. Without loss of generality, we will
shift the domain such that the rest length of the fast variable is at the
center of the domain, namely $x_{\ell,0}=0$.

The slow particle may transiently bind to the linker (see Fig. 1-b, orange
bond). In this entire paper, we will consider that when the bond is formed, it
corresponds to a stiff spring with spring constant $k_{b}$ added between the
particle and the linker. Hence, in the bound state, the forces on the slow
particle are $\mathcal{F}_{\rm
b}=-k_{b}(x_{s}-x_{\ell})-\partial_{s}\mathcal{U}_{s}(x_{s})$. The bound
dynamics are thus

$\begin{cases}\frac{dx_{s}}{dt}&=\frac{\mathcal{F}_{\rm
b}(x_{s},x_{\ell})}{\gamma_{s}}+\sqrt{\frac{2k_{B}T}{\gamma_{s}}}\eta_{s}(t)\\\
&=-\frac{k_{b}}{\gamma_{s}}(x_{s}-x_{\ell})-\frac{\partial_{s}\mathcal{U}_{s}(x_{s})}{\gamma_{s}}+\sqrt{\frac{2k_{B}T}{\gamma_{s}}}\eta_{s}(t)\\\
\frac{dx_{\ell}}{dt}&=\frac{k_{b}}{\gamma_{\ell}}(x_{s}-x_{\ell})-\frac{k_{\ell}}{\gamma_{\ell}}(x_{\ell})+\sqrt{\frac{2k_{B}T}{\gamma_{\ell}}}\eta_{\ell}(t).\end{cases}$
(2)

Our model for the bound dynamics is not unique. For example, one could
consider that instead of a stiff spring, the bond formed is a rigid rod
constraining the dynamics [31, 18, 26, 25]. Yet, we choose this spring bond
model as it is a simple starting point and because it is physically
satisfying. Indeed, with the spring model, the linker and the particle relax
towards each other, while in the rigid rod model, they can stay unnaturally
far apart. Ultimately we will consider the dynamics in the limit where the
bond is very stiff, corresponding to a so-called soft constraint [32, 33, 34].

The equilibrium distribution corresponding to these choices of dynamics can be
decomposed over the 2 states (bound and unbound) as $\pi=\left(\pi_{\rm
u},\pi_{\rm b}\right)^{T}$. The component of the equilibrium distribution
corresponding to the unbound state is

$\pi_{\rm
u}(x_{s},x_{\ell})=\frac{1}{Z_{u}}e^{-\mathcal{U}_{s}(x_{s})/k_{B}T-k_{\ell}(x_{\ell})^{2}/2k_{B}T}$
(3)

and the bound one is

$\pi_{\rm
b}(x_{s},x_{\ell})=\frac{1}{Z_{b}}e^{-k_{b}(x_{s}-x_{\ell})^{2}/2k_{B}T-\mathcal{U}_{s}(x_{s})/k_{B}T-k_{\ell}(x_{\ell})^{2}/2k_{B}T},$
(4)

where $Z_{u}$ and $Z_{b}$ are constant prefactors that are set by a
normalization condition on the total equilibrium distribution
$\int\text{d}x_{\ell}\text{d}x_{s}\left(\pi_{\rm u}+\pi_{\rm b}\right)=1$ and
by detailed balance, which we turn to now.

###  I.2 Possible kinetic rates and detailed balance

We consider that the linker and the particle bind to each other with rate
$q_{\rm on}$ and unbind with rate $q_{\rm off}$. To be physically accurate, it
is reasonable to assume that both rates may depend on the spatial variables
($x_{s},x_{\ell}$), a priori. While the exact expression of the rates for our
coarse-graining approach does not matter, it is crucial to recall how these
rates should be specified to satisfy detailed balance.

If the system we consider is at equilibrium, the rates must satisfy detailed
balance [29, 18]. Here this means the probability flux at equilibrium of going
from one state to the other is equal to the inverse flux, namely

$\pi_{\rm u}(x_{s},x_{\ell})q_{\rm on}(x_{s},x_{\ell})=\pi_{\rm
b}(x_{s},x_{\ell})q_{\rm off}(x_{s},x_{\ell}).$ (5)

Here this relation simplifies to

$\frac{q_{\rm on}}{q_{\rm
off}}=\frac{Z_{u}}{Z_{b}}e^{-k_{b}(x_{s}-x_{\ell})^{2}/2k_{B}T}.$ (6)

To make this expression more explicit, we can redefine the constants $Z_{u}=Z$
and $Z_{b}=Zq_{\rm off}^{0}/q_{\rm on}^{0}$, where $Z$ is a global
normalization constant such that
$\int\text{d}x_{\ell}\text{d}x_{s}\left(\pi_{\rm u}+\pi_{\rm b}\right)=1$.
Here $q_{\rm off}^{0}$ and $q_{\rm on}^{0}$ set the typical range of the
kinetic rates and are related via the typical free energy of bond formation
$E_{0}$ such that $q_{\rm on}^{0}/q_{\rm off}^{0}\equiv e^{-E_{0}/k_{B}T}$. We
obtain

$\frac{q_{\rm on}}{q_{\rm off}}=\frac{q_{\rm on}^{0}}{q_{\rm
off}^{0}}e^{-k_{b}(x_{s}-x_{\ell}+\ell_{b})^{2}/2k_{B}T}.$ (7)

Since this is the only relation that constrains $q_{\rm on}$ and $q_{\rm
off}$, it is clear that the choice of $q_{\rm on}$ and $q_{\rm off}$ is not
unique and that at least one of the rates has to depend on space.

Figure 2: Possible choices of binding and unbinding kinetics agreeing with
detailed balance. (a) constant binding rate, (b) constant unbinding rate and
(c) both rates varying; see text for detailed expressions. Here we chose a
bond spring constant $k_{b}L^{2}/k_{B}T=40$ and a fixed macroscopic
probability $\Pi_{b}=0.5$.

####  I.2.1 Possible expressions of the rates

Therefore, there are infinite possibilities in how we can specify rates
consistent with detailed balance, that reflects the diversity of choices made
in the literature [20, 21, 22, 23, 16, 24, 25, 26, 27]. However, we can
catalog a few simple commonly chosen examples, see Fig. 2, and discuss to what
extent they are consistent with physical intuition.

  * •

(model 0) Unbinding is faster further away from the target, but the binding
rate is constant,

$\begin{cases}&q_{\rm on}=q_{\rm on}^{0},\\\ &q_{\rm
off}(x_{s},x_{\ell})=q_{\rm
off}^{0}e^{k_{b}(x_{s}-x_{\ell}+\ell_{b})^{2}/2k_{B}T}\end{cases}$ (8)

This form is taken often in studies involving molecular motors, where force-
dependent binding has been observed directly [35, 36, 37]. However, we note
that this choice suffers from the numerically undesirable feature of the off
rate increasing exponentially as a function of distance.

  * •

(model 1) Binding is faster closer to the target, but unbinding is constant,

$\begin{cases}&q_{\rm on}(x_{s},x_{\ell})=q_{\rm
on}^{0}e^{-k_{b}(x_{s}-x_{\ell})^{2}/2k_{B}T},\\\ &q_{\rm off}=q_{\rm
off}^{0},\end{cases}$ (9)

which is a model used in Refs. 21, 24. Constant unbinding seems quite
unphysical since one would expect the bond to break if the linker and the
particle are brought too far apart. We can instead opt for an expression where
both rates depend on space, for example:

  * •

(model 2) Binding is faster closer to the target, unbinding is faster further
away from the target,

$\begin{cases}&q_{\rm on}(x_{s},x_{\ell})=\displaystyle\frac{q_{\rm
on}^{0}}{1+e^{k_{b}(x_{s}-x_{\ell})^{2}/2k_{B}T}},\\\ &q_{\rm
off}(x_{s},x_{\ell})=\displaystyle\frac{q_{\rm
off}^{0}}{1+e^{-k_{b}(x_{s}-x_{\ell})^{2}/2k_{B}T}}\end{cases}$ (10)

which solves the issues raised above.

While these are possible choices of the binding rates that agree with detailed
balance, these are not the only ones, and one could consider various other
kinetics, possibly with a kinetic barrier to overcome to unbind or to bind.

One might then wonder if the choice of the microscopic kinetic rates affects
the long-time dynamics of the system. We raise the question here but we will
not attempt to answer it thoroughly. Rather, our goal is, once the microscopic
dynamics are properly chosen, to show how to systematically coarse-grain the
dynamics of the fast linker.

####  I.2.2 Binding models with uniform kinetic rates

Inspired by previous models [38], we also discuss an alternative binding
model. Specifically, if one wanted the binding and unbinding rates to be
uniform, at least over a certain lengthscale, the only option is that the
dynamics are not specified via Eqs. 1 and 2 but have to be changed. We
illustrate this point briefly below.

For example, one could define uniform rates, as

$\begin{cases}&q_{\rm on}=q_{\rm on}^{0},\\\ &q_{\rm off}=q_{\rm
off}^{0}\end{cases}$ (11)

where the rates $q_{\rm on}^{0}$ and $q_{\rm off}^{0}$ are non zero. In that
case, the bound $\pi_{\rm b}$ and unbound $\pi_{\rm u}$ parts of the
equilibrium distribution must be the same, which constrains the forces to be
the same in each state. Otherwise, detailed balance is broken. For example,
the bound and unbound dynamics could both satisfy

$\begin{cases}\frac{dx_{s}}{dt}&=-\frac{k_{b}}{\gamma_{s}}(x_{s}-x_{\ell})-\frac{\partial_{x}\mathcal{U}_{s}(x_{s})}{\gamma_{s}}+\sqrt{\frac{2k_{B}T}{\gamma_{s}}}\eta_{s}(t)\\\
\frac{dx_{\ell}}{dt}&=\frac{k_{b}}{\gamma_{\ell}}(x_{s}-x_{\ell})-\frac{k_{\ell}}{\gamma_{\ell}}(x_{\ell})+\sqrt{\frac{2k_{B}T}{\gamma_{\ell}}}\eta_{\ell}(t).\end{cases}$
(12)

As a result, the dynamics are not very interesting as they simply amount to a
global quadratic confining potential with spring constant $k_{b}$.

Tethered dynamics are more in line with physical intuition if the confining
potential associated with the bond is restricted to a patch in space.
Following Ref. 38, 39, this can be achieved by defining kinetic rates that are
non-zero only when the linker and the particle are close, typically

$\begin{cases}&q_{\rm on}=q_{\rm on}^{0}H(|x_{s}-x_{\ell}|<R),\\\ &q_{\rm
off}=q_{\rm off}^{0}H(|x_{s}-x_{\ell}|<R)\end{cases}$ (13)

where $R$ is a characteristic distance that sets a reasonable maximum binding
distance [25, 26]. One can define $R=\sqrt{2k_{B}T/k_{b}}$ where $k_{b}$ is
the spring constant of the bond, and $R$ corresponds, therefore, to thermal
vibrations around that bond. When the linker and the particle are nearby
$|x_{s}-x_{\ell}|<R$, the dynamics follow Eq. (12) while they are given by
Eqs. (1) and (2) otherwise. This ensures that detailed balance is satisfied in
both regions, since when $|x_{s}-x_{\ell}|\geq R$ the kinetic rates are zero.
We refer to models with uniform kinetic rates as in Eq. (13), as Doi models
[40, 41, 29] in analogy with reaction models without linkers.

However, such uniform kinetic rates are unphysical in 2 ways: (i) bond
dissociation is not allowed when the bond is stretched beyond $R$, which seems
rather unlikely; and (ii) when they are close, the interaction between the
linker and particle is the same regardless of the state of the bond – see Eq.
(12)), which also seems rather unlikely.

We will briefly show in Sec. V how also at the coarse-grained level, important
physical differences arise between Doi models with uniform kinetic rates as in
Eq. (13), and models with spatially dependent kinetic rates as say in Eq.
(10). This means, again, that the choice of kinetic rates affects short-time
dynamics at least in non-trivial ways, and hence that is an important
assumption of any model.

####  I.2.3 Out-of-equilibrium dynamics

If the system is driven out-of-equilibrium, with some consumption of energy,
then detailed balance Eq. (5) can be broken: it is possible that probability
fluxes are not in equilibrium. One can then choose a priori any functional
form for the rates, in some way that is consistent with energy production
[42]. In the future, it would be interesting to quantify how far from
equilibrium a system is, relative to detailed balance, in such studies. We
leave this question open.

##  II Coarse-grained dynamics with fast linkers

Our aim is now to rigorously coarse-grain the linker dynamics to obtain the
effective long-time dynamics of the particle. This is useful both to gain
analytic insight, but also to propose consistent and fast simulation schemes.
We use multiscale averaging [43] to coarse-grain the dynamics, a technique
which is broadly used in the field to properly average over short length
scales and timescales [31, 18, 20, 21, 36]. We will show the approach is valid
over a broad range of parameters in Sec. III. We provide Table 2 as a summary
of the main notations used throughout the discussion.

###  II.1 Set up of the dynamics

The set of stochastic Eqns. (1)-(2) defines a Markov process that is
conveniently studied via the Kolmogorov backward equation [44, 43] on the
functions
$f(x_{s},x_{\ell},t)=\left(f_{u}(x_{s},x_{\ell},t),f_{b}(x_{s},x_{\ell},t)\right)^{T}$

$\partial_{t}f=\mathcal{L}f\,,\quad f(x_{s},x_{\ell},0)=g(x_{s},x_{\ell})\,,$
(14)

where $\mathcal{L}$ is the generator of the system and $g$ is any scalar
function. Here the functions $f(x_{s},x_{\ell},t)=\int
p(x_{s}^{\prime},x_{\ell}^{\prime},t|x_{s},x_{\ell})g(x_{s}^{\prime},x_{\ell}^{\prime})\text{d}x_{\ell}^{\prime}\text{d}x_{s}^{\prime}$
give the expectation of any scalar function $g(x_{s}(t),x_{\ell}(t))$, given
an initial condition $x_{s}(0)=x_{s},x_{\ell}(0)=x_{\ell}$, where $p$ is the
probability density that the system evolved from the initial condition to
$(x_{s}^{\prime},x_{\ell}^{\prime})$ at time $t$. Once we know how such
functions $f$ evolve, we may calculate any statistic $g$ of our stochastic
process.

The generator $\mathcal{L}$ can be calculated from the forward equation, the
Fokker Planck equation, associated with the dynamics Eqns. (1)-(2), see
Appendix B.1. It is

$\mathcal{L}=\mathcal{Q}+\mathcal{V}$ (15)

where

$\displaystyle\mathcal{Q}$ $\displaystyle=\begin{pmatrix}-q_{\rm on}&q_{\rm
on}\\\ q_{\rm off}&-q_{\rm
off}\end{pmatrix},\,\,\,\text{and}\,\,\mathcal{V}=\begin{pmatrix}\mathcal{V}_{u}&0\\\
0&\mathcal{V}_{b}\end{pmatrix}\,\,\text{with}$ $\displaystyle\mathcal{V}_{u}$
$\displaystyle=-\frac{k_{\ell}}{\gamma_{\ell}}x_{\ell}\partial_{\ell}+\frac{k_{B}T}{\gamma_{\ell}}\partial_{\ell\ell}-\frac{\partial_{x}\mathcal{U}(x_{s})}{\gamma_{s}}x_{s}\partial_{s}+\frac{k_{B}T}{\gamma_{s}}\partial_{ss}$
$\displaystyle\mathcal{V}_{b}$
$\displaystyle=-\frac{k_{\ell}}{\gamma_{\ell}}x_{\ell}\partial_{\ell}+\frac{k_{B}T}{\gamma_{\ell}}\partial_{\ell\ell}-\frac{\partial_{x}\mathcal{U}(x_{s})}{\gamma_{s}}x_{s}\partial_{s}+\frac{k_{B}T}{\gamma_{s}}\partial_{ss}$
$\displaystyle\qquad-\frac{k_{b}}{\gamma_{\ell}}(x_{\ell}-x_{s})\partial_{\ell}-\frac{k_{b}}{\gamma_{s}}(x_{s}-x_{\ell})\partial_{s}.$

###  II.2 Nondimensionalization

To highlight the ratio between the different temporal and spatial scales at
play, we non-dimensionalize our equations. Since the particle’s motion is much
slower than that of the linker, we can identify a small number
$\varepsilon=\gamma_{\ell}/\gamma_{s}$. We seek the dynamics of the slow
particle over a typical timescale $\tau_{0}$ such that its motion is
diffusive, extending over the range $x_{0}=\sqrt{D_{0}t_{0}}$. We now can call
$q_{\rm on}^{\prime}=q_{\rm on}\tau_{0}$ and similarly $q_{\rm
off}^{\prime}=q_{\rm off}\tau_{0}$, remembering that these are both functions
of space. We consider for now that we observe the dynamics over timescales
$\tau_{0}$ where transient binding and unbinding are still relevant such that
$q_{\rm on}^{\prime}=O(1)$. Since $\gamma_{s}/\gamma_{\ell}=1/\varepsilon$, we
have
$k_{\ell}\tau_{0}/\gamma_{\ell}=k_{\ell}\tau_{0}/\gamma_{s}=1/\varepsilon$.
Typical slow dynamics are captured by the timescale $\tau_{0}$ so again we
consider $\kappa=k_{\ell}\tau_{0}/\gamma_{s}=O(1)$. We also write
$\lambda=k_{b}\tau_{0}/\gamma_{s}=O(1)$ for now.

In this scaling, we have the nondimensional generator
$\mathcal{L}^{\prime}=\mathcal{Q}^{\prime}+\mathcal{V}^{\prime}$ such that

$\mathcal{Q}^{\prime}=\begin{pmatrix}-q_{\rm on}^{\prime}&q_{\rm
on}^{\prime}\\\ q_{\rm off}^{\prime}&-q_{\rm off}^{\prime}\end{pmatrix}$

and

$\displaystyle\mathcal{V}_{u}^{\prime}$
$\displaystyle=-\frac{\kappa}{\varepsilon}x_{\ell}\partial_{\ell}+\frac{1}{\varepsilon}\partial_{\ell\ell}-\frac{\partial_{s}\mathcal{U_{s}}(x_{s})}{k_{B}T}\partial_{s}+\partial_{ss}$
$\displaystyle\mathcal{V}_{b}^{\prime}$
$\displaystyle=-\frac{\kappa}{\varepsilon}x_{\ell}\partial_{\ell}+\frac{1}{\varepsilon}\partial_{\ell\ell}-\frac{\partial_{s}\mathcal{U_{s}}(x_{s})}{k_{B}T}\partial_{s}+\partial_{ss}$
$\displaystyle\qquad-\frac{\lambda}{\varepsilon}(x_{\ell}-x_{s})\partial_{\ell}-\lambda(x_{\ell}-x_{s})\partial_{s}.$

In the following, we will drop the $\cdot^{\prime}$ notations for simplicity.

###  II.3 Averaging procedure

We then look for a solution to $\partial_{t}f=\mathcal{L}f$ of the form
$f=f_{0}+\epsilon f_{1}+\epsilon^{2}f_{2}$. To first order, we have

$\displaystyle\begin{pmatrix}-\kappa
x_{\ell}\partial_{\ell}+\partial_{\ell\ell}&0\\\ 0&-\kappa
x_{\ell}\partial_{\ell}-\lambda(x_{\ell}-x_{s})\partial_{\ell}+\partial_{\ell\ell}\end{pmatrix}f_{0}=0$
(16)

Notice that $-\kappa
x_{\ell}-\lambda(x_{\ell}-x_{s})=-(\kappa+\lambda)\left(x_{\ell}-x_{s}\frac{\lambda}{\kappa+\lambda}\right)$
such that the general solution to Eq. (16) is

$\begin{split}f_{0}&=\begin{pmatrix}g_{1}(x_{s},t)\\\
g_{2}(x_{s},t)\end{pmatrix}+\begin{pmatrix}h_{1}(x_{s},t)\\\
0\end{pmatrix}\int_{0}^{x_{\ell}}e^{-\kappa x^{2}}\,\text{d}x\\\
&\qquad+\begin{pmatrix}0\\\
h_{2}(x_{s},t)\end{pmatrix}\int_{0}^{x_{\ell}}e^{-(\kappa+\lambda)(x-\frac{\lambda}{\kappa+\lambda}x_{s})^{2}}\,\text{d}x\end{split}$
(17)

and for which the associated equilibrium distribution is of the form

$\pi_{0}=\frac{1}{\sqrt{2\pi}}\begin{pmatrix}\alpha e^{-\kappa
x_{\ell}^{2}/2}\\\ \beta
e^{-(\kappa+\lambda)(x_{\ell}-\frac{\lambda}{\kappa+\lambda}x_{s})^{2}/2}\end{pmatrix}.$
(18)

Here $\alpha$ and $\beta$ are free parameters and are not constrained by
detailed balance; they can be any real numbers. In fact, there is no exchange
between the bound and the unbound times at these very short timescales. Notice
that this means that the averaging approach does not know a priori that it
should preserve detailed balance.

In any case, we require $\langle f_{0},\pi_{0}\rangle_{\ell}$ to be bound;
$\langle f,p\rangle_{\ell}$ is the inner product where the integration is only
carried over the fast variable $x_{\ell}$. This imposes $h_{1}\equiv 0$ and
$h_{2}\equiv 0$, and $f_{0}=\left(g_{1}(x_{s},t),g_{2}(x_{s},t)\right)^{T}$ is
actually independent of the fast variable.

Seeking the next order
$\mathcal{L}_{0}f_{1}=-\mathcal{L}_{1}f_{0}+\partial_{t}f_{0}$. A solution
exists for $f_{1}$ if the Fredholm alternative is satisfied [43], namely if
$\langle(\partial_{t}f_{0}-\mathcal{L}_{1}f_{0}).\pi_{0}\rangle_{\ell}=0$ is
true for any $\pi_{0}$ in the nullspace of $\mathcal{L}_{0}^{\star}$. This
corresponds to any real value combinations of $\alpha$ and $\beta$, and hence
we may pick the convenient choice of $(\alpha,\beta)=(1,0)$ and
$(\alpha,\beta)=(0,1)$. We obtain

$\begin{split}\partial_{t}g_{1}&=-\frac{\int e^{-\kappa x_{\ell}^{2}/2}q_{\rm
on}\text{d}x_{\ell}}{\int e^{-\kappa
x_{\ell}^{2}/2}\text{d}x_{\ell}}(g_{1}-g_{2})-x_{s}\partial_{s}g_{1}+\partial_{ss}g_{1}\\\
\partial_{t}g_{2}&=\frac{\int
e^{-(\kappa+\lambda)(x_{\ell}-\frac{\lambda}{\kappa+\lambda}x_{s})^{2}/2}q_{\rm
off}\text{d}x_{\ell}}{\int
e^{-(\kappa+\lambda)(x_{\ell}-\frac{\lambda}{\kappa+\lambda}x_{s})^{2}/2}\text{d}x_{\ell}}(g_{1}-g_{2})-x_{s}\partial_{b}g_{2}+\partial_{ss}g_{2}\\\
&-\frac{\lambda\left(\int(x_{s}-x_{\ell})e^{-(\kappa+\lambda)(x_{\ell}-\frac{\lambda}{\kappa+\lambda}x_{s})^{2}/2}dx_{\ell}\right)}{\int
e^{-(\kappa+\lambda)(x_{\ell}-\frac{\lambda}{\kappa+\lambda}x_{s})^{2}/2}\text{d}x_{\ell}}\partial_{s}g_{2}.\end{split}$
(19)

###  II.4 Effective kinetic rates and dynamics

We have just obtained the coarse-grained backward equations for the dynamics.
We can carry over the last line’s integral and return to dimensional units.
This allows us to directly read off the effective on and off rates

$\begin{cases}q_{\rm on}^{\rm eff}(x_{s})&=\frac{\int
e^{-k_{\ell}x_{\ell}^{2}/2k_{B}T}q_{\rm
on}(x_{\ell},x_{s})\text{d}x_{\ell}}{\int
e^{-k_{\ell}x_{\ell}^{2}/2k_{B}T}\text{d}x_{\ell}},\\\ q_{\rm off}^{\rm
eff}(x_{s})&=\frac{\int
e^{-(k_{\ell}+k_{b})(x_{\ell}-\frac{k_{b}}{k_{\ell}+k_{b}}x_{s})^{2}/2k_{B}T}q_{\rm
off}(x_{\ell},x_{s})\text{d}x_{\ell}}{\int
e^{-(k_{\ell}+k_{b})(x_{\ell}-\frac{k_{b}}{k_{\ell}+k_{b}}x_{s})^{2}/2k_{B}T}\text{d}x_{\ell}}.\end{cases}$
(20)

We find that the effective on and off rates are weighted averages over the
distribution of positions of the fast variable in the respective states. We
can also read off the effective dynamics of the slow particle in the unbound
and bound states

$\begin{cases}\frac{dx_{s}}{dt}&=-\frac{\partial_{x}\mathcal{U}(x_{s})}{\gamma_{s}}+\sqrt{\frac{2k_{B}T}{\gamma_{s}}}\eta_{u}(t)\,\,\text{(unbound)}\\\
\frac{dx_{s}}{dt}&=-\frac{\partial_{x}\mathcal{U}(x_{s})}{\gamma_{s}}-\frac{K}{\gamma_{s}}x_{s}+\sqrt{\frac{2k_{B}T}{\gamma_{s}}}\eta_{b}(t)\,\,\text{(bound).}\end{cases}$
(21)

where $K=\frac{k_{\ell}k_{b}}{k_{\ell}+k_{b}}$ is an effective spring
constant. We find the unbound dynamics of the fast particle are not changed by
the coarse-grained approach. However, in the bound state, an extra recoil
force is exerted on the slow particle, corresponding to a force averaged over
the fast-moving linker. Interestingly, the spring constant $K$ of the bond
formed at the coarse-grained level corresponds to the effective spring
constant corresponding to two springs in series, with spring constants
$k_{\ell}$ and $k_{b}$. This is precisely what is expected physically and from
the sketch of the setup; see Fig. 1-b. Note that this is a new result that
allows one to give meaning to the spring constant $K$ used in effective models
such as those used in Refs. [21, 24].

###  II.5 Detailed balance at the coarse-grained level

Finally, for physical consistency, we need to check that the marginal
equilibrium distribution, i.e. the equilibrium distribution integrated over
the fast degrees of freedom $\pi^{\rm
eff}(x_{s})=\int\pi(x_{s},x_{\ell})\text{d}x_{\ell}$, is indeed a stationary
solution of the effective dynamics obtained. The marginal distribution is

$\begin{split}\pi^{\rm eff}(x_{s})&=\begin{pmatrix}\pi_{\rm u}^{\rm
eff}(x_{s})\\\ \pi_{\rm b}^{\rm eff}(x_{s})\end{pmatrix}\\\
&=\frac{\sqrt{2\pi}e^{-\mathcal{U}_{s}(x_{s})/k_{B}T}}{Z}\begin{pmatrix}\frac{1}{\sqrt{k_{\ell}}}\\\
\frac{q_{\rm on}^{0}}{q_{\rm
off}^{0}\sqrt{k_{\ell}+k_{b}}}e^{-K\frac{x_{s}^{2}}{2k_{B}T}}.\end{pmatrix}\end{split}$
(22)

It is clear that the marginal distribution of either state is indeed a
stationary solution of the dynamics in each state as specified in Eq. (21).

To check that the marginal distribution is a stationary distribution of the
dynamics as a whole, we still need to check that detailed balance is satisfied
at the coarse-grained level. This is not guaranteed a priori since the
averaging technique does not use at any point that it should preserve detailed
balance. At this point, it is important to notice that in fact the effective
rates are related to the equilibrium probability distribution as

$q_{\rm on}^{\rm eff}(x_{s})=\frac{\int q_{\rm on}(x_{s},x_{\ell})\pi_{\rm
u}(x_{s},x_{\ell})dx_{\ell}}{\int\pi_{\rm u}(x_{s},x_{\ell})dx_{\ell}}$ (23)

and similarly for $q_{\rm off}^{\rm eff}$. Hence, we simply have that

$\displaystyle\pi_{\rm u}^{\rm eff}q_{\rm on}^{\rm eff}(x_{s})$
$\displaystyle=\pi_{\rm u}^{\rm eff}\frac{\int q_{\rm
on}(x_{s},x_{\ell})\pi_{\rm u}(x_{s},x_{\ell})dx_{\ell}}{\int\pi_{\rm
u}(x_{s},x_{\ell})dx_{\ell}}$ $\displaystyle=\int q_{\rm
on}(x_{s},x_{\ell})\pi_{\rm u}(x_{s},x_{\ell})dx_{\ell}$ $\displaystyle=\int
q_{\rm off}(x_{s},x_{\ell})\pi_{\rm b}(x_{s},x_{\ell})dx_{\ell}$
$\displaystyle=\int q_{\rm off}(x_{s},x_{\ell})\pi_{\rm
b}(x_{s},x_{\ell})dx_{\ell}$ $\displaystyle\qquad\times\frac{\pi_{\rm b}^{\rm
eff}}{\int\pi_{\rm b}(x_{s},x_{\ell})dx_{\ell}}$ $\displaystyle=\pi_{\rm
b}^{\rm eff}q_{\rm off}^{\rm eff}(x_{s}).$

Detailed balance is therefore also true at the coarse-grained level. Since the
marginal distribution is consistent with the dynamics in each state and with
detailed balance, it is indeed the stationary solution to the effective
dynamics.

###  II.6 General coarse-grained dynamics

The averaging approach can be extended in a straightforward way, following the
averaging steps above, to more arbitrary dynamics, and we summarize effective
dynamics in full generality in Table 1. The initial forces on the slow
particle in the bound $\mathcal{F}_{\rm b}(x_{s},x_{\ell})$ and the unbound
$\mathcal{F}_{\rm u}(x_{s},x_{\ell})$ states can be arbitrary forces. All
forces on the particle and linker should be conservative so as to define an
equilibrium distribution. Table 1 then reports the general formulas for the
effective binding and unbinding rates as well as the effective force on the
particle in the unbound and bound states. The diffusive part of the slow
particle motion is not affected by the coarse-graining procedure, since we
assume diffusion coefficients do not depend on space. Finally, the formulas
can be extended in a straightforward way to 3D dynamics, and to multiple fast
linkers. We will show in Sec. IV how to use these formulas with specific
examples.

Table 1: Summary of the effective dynamics considering a coarse-grained fast
linker ($x_{\ell}$) that can bind to a slow particle ($x_{s}$). For 3D
coordinates, the integrals have to be carried over all the spatial dimensions
of the fast variable $x_{\ell}$ and the force field on the slow particle has
to be obtained coordinate by coordinate.

effective binding rate $\displaystyle\displaystyle q_{\rm on}^{\rm
eff}(x_{s})=\frac{\int q_{\rm on}(x_{s},x_{\ell})\pi_{\rm
u}(x_{s},x_{\ell})dx_{\ell}}{\int\pi_{\rm u}(x_{s},x_{\ell})dx_{\ell}}$ (24)

effective unbinding rate $\displaystyle\displaystyle q_{\rm off}^{\rm
eff}(x_{s})=\frac{\int q_{\rm off}(x_{s},x_{\ell})\pi_{\rm
b}(x_{s},x_{\ell})dx_{\ell}}{\int\pi_{\rm b}(x_{s},x_{\ell})dx_{\ell}}$ (25)

effective force on the slow unbound particle $\displaystyle\mathcal{F}_{\rm
u}^{\rm eff}(x_{s})=\frac{\int\mathcal{F}_{\rm u}(x_{\ell},x_{s})\pi_{\rm
u}(x_{s},x_{\ell})dx_{\ell}}{\int\pi_{\rm u}(x_{s},x_{\ell})dx_{\ell}}$ (26)

effective force on the slow bound particle
$\displaystyle\displaystyle\mathcal{F}_{\rm b}^{\rm
eff}(x_{s})=\frac{\int\mathcal{F}_{\rm b}(x_{\ell},x_{s})\pi_{\rm
b}(x_{s},x_{\ell})dx_{\ell}}{\int\pi_{\rm b}(x_{s},x_{\ell})dx_{\ell}}$ (27)

Notice how each formula from Eq. (24-25) can be interpreted intuitively. The
effective force in a given state or the rate of switching from that state is
simply the spatial average weighted by the local probability distribution to
be in that state. While this seems consistent a posteriori, and consistent
with detailed balance at the coarse grained level, the formulas in Eq. (24-25)
are not the only possible expressions for the effective rates that obey
detailed balance. The results of Eq. (24-25) are therefore not trivial.

Interestingly, formulas in Table 1 are straightforward to compute. One simply
needs to know the equilibrium probability distribution of the free and bound
linker to obtain the effective dynamics 111Note that one only needs to know
the dependence of the equilibrium distribution on the linker coordinate
$x_{\ell}$, and not the details of the landscape for the slow particle
$x_{s}$. Formally it is sufficient to know $\pi_{\rm
u}(x_{s},x_{\ell})/\pi_{\rm u}(x_{s},0)$ and similarly for $\pi_{\rm b}$, to
obtain the formulas in Eq. (24-25).. In practice, one could then simulate (or
calculate) a free and bound linker, obtain their local probability
distributions, and integrate them to obtain the effective dynamics. We note
the caveat that the full probability distribution needs to be evaluated if
direct interactions between linkers exist.

###  II.7 Limit regimes

We will now comment on the effective dynamics and kinetic rates obtained for
the minimal system of Fig. 1-b (Eqs. (1) and (2)) in a few limiting regimes of
the system parameters. In the following, it will be useful to specify the
expression of the rates and we will use for example Eq. (9).

In all limiting regimes, some variables such as $q_{\rm on}^{0}/q_{\rm
off}^{0}$ have to be specified with respect to the limits. In practice, the
macroscopic probability of being bound, $\Pi_{\rm b}$, (or unbound, $\Pi_{\rm
u}$) can be measured experimentally [46, 18], and is easier to probe than
spatially dependent kinetic rates. Hence, we constrain the different
functional forms for the kinetic rates to predict the same $\Pi_{\rm b}$. The
macroscopic probability of being bound (respectively unbound) is $\Pi_{\rm
b}=\int dx_{s}dx_{\ell}\pi_{\rm b}(x_{s},x_{\ell})$ (resp. $\Pi_{\rm u}=\int
dx_{s}dx_{\ell}\pi_{\rm u}(x_{s},x_{\ell})$). It is easy to show that here

$\frac{\Pi_{\rm b}}{\Pi_{\rm
u}}=e^{-E_{0}/k_{B}T}\sqrt{\frac{K}{k_{b}}}\frac{\int
e^{-\mathcal{U}_{s}(x_{s})/k_{B}T}e^{-k_{\rm
eff}x_{s}^{2}/2k_{B}T}\text{d}x_{s}}{\int
e^{-\mathcal{U}_{s}(x_{s})/k_{B}T}\text{d}x_{s}}$ (28)

such that the macroscopic bound and unbound probabilities measure in
particular the energy of the bond, weighted by the geometry of the system. Of
course, this does not constrain the local values of the kinetic rates. Rather,
it sets the value of some parameters, here of $E_{0}$.

####  II.7.1 Stiff bond.

We first investigate the limit regime where the bond is very stiff, i.e.
$k_{b}\gg k_{\ell}$. This is the so-called “soft constraint” limit [32, 33,
34]. In that case, we expect the fast and slow particles are constrained to
move synchronously in the bound state. For this limit to make sense, we need,
according to Eq. (28), to constrain $q_{\rm on}^{0}\sim\sqrt{k_{b}}q_{\rm
off}^{0}$.

The effective force in the bound state converges to $\mathcal{F}_{\rm u}^{\rm
eff}=-k_{l}x_{s}-\partial_{s}\mathcal{U}_{s}$, i.e. simply to a recoil force
exerted with a spring constant corresponding to that of the fast linker (and
the force deriving from the external potential). This makes sense since in the
limit $k_{b}\gg k_{\ell}$, since the springs are in series, we expect the
weakest spring to take over and $K\sim k_{\ell}$.

The effective kinetic rates, according to Eq. (20) (and choosing the kinetic
rates as in Eq. (9)), are

$q_{\rm on}^{\rm eff}=q_{\rm
on}^{0}\sqrt{\frac{K}{k_{b}}}e^{-Kx_{s}^{2}/2k_{B}T},\,\,q_{\rm off}^{\rm
eff}=q_{\rm off}^{0}.$ (29)

In the limit regime where $k_{b}\gg k_{\ell}$, we have $q_{\rm on}\sim q_{\rm
off}^{0}e^{-k_{\ell}x_{s}^{2}/2k_{B}T}$ and $q_{\rm off}=q_{\rm off}^{0}$.
Hence the on rate remains spatially dependent, and the particle binds more
likely when it is closer to the average linker position. It unbinds though at
the same rate regardless of the linker position. Similar results may be found
with other initial choices of kinetic rates such as with Eq. (8) or (10).

A few models in the literature actually take the spring constant for the
binding kinetics to be the linker’s spring constant $K=k_{\ell}$ [21, 24],
making the implicit, albeit rather physical, assumption that the bond’s spring
constant is much stiffer than the linker’s. This underlines that the meaning
of the parameters in the kinetic rates is underappreciated in the field.

Notice that here we explored a limit regime after taking the limit of fast
linker dynamics. This is not an issue here since the limits commute. Indeed,
one could initially consider an infinitely stiff bond, and then take the limit
of fast linker dynamics, and get the same result – see Appendix B.2.

####  II.7.2 Stiff linker.

When the fast linker is stiff, then $k_{\ell}\gg k_{b}$. We then simply have
that the force in the bound state is determined by the spring constant of the
bond $\mathcal{F}_{\rm u}^{\rm eff}=-k_{b}x_{s}-\partial_{s}\mathcal{U}_{s}$,
as expected since now $K\sim k_{b}$ in that limit. With the same choice of
kinetic rates as in Eq. (9), we simply have that the effective on rate is
faster near the average linker position $q_{\rm on}^{\rm eff}=q_{\rm
on}^{0}e^{-\frac{k_{b}x_{s}^{2}}{2k_{B}T}}$ and the effective off rate is
constant. All in all, this is a simple consequence of the dynamics of 2
springs in series, when one of the spring constants is stiff compared to the
other.

##  III Validation of the coarse-graining approach

We now use numerical simulations to test the derived effective forces and
kinetic rates, and determine the range of parameters over which the averaging
procedure is valid.

###  III.1 Simulation set up

We simulate the dynamics specified through Eqs. (1)-(2) (with no confining
potential on the slow particle $\mathcal{U}(x_{s})\equiv 0$). We use the same
nondimensional variables $\tau_{0}$ and
$x_{0}=\sqrt{D_{0}\tau_{0}}=\sqrt{k_{B}T\tau_{0}/\gamma_{s}}$ such that the
problem is fully characterized by 5 non-dimensional numbers
$\varepsilon,\kappa,\lambda,q_{\rm off}^{\prime}$ and $\Pi_{\rm b}/\Pi_{\rm
u}$. Here, $\varepsilon=\gamma_{\ell}/\gamma_{s}$ is not necessarily small and
represents the ratio between friction coefficients;
$\kappa=k_{\ell}\tau_{0}/\gamma_{s}$ is the non-dimensional spring constant of
the linker; $\lambda=k_{b}\tau_{0}/\gamma_{s}$ is that of the bond,
$q^{\prime}_{\rm off}=q_{\rm off}^{0}\tau_{0}$ the non-dimensional typical
off-rate. Finally, the on rate is set by the conservation of macroscopic
probability through Eq. (28), so by the ratio $\Pi_{\rm b}/\Pi_{\rm u}$, once
a choice of functional forms for the kinetic rates has been made. Here, we
will keep Eq. (10) as an example.

We discretize the dynamics of both slow and fast particles with a standard
Euler-Maruyama scheme with time step $\Delta t$ until a terminal time of
$T=1000\tau_{0}$ with $M=1000$ simulations for each set of parameters. We
impose periodic boundary conditions on the slow particle at a non-dimensional
distance $L$ so that the slow particle does not escape far away from the
domain. The domain size $L=10x_{0}$ and $\Delta t=0.01\tau_{0}$ are chosen
sufficiently large and small respectively such that our results do not depend
on the specific value.

To estimate the effective rates from simulation, we note that this is a
doubly-stochastic Poisson process, or a Cox process[47], because the
stochastic dynamics of the particle positions drive the stochastic Poisson
events of binding and unbinding. While sophisticated methods for inference on
Cox intensities exist [47], a simple binning approach suffices here. We
discretize $x_{s}$ into bins of width $\Delta x$ and count the number of
occurrences of each event in each bin. For bin $i$ with center $x_{i}$, the
estimated flux to the other state is then $\hat{J}(x_{i})=N_{i}/(TM\Delta x)$,
where $\hat{\cdot}$ notation means estimate and $N_{i}$ is the number of
events (either binding or unbinding) occurring in bin $i$. Notably, this
estimates the macroscopic kinetic rates, for instance, for unbinding
$\hat{J}_{\mathrm{off}}^{\rm eff}(x_{s})\approx q^{\rm eff}_{\rm
off}(x_{s})\pi_{\rm on}^{\rm eff}(x_{s})$. The marginal densities of being
bound and unbound $\pi_{\rm on}^{\rm eff}(x_{s})$ and $\pi_{\rm off}^{\rm
eff}(x_{s})$ are straightforwardly computed by the fraction of time in each
state and bin for $x_{s}$. Then, the microscopic rates are estimated by
$\hat{q}^{\rm eff}(x_{s})=\hat{J}^{\rm eff}(x_{s})/\hat{\pi}^{\rm
eff}(x_{s})$. Lastly, the forces $\mathcal{F}^{\rm eff}(x_{s})$ are taken as
the average $\langle x_{s}(t+\Delta t)-x_{s}(t)\rangle$ and accumulated in the
corresponding bin for $x_{s}(t)$.

Figure 3: Numerical validation of the coarse-graining procedure, here in the
case of a fast linker transiently binding to a slow particle in 1D. Both
binding and unbinding are chosen to be spatially dependent, as in model 2
(10), with numerical parameters $\epsilon=0.1$, $\kappa=1$, $\lambda=10$,
$q_{\rm off}^{0}=1/\tau_{0}$, and $q_{\rm on}^{0}$ chosen such that $\Pi_{\rm
b}/\Pi_{\rm u}=1$ determined by the relation (28). 1000 Simulations are run
until $T=1000\tau_{0}$. a: microscropic binding and unbinding effective rates
b: effective force. c: effective marginal probabilities of being bound or
unbound. d: macroscopic transition rates. e (inset): error for effective
microscopic rates and force, defined to be
$\int(\text{theory}-\text{simulated})^{2}\mathrm{d}x$ as a function of
$\varepsilon$, showing increasing error as $\varepsilon$. Theory curves are
obtained with the expressions summarized in Table 1.

###  III.2 Agreement between simulations and averaging approach

We compare in Fig. 3-a and b, the kinetic rates and forces obtained
numerically and with averaging theory Eq. (20) and (21) (alternatively with
the formulas in Table. 1) and find excellent agreement. Interestingly, the
agreement holds for values of $\epsilon\lesssim 10$ as long as dynamics are
observed at long enough times (Fig. 3-b, inset e). The effective probability
distribution function is also in agreement with the marginal distribution Eq.
(22) (Fig. 3-c). Overall, at the coarse-grained level, detailed balance holds
numerically as well as analytically (Fig. 3-d). Our numerical results thus
show the averaging approach is robust to infer coarse-grained effective
dynamics.

##  IV Applications

Having shown the validity of the averaging approach, we now explore how to use
the formulas in Table 1 in specific situations. We investigate (i) a pair of
connecting linkers, (ii) a linker stiffening upon binding, and (iii) a slip
bond with force-dependent unbinding. According to the specific example under
scrutiny, we will focus either more on the effective forces or on the
effective kinetic rates, and comment on the physical meaning of the results.

###  IV.1 Pair of connecting linkers

We first consider 2 fast linkers that can connect to each other (see Fig. 4).
This could represent for example two complementary DNA strands transiently
hybridizing, which finds some applications for example in the field of DNA-
coated colloids[31, 18, 22, 14, 48, 49]. We consider that one of the linkers,
in position $x_{2}$, is tethered to a fixed plate while the other, in position
$x_{1}$, is tethered to a mobile slow particle, itself in position $x_{s}$.
Both linkers are described by springs with the same spring constant $k_{\ell}$
and fluctuate rapidly.

Figure 4: (a) Cartoon of 2 fast linkers that can connect to each other,
representing for example (2) 2 complementary DNA strands transiently
hybridizing, one of them being attached to a colloid whose motion is of
interest.

##### Unbound dynamics

The unbound dynamics are

$\begin{cases}\frac{dx_{s}}{dt}&=-\frac{k_{\ell}}{\gamma_{s}}(x_{s}-x_{1})+\sqrt{\frac{2k_{B}T}{\gamma_{s}}}\eta_{s}(t)\\\
\frac{dx_{1}}{dt}&=-\frac{k_{\ell}}{\gamma_{\ell}}(x_{1}-x_{s})+\sqrt{\frac{2k_{B}T}{\gamma_{\ell}}}\eta_{1}(t)\\\
\frac{dx_{2}}{dt}&=-\frac{k_{\ell}}{\gamma_{\ell}}(x_{2})+\sqrt{\frac{2k_{B}T}{\gamma_{\ell}}}\eta_{2}(t)\end{cases}$
(30)

where the $\eta_{i}(t)$ are uncorrelated Gaussian white noises and here we
identify the force on the slow particle in the unbound state as

$\mathcal{F}_{\rm u}=-k_{\ell}(x_{s}-x_{1}).$ (31)

The unbound term of the equilibrium distribution corresponding to this choice
of dynamics is

$\pi_{\rm
u}(x_{s},x_{1},x_{2})=\frac{1}{Z_{u}}e^{-\frac{k_{\ell}x_{2}^{2}+k_{\ell}(x_{s}-x_{1})^{2}}{2k_{B}T}}.$
(32)

##### Bound dynamics

When the linkers can transiently form a bond with spring constant $k_{b}$.

$\begin{cases}\frac{dx_{s}}{dt}&=-\frac{k_{\ell}}{\gamma_{s}}(x_{s}-x_{1})+\sqrt{\frac{2k_{B}T}{\gamma_{s}}}\eta_{s}(t)\\\
\frac{dx_{1}}{dt}&=-\frac{k_{\ell}}{\gamma_{\ell}}(x_{1}-x_{s})+\sqrt{\frac{2k_{B}T}{\gamma_{\ell}}}\eta_{1}(t)+\frac{k_{b}}{\gamma_{\ell}}(x_{2}-x_{1})\\\
\frac{dx_{2}}{dt}&=-\frac{k_{\ell}}{\gamma_{\ell}}(x_{2})+\sqrt{\frac{2k_{B}T}{\gamma_{\ell}}}\eta_{2}(t)-\frac{k_{b}}{\gamma_{\ell}}(x_{2}-x_{1})\end{cases}$
(33)

and we identify the force on the slow particle in the bound state as identical
to that in the unbound state

$\mathcal{F}_{\rm b}=-k_{\ell}(x_{s}-x_{1}).$ (34)

The bound term of the equilibrium distribution corresponding to this choice of
dynamics is

$\pi_{\rm
bound}(x_{s},x_{1},x_{2})=\frac{1}{Z_{b}}e^{-\frac{k_{\ell}x_{2}^{2}+k_{\ell}(x_{s}-x_{1})^{2}+k_{b}(x_{2}-x_{1})^{2}}{2k_{B}T}}.$
(35)

##### Kinetic rates and detailed balance

Here we will not specify the kinetic rates in detail. However, since we
specify the dynamics, the binding and unbinding rates must satisfy detailed
balance and hence

$\frac{q_{\rm on}}{q_{\rm off}}=\frac{q_{\rm on}^{0}}{q_{\rm
off}^{0}}e^{-k_{\rm\ell}(x_{2}-x_{1})^{2}/2k_{B}T}.$

##### Effective force in the unbound state

We can now use the expressions in Table 1 to obtain the effective force. Here,
compared to our foundational example with 1 fast linker in Sec. I, we have 2
fast linkers, hence we need to carry the integral over those 2 fast degrees of
freedom. We find with Eq. (26)

$\begin{split}&\mathcal{F}_{\rm u}^{\rm eff}(x_{s})\\\
&\quad=\frac{\int\mathcal{F}_{\rm u}(x_{1},x_{2},x_{s})\pi_{\rm
u}(x_{1},x_{2},x_{s})dx_{1}dx_{2}}{\int\pi_{\rm
u}(x_{1},x_{2},x_{s})dx_{1}dx_{2}}\\\
&\quad=\frac{\int\left(-k_{\ell}(x_{s}-x_{1})\right)e^{-k_{\ell}x_{2}^{2}/2k_{B}T}e^{-k_{\ell}(x_{s}-x_{1})^{2}/2k_{B}T}dx_{1}dx_{2}}{\int
e^{-k_{\ell}x_{2}^{2}/2k_{B}T}e^{-k_{\ell}(x_{s}-x_{1})^{2}/2k_{B}T}dx_{1}dx_{2}}\\\
&\quad=0\end{split}$ (36)

for symmetry reasons. In the unbound state, quite logically, at the coarse-
grained level the particle doesn’t feel any effective force from its unbound
fast linker.

##### Effective force in the bound state

In the bound state, using Eq. (27) we find

$\begin{split}\mathcal{F}_{\rm b}^{\rm eff}(x_{s})&=\frac{\int\mathcal{F}_{\rm
b}(x_{1},x_{2},x_{s})\pi_{\rm
bound}(x_{1},x_{2},x_{s})dx_{1}dx_{2}}{\int\pi_{\rm
bound}(x_{1},x_{2},x_{s})dx_{1}dx_{2}}\\\
&=-\frac{\frac{k_{\ell}}{2}k_{b}}{\frac{k_{\ell}}{2}+k_{b}}x_{s}.\end{split}$
(37)

In the bound state we thus obtain that the effective force on the particle is
a spring force. The force is centered around 0 as this is the average position
of both springs. The spring constant at the coarse-grained level is
$[(k_{\ell}/2)k_{b}]/[(k_{\ell}/2)+k_{b}]$ which corresponds, logically, to
the effective spring constant of $3$ springs in parallel, with spring
constants $k_{\ell},k_{b},k_{\ell}$.

###  IV.2 Stiffening linker upon binding

We now turn to another example where the linker stiffens when bound – see Fig.
5-a. Such stiffening occurs when the linker undergoes conformational changes
upon binding [21] or else upon single-stranded DNA hybridizing into double-
stranded, resulting in a stiffer connection [11, 50, 18, 51, 13, 15].

Figure 5: (a) Cartoon of a fast linker stiffening upon binding and (b) a fast
linker connecting to the particle via a slip bond with force-dependent
unbinding.

##### Unbound dynamics

The unbound dynamics are still given by Eq. (1), which we recall here

$\begin{cases}\frac{dx_{s}}{dt}&=+\sqrt{\frac{2k_{B}T}{\gamma_{s}}}\eta_{s}(t)\\\
\frac{dx_{\ell}}{dt}&=-\frac{k_{\ell}}{\gamma_{\ell}}x_{\ell}+\sqrt{\frac{2k_{B}T}{\gamma_{\ell}}}\eta_{\ell}(t)\end{cases}$
(38)

where the $\eta_{i}(t)$ are uncorrelated Gaussian white noises. The
corresponding unbound part of the equilibrium distribution is

$\pi_{\rm
u}(x_{s},x_{\ell})=\frac{1}{Z_{u}}e^{-k_{\ell}x_{\ell}^{2}/2k_{B}T}.$ (39)

##### Bound dynamics

The bound dynamics are now changed compared to Eq. (2), since the linker now
has a stiffer spring constant say $k_{\ell}^{\prime}>k_{\ell}$,

$\begin{cases}\frac{dx_{s}}{dt}&=-\frac{k_{b}}{\gamma_{s}}(x_{s}-x_{\ell})+\sqrt{\frac{2k_{B}T}{\gamma_{s}}}\eta_{s}(t)\\\
\frac{dx_{\ell}}{dt}&=-\frac{k_{\ell}^{\prime}}{\gamma_{\ell}}x_{\ell}-\frac{k_{b}}{\gamma_{\ell}}(x_{\ell}-x_{s})+\sqrt{\frac{2k_{B}T}{\gamma_{\ell}}}\eta_{\ell}(t).\end{cases}$
(40)

The bound term of the equilibrium distribution corresponding to this choice of
bound dynamics is

$\pi_{\rm
b}(x_{s},x_{\ell})=\frac{1}{Z_{b}}e^{-k_{\ell}^{\prime}x_{\ell}^{2}/2k_{B}T}e^{-k_{b}(x_{s}-x_{\ell})^{2}/2k_{B}T}.$
(41)

##### Kinetic rates

Again without specifying the kinetic rates in detail, since we set the
dynamics, the binding and unbinding rates must satisfy detailed balance and
hence are related via

$\frac{q_{\rm on}}{q_{\rm off}}=\frac{q_{\rm on}^{0}}{q_{\rm
off}^{0}}e^{-k_{b}(x_{s}-x_{\ell})^{2}/2k_{B}T}e^{-(k_{\ell}^{\prime}-k_{\ell})x_{\ell}^{2}/2k_{B}T}.$

This relation favors binding when the tether is not too extended. Otherwise,
the energetic price to pay to “stiffen” is too costly. Reciprocally the bond
is more likely to break when the tether is quite extended, and the energetic
gain to loosen the bond is quite high. One could thus choose, in line with
physical intuition,

$\begin{cases}&q_{\rm on}\sim q_{\rm
on}^{0}e^{-k_{b}(x_{s}-x_{\ell})^{2}/2k_{B}T},\\\ &q_{\rm off}\sim q_{\rm
off}^{0}e^{(k_{\ell}^{\prime}-k_{\ell})x_{\ell}^{2}/2k_{B}T}.\end{cases}$ (42)

The detailed prefactors should be specified in agreement with detailed
balance.

##### Effective force in the bound state

The effective force in the unbound state naturally vanishes. Hence we focus on
the effective force in the bound state, again obtained via Eq. (27)

$\begin{split}\mathcal{F}_{\rm b}^{\rm
eff}(x_{s})&=-\frac{k_{\ell}^{\prime}k_{b}}{k_{\ell}^{\prime}+k_{b}}x_{s}.\end{split}$
(43)

Again, it makes sense that the coarse-grained force is that associated with 2
springs in series with spring constants $k_{\ell}^{\prime}$ and $k_{b}$.
However, here it was not obvious a priori which of the spring constants for
the linker, either $k_{\ell}^{\prime}$ or $k_{\ell}$ or a mix of both, should
contribute to the force.

##### Effective rates

We now use the rates defined in Eq. (42) to study coarse-grained kinetic rates
as well. Let $K=\frac{k_{\ell}k_{b}}{k_{\ell}+k_{b}}$ and
$K^{\prime}=\frac{k_{\ell}^{\prime}k_{b}}{k_{\ell}^{\prime}+k_{b}}$. Then we
obtain

$\begin{cases}&q_{\rm on}^{\rm eff}(x_{s})\sim e^{-Kx_{s}^{2}/2k_{B}T}\\\
&q_{\rm off}^{\rm eff}(x_{s})\sim e^{(K^{\prime}-K)x_{s}^{2}/2k_{B}T}.\\\
\end{cases}$ (44)

The effective binding rate is determined by the typical, unstiff, radius of
the interaction, $\sqrt{k_{B}T/K}$, similarly as in our foundational example
in Sec. I. The unbinding rate is now larger at larger distances, according to
how much stiffening occurred. Notice that even if the bond is really stiff,
meaning $k_{b}\gg k_{\ell},k_{\ell}^{\prime}$ then $K^{\prime}-K\simeq
k_{\ell}^{\prime}-k_{\ell}$ still converges to a finite value.

###  IV.3 Slip bonds with force-dependent unbinding

Another quite common situation is that of slip bonds with force-dependent
unbinding [21, 52] – see Fig. 5-b. In these cases the unbinding rate scales as
[52]

$q_{\rm off}(x_{s},x_{\ell})=q_{\rm off}^{0}e^{k_{\rm
slip}|x_{s}-x_{\ell}|/F_{0}}$ (45)

where $k_{\rm slip}$ is a characteristic bond spring constant and $F_{0}$ a
characteristic force threshold. Above this threshold, the unbinding rate is
indeed greatly enhanced.

##### Satisfying detailed balance

How shall we proceed to model the dynamics with such force-dependent unbinding
rate, when the system is still at equilibrium? Compared to our foundational
example in Sec. I, here we might change the binding rate to scale as $q_{\rm
on}(x_{s},x_{\ell})=q_{\rm on}^{0}e^{-k_{\rm
slip}|x_{s}-x_{\ell}|/F_{0}}e^{-k_{b}(x_{s}-x_{\ell})^{2}/2k_{B}T}$, in which
case the binding rate is strongly depressed far from the target. Notice that
this is not the only possibility to satisfy detailed balance and one could
also consider alternative bound dynamics.

For the sake of the example, we will make the second choice above with an
additional force on the bound particle, and we recapitulate dynamics and
kinetic rates below before turning to effective dynamics.

##### Unbound dynamics

The unbound dynamics are still given by Eq. (1), which we recall here

$\begin{cases}\frac{dx_{s}}{dt}&=+\sqrt{\frac{2k_{B}T}{\gamma_{s}}}\eta_{s}(t)\\\
\frac{dx_{\ell}}{dt}&=-\frac{k_{\ell}}{\gamma_{\ell}}x_{\ell}+\sqrt{\frac{2k_{B}T}{\gamma_{\ell}}}\eta_{\ell}(t)\end{cases}$
(46)

where the $\eta_{i}(t)$ are uncorrelated Gaussian white noises. The
corresponding unbound term of the equilibrium distribution is

$\pi_{\rm
u}(x_{s},x_{\ell})=\frac{1}{Z_{u}}e^{-k_{\ell}x_{\ell}^{2}/2k_{B}T}.$ (47)

##### Bound dynamics

The bound dynamics are now changed compared to Eq. (2), since the linker and
particle are now subjected to additional forces.

$\begin{cases}\frac{dx_{s}}{dt}&=-\frac{k_{b}}{\gamma_{s}}(x_{s}-x_{\ell})\\\
&\,\,-\frac{k_{\rm
slip}}{\gamma_{s}}\frac{k_{B}T}{F_{0}}\text{sgn}(x_{s}-x_{\ell})+\sqrt{\frac{2k_{B}T}{\gamma_{s}}}\eta_{s}(t)\\\
\frac{dx_{\ell}}{dt}&=-\frac{k_{\ell}}{\gamma_{\ell}}x_{\ell}-\frac{k_{b}}{\gamma_{\ell}}(x_{\ell}-x_{s})\\\
&\,\,+\frac{k_{\rm
slip}}{\gamma_{\ell}}\frac{k_{B}T}{F_{0}}\text{sgn}(x_{s}-x_{\ell})+\sqrt{\frac{2k_{B}T}{\gamma_{\ell}}}\eta_{\ell}(t).\end{cases}$
(48)

The bound term of the equilibrium distribution corresponding to this choice of
bound dynamics is

$\pi_{\rm
b}(x_{s},x_{\ell})=\frac{1}{Z_{b}}e^{-k_{\ell}x_{\ell}^{2}/2k_{B}T}e^{-k_{b}(x_{s}-x_{\ell})^{2}/2k_{B}T}e^{-k_{s}|x_{s}-x_{\ell}|/F_{0}}.$
(49)

##### Kinetic rates

We choose

$\begin{cases}&q_{\rm on}\sim q_{\rm
on}^{0}e^{-k_{b}(x_{s}-x_{\ell})^{2}/2k_{B}T},\\\ &q_{\rm off}\sim q_{\rm
off}^{0}e^{k_{\rm slip}|x_{s}-x_{\ell}|/F_{0}}\end{cases}$ (50)

which is rather satisfactory since it is physically coherent with faster
binding and slower unbinding when the particle and linker are closer. The
detailed prefactors should be specified in agreement with detailed balance.

##### Effective force in the bound state

The effective force in the unbound state naturally vanishes. Hence we focus on
the effective force in the bound state, which has a lengthy expression that we
do not report here. In the case of a small slip force $k_{\rm
slip}k_{B}T/F_{\rm unbinding}\ll k_{\ell}|x_{s}|$ we obtain

$\begin{split}\mathcal{F}_{\rm b}^{\rm
eff}(x_{s})&\simeq-\frac{k_{\ell}k_{b}}{k_{\ell}+k_{b}}x_{s}-\\\
&\frac{k_{\ell}}{k_{\ell}+k_{b}}\frac{k_{\rm slip}k_{B}T}{F_{\rm
unbinding}}\text{erf}\left(\sqrt{\frac{k_{\ell}}{k_{\ell}+k_{b}}\frac{k_{\ell}x_{s}^{2}}{2k_{B}T}}\right).\end{split}$
(51)

The coarse-grained force contains now 2 contributions. The first one is that
associated with 2 springs in series with spring constants $k_{\ell}$ and
$k_{b}$ that we have seen before. The second one corresponds to the slip bond,
which grows stronger when the particle goes further away from the target.
Notice that this latter slip bond friction force is screened by a factor
$\frac{k_{\ell}}{k_{\ell}+k_{b}}$ in the coarse-grained state. In the case of
a stiff bond, $k_{b}\gg k_{\ell}$, the slip force is entirely screened, and
the particle “slips”; the linker essentially accommodates changing
configurations by rapidly adjusting its length.

##### Effective rates

We now use the rates defined in Eq. (42) to study coarse-grained kinetic rates
as well. Let $K=\frac{k_{\ell}k_{b}}{k_{\ell}+k_{b}}$. Then we obtain, in the
case of a small slip force $k_{\rm slip}k_{B}T/F_{\rm unbinding}\ll
k_{\ell}|x_{s}|$,

$\begin{cases}&q_{\rm on}^{\rm eff}(x_{s})\sim e^{-Kx_{s}^{2}/2k_{B}T}\\\
&q_{\rm off}^{\rm eff}(x_{s})\sim
e^{\frac{k_{\ell}}{k_{\ell}+k_{b}}\frac{k_{\rm slip}|x_{s}|}{F_{\rm
unbinding}}}.\\\ \end{cases}$ (52)

The effective binding rate is determined by the typical, unstiff, radius of
the interaction, $\sqrt{k_{B}T/K}$, similarly as in our foundational example
in Sec. I. The unbinding rate is now larger at larger distances, with force-
dependent unbinding.

##  V Macroscopic consequences for the choice of binding kinetics

###  V.1 Coarse-graining various microscopic binding models: the Doi model

How might the effective dynamics change from our foundational example in Sec.
I when we consider the Doi model for binding?

##### Kinetic rates

In the Doi model, we recall that the kinetic rates are such that the states
can only change if the linker and the particle are within a distance $R$ from
each other.

$\begin{cases}&q_{\rm on}=q_{\rm on}^{0}H(R-|x_{s}-x_{\ell}|)\\\ &q_{\rm
off}=q_{\rm off}^{0}H(R-|x_{s}-x_{\ell}|).\end{cases}$ (53)

##### Unbound dynamics

In the Doi model, to satisfy detailed balance, when the linker and the
particle are near each other, they feel an attractive recoil force, regardless
of their state. The unbound dynamics are therefore

$\begin{cases}\frac{dx_{s}}{dt}&=-\frac{k_{b}}{\gamma_{s}}(x_{s}-x_{\ell})H(R-|x_{s}-x_{\ell}|)+\sqrt{\frac{2k_{B}T}{\gamma_{s}}}\eta_{s}(t)\\\
\frac{dx_{\ell}}{dt}&=-\frac{k_{\ell}}{\gamma_{\ell}}x_{\ell}-\frac{k_{b}}{\gamma_{\ell}}(x_{\ell}-x_{s})H(R-|x_{s}-x_{\ell}|)+\sqrt{\frac{2k_{B}T}{\gamma_{\ell}}}\eta_{\ell}(t).\end{cases}$
(54)

The unbound term of the equilibrium distribution corresponding to this choice
of unbound dynamics is

$\begin{split}\pi_{\rm
u}(x_{s},x_{\ell})=\frac{1}{Z_{u}}&e^{-k_{\ell}x_{\ell}^{2}/2k_{B}T}\big{[}H(|x_{s}-x_{\ell}|-R)\\\
&+H(R-|x_{s}-x_{\ell}|)e^{-k_{b}(x_{s}-x_{\ell})^{2}/2k_{B}T}\big{]}.\end{split}$
(55)

##### Bound dynamics

The bound dynamics are unchanged compared to our foundational model and
satisfy

$\begin{cases}\frac{dx_{s}}{dt}&=-\frac{k_{b}}{\gamma_{s}}(x_{s}-x_{\ell})+\sqrt{\frac{2k_{B}T}{\gamma_{s}}}\eta_{s}(t)\\\
\frac{dx_{\ell}}{dt}&=-\frac{k_{\ell}}{\gamma_{\ell}}x_{\ell}-\frac{k_{b}}{\gamma_{\ell}}(x_{\ell}-x_{s})+\sqrt{\frac{2k_{B}T}{\gamma_{\ell}}}\eta_{\ell}(t).\end{cases}$
(56)

The bound term of the equilibrium distribution corresponding to this choice of
bound dynamics is

$\pi_{\rm
b}(x_{s},x_{\ell})=\frac{1}{Z_{b}}e^{-k_{\ell}x_{\ell}^{2}/2k_{B}T}e^{-k_{b}(x_{s}-x_{\ell})^{2}/2k_{B}T}.$
(57)

##### Effective force in the unbound state

According to Eq. (26) the unbound friction force is simply

$\mathcal{F}_{\rm u}^{\rm
eff}(x_{s})=-k_{b}\frac{\int_{x_{s}-R}^{x_{s}+R}(x_{s}-x_{\ell})e^{-\frac{k_{b}(x_{s}-x_{\ell})^{2}+k_{\ell}x_{\ell}^{2}}{2k_{B}T}}\text{d}x_{\ell}}{\int\pi_{\rm
u}(x_{s},x_{\ell})\text{d}x_{\ell}}$ (58)

which has a cumbersome, non-vanishing, expression that we do not report here.
However we can make a finite perturbation of the obtained expression in the
limit of a stiff bond, when $k_{b}\gg k_{\ell}$, and assuming
$R=\sqrt{2k_{B}T/k_{b}}$

$\mathcal{F}_{\rm u}^{\rm
eff}(x_{s})\simeq-0.43\sqrt{\frac{k_{\ell}}{k_{b}}}k_{\ell}x_{s}$ (59)

Since $x_{\ell}$ wiggles around 0 and even in the unbound state, close to the
linker, the particle feels a recoil force, then it makes sense that the
particle can now feel a recoil force everywhere. The magnitude of this force
is slightly decreased because the bond can only form when the particle is
close enough to the linker. Notice how the scaling of the effective spring
constant is entirely nontrivial as $\sqrt{\frac{k_{\ell}}{k_{b}}}k_{\ell}$.

##### Effective force in the bound state

According to Eq. (27) the bound friction force is simply

$\mathcal{F}_{\rm b}^{\rm
eff}(x_{s})=-\frac{k_{\ell}k_{b}}{k_{\ell}+k_{b}}x_{s}$ (60)

which is similar to the results obtained with our foundational example.

##### Effective kinetic rates

Again, the expression for the effective binding and unbinding rates obtained
from Eqs. (24) and (25) is rather cumbersome. When $k_{b}\gg k_{\ell}$, and
assuming $R=\sqrt{2k_{B}T/k_{b}}$, the kinetic rates are smoothly decaying to
0 (instead of a sharp Heaviside function as in the microscopic equations), at
a characteristic distance
$x_{s}\simeq\sqrt{\frac{k_{B}T}{k_{\ell}}\sqrt{\frac{k_{\ell}}{k_{b}}}}$.
Again, one sees how this scaling is entirely non trivial. Systematic coarse-
graining is therefore essential for faster numerical simulations and enhanced
theoretical investigations.

###  V.2 Macroscopic consequences

We finish by briefly exploring how the microscopic choices for $q_{\rm
on}(x_{s},x_{\ell})$ and $q_{\rm off}(x_{s},x_{\ell})$ can affect the
macroscopic dynamics – see Sec. I.2. To simplify the exploration, here we will
consider a particle and a linker. where the linker is very stiff ($k_{\ell}\gg
k_{b}$) which simply means we can neglect linker dynamics and such that it
stays at a fixed position in $x_{\ell}=0$. We use the coarse-grained forces
and kinetic rates in Table. 1 to simulate the motion of the particle with
different initial choices of microscopic binding, specifically with model 1
corresponding to Eq. (9) with spatially dependent binding only and model 2
from Eq. (10) with both spatially dependent binding and unbinding. To compare
different microscopic models in a sensible fashion, we constrain the
macroscopic bound probability $\Pi_{b}=0.5$ (see Fig. 6-c) for all microscopic
models. We then calculate the macroscopic mean binding and unbinding times for
the particle (Fig. 6-a and b), as well as the particle’s long-time diffusion
coefficient (Fig. 6-d), determined by its mean-squared displacement.
Importantly, we vary the effective bond spring constant
$K=\frac{k_{\ell}k_{b}}{k_{b}+k_{\ell}}\simeq k_{b}$ to probe how model
parameters affect macroscopic dynamics.

We find that both the choice of microscopic binding model and the parameter
$K$ significantly affect the macroscopic binding and unbinding times (Figs.
3-a and b). Since the macroscopic binding probability $\Pi_{b}$ is fixed, the
mean binding and unbinding times are identical, and hence Fig. 6-a and b
appear the same. At long times, we expect, as for the coarse-graining over
$x_{\ell}$, that the kinetic rates do not depend on the position $x_{s}$
anymore and would verify

$Q_{\rm off}=\frac{\int q_{\rm off}^{\rm eff}(x_{s})\pi_{\rm b}^{\rm
eff}(x_{s})\text{d}x_{s}}{\int\pi_{\rm b}^{\rm eff}(x_{s})\text{d}x_{s}},$
(61)

and similarly for the binding rate $Q_{\rm on}$. Eq. (61) gives the mean
unbinding time as $1/Q_{\rm off}$ and reproduces perfectly the numerical
results (lines in Fig. 3-a).

For the different microscopic models, the response to various binding
constants $K$ is non-trivial. For model 1, the macroscopic binding and
unbinding times remain constant. In model 1, $q_{\rm off}^{\rm eff}(x_{s})$
does not depend on space, and, therefore, its macroscopic counterpart $q_{\rm
off}^{\rm eff}$ which verifies Eq. (61), seems to have no direct dependence on
$K$. However, $q_{\rm on}^{\rm eff}(x_{s})$ is inherently spatially dependent
and hence its macroscopic counterpart may depend on $K$ as well, via the
associated length scale $\sqrt{k_{B}T/K}$. In contrast, with model 2 the
macroscopic kinetic rates both increase with the binding constant $K$. When
$K$ is larger, the bond is stronger, which decreases the binding time but
increases the unbinding time. Hence, the macroscopic dependence where both
kinetic rates increase with $K$ is interesting. In fact, to properly coarse-
grain to this now macroscopic scale, one should also take into account the
probability distribution of being in each state with a given extension. Such
entangled behavior requires proper integration beyond simple intuition.
Eventually, we find that the macroscopic binding times strongly depend on
microscopic choices for the kinetic rates, which means one should use caution
when designing such models.

In addition, for both microscopic model choices, the long-time self-diffusion
coefficient of the particle depends on $K$ (Fig. 3-d). In fact, at small $K$
we find $D\rightarrow k_{B}T/\gamma_{s}$ since the bond is weak enough that it
barely affects particle motion. At larger $K$ values, we find $D\rightarrow
k_{B}T/2\gamma_{s}$ since when it is bound, the bond is strong enough that it
prevents any motion, and the particle is bound half of the time
($\Pi_{b}=0.5$). The dependence on $K$ appears to be similar for both
microscopic binding models 1 and 2. In more varied models for binding, there
is no reason this should stay true. We leave the general exploration of
macroscopic transport properties, such as diffusion coefficients, and their
dependence on microscopic binding kinetics, for future work.

Figure 6: Different choices of $q_{\rm on}(x)$ and $q_{\rm off}(x)$ yield
different dynamics. Model 1 corresponds to Eq. (9) with spatially dependent
binding only and model 2 corresponds to Eq. (10) with both spatially dependent
binding and unbinding. Simulation parameters are $q_{\rm
off}^{0}=0.1\tau_{0}^{-1}$ and $q_{\rm on}^{0}$ chosen such that
$\Pi_{b}/\Pi_{u}=1$. 1000 simulations are run until $T=1000\tau_{0}$. a: mean
unbinding time as a function of the bond constant $k_{\ell}$, showing
divergence between models 1 and 2. Theory curves are obtained with Eq. (61).
b: mean binding time as a function of the bond constant $k_{\ell}$ with
similar trend. c: Macroscopic bound probabilities $\Pi_{b}$ are set to be
fixed and are the same for both models. d: Macroscopic diffusion coefficient
$D=\lim_{t\to\infty}\langle x_{s}(t)^{2}\rangle/(2t)$ for both models.

## Conclusion

In this work, we have attempted to unify and justify various coarse-graining
approaches for linker dynamics [20, 21, 22, 23, 16, 24, 25, 26, 27]. In these
earlier approaches, we have identified many different choices for how binding
and unbinding depend on distance, yet (1) these are all heuristically
motivated, and (2) it is not clear how they relate to each other. Here, we
have addressed all of these issues by providing a systematic derivation of
effective forces and kinetics for linker molecules that obey any detailed
microscopic descriptions. Our coarse-graining approach is based on averaging
techniques and preserves detailed balance, in line with our assumption of
equilibrium dynamics. We have verified our approach with numerical simulations
and found excellent agreement. We showed how our general framework may be
applied to diverse microscopic scenarios, with 2 linkers binding to each
other, a linker stiffening upon binding or a slip bond with force-dependent
unbinding. Finally, we showed that different microscopic kinetic rates result
in fundamentally different dynamics at the macroscopic scale, raising caution
in making these choices without care.

Many choices for effective dynamics in the literature clearly violate detailed
balance and hence operate out-of-equilibrium [42, 21]. While one could a
priori investigate coarse-grained out-of-equilibrium dynamics, such details
are often barely discussed. So far, there does not exist a framework that can
properly address coarse-graining beyond equilibrium. The averaging approach
could constitute a starting point since it does not rely on detailed balance.
The authors in Ref. 53 interestingly note that effective equilibria can arise
by switching between non-conservative systems, providing hope that this
pursuit is both fruitful and interesting. Thus, extending our method to out-
of-equilibrium systems would be of broad relevance, in particular, to explore
transient bonds with activated cleaving that are common in viral linker-
mediated motion [54, 55] and also artificial nano-motors [6].

Although we have investigated relatively simple setups here, the tools we
introduce and the lessons learned are applicable across many more complex
systems. The formulas we derived for effective dynamics can be applied with
ease via Table 1 and provide a baseline for more systematic coarse-grainings
found in simulations and theoretical studies across the literature. One such
example is molecular motor binding in intracellular transport. There, coarse-
graining ranges from non-spatial effective kinetic rates [56, 57], motor
linkers obeying a worm-like chain model [58, 59], or Doi-like motors that bind
when within a specific radius of interaction [37]. Coarse-grained cross-linked
cytoskeletal networks are studied extensively [60, 4, 5, 61, 62], especially
in self-organization in the mitotic spindle [63, 64, 65, 66] and actomyosin
network mechanics [67, 68]. Transient dynamics of (typically coarse-grained)
cross-linkers are also fundamental in controlling viral responses in bio-gels
[69, 70] and building chromosomal territories[71]. Our framework also readily
extends to systems that are not "cross"-linked, such as membrane-filament
interactions that drive cell protrusion[72, 73].

###### Acknowledgements.

We wish to acknowledge fruitful discussions with Aleksandar Donev and Miranda
Holmes-Cerfon. S.M. received funding from the European Union’s Horizon 2020
research and innovation programme under the Marie Skłodowska-Curie grant
agreement 839225, MolecularControl.

## Data Availability Statement

The data generated by this paper is available upon reasonable request to the
authors.

##  Appendix A Main notations

We report in Table 2 the main notations used in the paper.

Table 2: Summary of the main notations used.

Notation Meaning $x_{s}$ coordinate of the slow particle $x_{\ell}$ coordinate
of the fast linker $x_{0}$ characteristic length scale for
nondimensionalisation $\gamma_{s}$ friction coefficient on the slow particle
$\gamma_{\ell}$ friction coefficient on the fast linker
$\varepsilon=\gamma_{\ell}/\gamma_{s}$ ratio of friction coefficients
$k_{\ell}$ spring constant describing the linker $k_{b}$ spring constant
describing the bond between the particle and the linker $k_{B}T$ thermal
energy $\tau_{0}$ characteristic timescale for nondimensionalisation $\pi_{\rm
u/b}(x_{s},x_{\ell})$ microscopic unbound (resp. bound) probability
distribution $\pi_{\rm u/b}^{\rm eff}(x_{s})$ coarse-grained unbound (resp.
bound) probability distribution $\Pi_{\rm u/b}$ macroscopic unbound (resp.
bound) probability $\mathcal{F}_{\rm u/b}(x_{s},x_{\ell})$ microscopic force
on the slow particle in the unbound (resp. bound) state $\mathcal{F}^{\rm
eff}_{\rm u/b}(x_{s})$ coarse-grained force on the slow particle in the
unbound (resp. bound) state $q_{\rm on/off}(x_{s},x_{\ell})$ microscopic
binding (resp. unbinding) rate $q^{\rm eff}_{\rm on/off}(x_{s})$ coarse-
grained binding (resp. unbinding) rate $Q_{\rm on/off}$ macroscopic binding
(resp. unbinding) rate

##  Appendix B Extended coarse-graining calculations

###  B.1 Obtaining the generator from the dynamics

The set of stochastic Eqns. (1)-(2) defines a Markov process that is
conveniently studied via the Fokker-Planck equation and its adjoint, the
Kolmogorov backward equation [44, 43]. Let
$p(x_{s},x_{\ell},t)=\left(p_{u}(x_{s},x_{\ell},t),p_{b}(x_{s},x_{\ell},t)\right)^{T}$
be the probability density function of finding the system at time $t$ and
positions $x_{s},x_{\ell}$ in the unbound or bound states. We obtain from
Eqns. (1)-(2) the Fokker-Plank equation

$\partial_{t}p=\mathcal{L}^{\star}p\,,$ (62)

with $\mathcal{L}^{\star}=\mathcal{Q}^{\star}+\mathcal{V}^{\star}$ where

$\displaystyle\mathcal{Q}^{\star}$ $\displaystyle=\begin{pmatrix}-q_{\rm
on}&q_{\rm off}\\\ q_{\rm on}&-q_{\rm
off}\end{pmatrix},\mathcal{V}^{\star}=\begin{pmatrix}\mathcal{V}_{u}^{\star}&0\\\
0&\mathcal{V}_{b}^{\star}\end{pmatrix},$
$\displaystyle\mathcal{V}_{u}^{\star}$
$\displaystyle=\partial_{\ell}\left(\frac{k_{\ell}}{\gamma_{\ell}}x_{\ell}+\frac{k_{B}T}{\gamma_{\ell}}\partial_{\ell}\right)+\partial_{s}(\partial_{s}\frac{\mathcal{U}_{s}(x_{s})}{\gamma_{s}})+\frac{k_{B}T}{\gamma_{s}}\partial_{ss}$
$\displaystyle\mathcal{V}_{b}^{\star}$
$\displaystyle=\left(\frac{k_{\ell}}{\gamma_{\ell}}x_{\ell}-\frac{k_{b}}{\gamma_{\ell}}(x_{\ell}-x_{s})+\frac{k_{B}T}{\gamma_{\ell}}\partial_{\ell}\right)+$
$\displaystyle\,\,\,\partial_{s}(\partial_{s}\frac{\mathcal{U}_{s}(x_{s})}{\gamma_{s}}+\frac{k_{b}}{\gamma_{s}}(x_{\ell}-x_{s}))+\frac{k_{B}T}{\gamma_{s}}\partial_{ss}$

with an appropriate initial condition. Additionally, we require the flux in
either state to vanish at infinity, to conserve total probability. The
stationary solution of Eq. (62) is $\pi=\left(\pi_{\rm u},\pi_{\rm
b}\right)^{T}$, the equilibrium probability density of the system. As we have
seen, it satisfies detailed balance.

While probability densities have an intuitive physical meaning, when averaging
it is often easier – and mathematically better posed – to consider the adjoint
of the Fokker-Planck equation and the corresponding dual functions. These are
functions $f(x_{s},x_{\ell},t)=$$\int
p(x_{s}^{\prime},x_{\ell}^{\prime},t|x_{s},x_{\ell})g(x_{s}^{\prime},x_{\ell}^{\prime})\text{d}x_{\ell}^{\prime}\text{d}x_{s}^{\prime}$
that give the expectation of any scalar function $g(x_{s}(t),x_{\ell}(t))$,
given an initial condition $x_{s}(0)=x_{s},x_{\ell}(0)=x_{\ell}$. Writing
$f(x_{s},x_{\ell},t)=\left(f_{u}(x_{s},x_{\ell},t),f_{b}(x_{s},x_{\ell},t)\right)^{T}$,
we have that $f$ satisfies the Kolmogorov backward equation [44] Eq. (14).
Here, $\mathcal{L}$ is the adjoint operator of $\mathcal{L}^{\star}$, defined
by the operator that satisfies $\langle
f,\mathcal{L}^{\star}p\rangle=\langle\mathcal{L}f,p\rangle$ for any
probability density $p$ and statistic $f$, where $\langle
f,p\rangle=\iint(f_{u}p_{u}+f_{b}p_{b})\text{d}x_{\ell}\text{d}x_{s}$ is the
inner product.

###  B.2 Commuting the limits of effective dynamics and stiff bond.

We now show that the limits of a stiff spring constant for the bond and of
coarse-graining fast linker dynamics commute. Since in the text we started
with coarse-graining, i.e. the limit of a fast linker, we here first take the
limit of stiff spring constant for the bond $k_{b}\gg k_{\ell}$. This limit
corresponds to a so-called soft constraint, where the linker and the slow
particle move synchronously when they are bound. The dynamics in the bound
state with this case can be readily obtained, by projecting the unbound
dynamics [32, 34, 33]

$\frac{dx_{s}}{dt}=\frac{dx_{\ell}}{dt}=-\frac{k_{\ell}x_{\ell}+\partial_{x}\mathcal{U}(x_{s})}{\gamma_{s}+\gamma_{\ell}}+\sqrt{\frac{2k_{B}T}{\gamma_{s}+\gamma_{\ell}}}\eta_{s}(t).$
(63)

We obtain the generator

$\mathcal{L}=\mathcal{Q}+\mathcal{V}$ (64)

where

$\displaystyle\mathcal{V}_{u}$
$\displaystyle=-\frac{k_{\ell}}{\gamma_{\ell}}x_{\ell}\partial_{\ell}+\frac{k_{B}T}{\gamma_{\ell}}\partial_{\ell\ell}-\frac{\partial_{x}\mathcal{U}(x_{s})}{\gamma_{s}}x_{s}\partial_{s}+\frac{k_{B}T}{\gamma_{s}}\partial_{ss}$
(65) $\displaystyle\mathcal{V}_{b}$
$\displaystyle=-\frac{k_{\ell}x_{\ell}+\partial_{x}\mathcal{U}(x_{s})}{\gamma_{\ell}+\gamma_{s}}(\partial_{\ell}+\partial_{s})+\frac{k_{B}T}{\gamma_{\ell}+\gamma_{s}}(\partial_{\ell}+\partial_{s})^{2}$
(66)

and we have to be careful that the bound state is only defined on the line
$x_{\ell}=x_{s}$. As a result the equilibrium probability distribution in the
unbound state contains a contribution $\delta(x_{\ell}-x_{s})$ and hence, the
force in the bound state is simply

$\mathcal{F}_{\rm b}^{\rm eff}(x_{s})=-k_{\ell}x_{s}.$ (67)

This corresponds exactly with the effective force in the bound state when the
limits are performed the other way around, see Sec. II.7.1.

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