# Information temperature as a measure of DNA’s and texts complexity

O. V. Usatenko  A. Ya. Usikov Institute for Radiophysics and Electronics
Ukrainian Academy of Science, 12 Proskura Street, 61805 Kharkov, Ukraine

###### Abstract

C. Shannon introduced the notion of entropy for random sequences. What about
their temperature? After discussing some methods for introducing information
temperature (IT) for binary random stationary ergodic sequence, we suggest
using IT as a characteristic of complexity and intelligence of an agent
capable of writing or generating meaningful texts. In particular, we discuss
the question of whether the temperature can characterize the academic level of
the text, or serve as an indicator of the quality of brain activity of the
text author.

###### pacs:

05.40.-a, 87.10+e

##  I Introduction

The description and understanding of the complexity of natural and artificial
dynamical systems remains an open problem in science. In connection with the
rapid development of neural networks and artificial intelligence, in order to
assess progress in the field of AI, the question inevitably arises of the
possibility of comparing various models and devices that simulate intelligence
and their comparison with the real human intelligence.

Quantitative evaluation of artificial intelligence faces currently several
important challenges. A fundamental problem in artificial intelligence is that
nobody really knows what intelligence is. There is no standard definition of
what exactly constitutes intelligence. We do not have unified models of an
artificially intelligent system. Others believe that intelligence encompasses
a range of aptitudes, skills, and talents. Despite all these complexities,
artificial intelligence systems are improving, and currently well-known
examples include self-driving cars, image recognizers and classifiers,
translators, chess and Go game simulations, and many others.

Human activity is so multifaceted that it is hardly possible to invent a
universal measure of his intellect. It is difficult to compare who is the more
gifted and brilliant writer or scientist, designer or artist, actor or
director.

On this path, the question inevitably arises about intelligence in what kind
of activity? We will consider intelligence as capability to produce meaningful
texts.

Currently, we can present and discuss briefly several way defining the level
of intelligence activity. We give here a number of well known informal
definitions of human intelligence that have been earlier given.

The classic approach to determining whether a machine is intelligent is the so
called Turing test Turing ; Living  which has been extensively debated over
the last 50 years. The idea is that if a machine can successfully imitate a
human, then this machine is AI. A computer program called Eugene Goostman,
which simulates a 13-year-old Ukrainian boy, is said to have passed the Turing
test at an event organized by the University of Reading.

IQ, or intelligence quotient test IQ , is a measure of ability to reason and
solve problems. It essentially reflects how well we does on a specific test as
compared to other people of some age group. Tests for the ability of
artificial intelligence are developing in parallel with the understanding of
what it really is.

Several authors Chaitin82 ; Chaitin92 ; Orallo ; CaDoOral  have suggested the
relevance of compression to intelligence, especially the inductive learning
part of intelligence.

Legg and Hutter LeggHutter05 ; LeggHutter07  gave ten definitions of human
intelligence and formulated its precise definition to the term _Universal_
Intelligence.

Among other papers let us mention _objective_ definition of intelligence
McCarthy  and integrated information theory Tononi  which has established
itself as one of the leading theories for the study of consciousness. Tegmark
Tegmark  suggests that there are as many types of consciousness as there are
types of matter.

In this work, we propose a new quantitative test for comparing the
intellectual capabilities of the AI simulator and living creatures using
information temperature. Our measure of intelligence is aimed at finding out
the ability to write meaningful texts. We will characterize the complexity of
written text by its informational temperature.

The structure of the rest of the article is as follows. In Section II we
provide brief description of the binary exactly solvable Markov chain of order
$N\leqslant 2$ with step-wise memory for which we introduce the information
temperature. In Section III we present two simplest examples of the
application of the concept of information temperature: the dependence of the
entropy and complexity of the system on the parameter of ordering of the
system. Section IV contain discussion of the use of the informational
temperature to characterize the informational and intellectual complexity of
people and artificial intelligence systems assuming that the latter, both
human and AI, are represented by written/generated texts.

##  II Information temperature

In this section we introduce briefly a definition of the information
temperature. More detailed description is contained in paper UsMPYa .

One of methods introducing the informational temperature (equivalent
correspondence (EC)) is based on the fact that every binary Markov chain is
equivalent to some binary two-sided chain AMUYa  which, in its turn, can be
viewed as the Ising sequence at a fixed temperature where the probability of a
configuration is given by the Boltzmann distribution.

The second method makes use of the traditional entropy based thermodynamic
definition of temperature with direct calculation of the block entropy and
energy of Markov chain in the pair-interaction approximation.

Both approaches give the same result for the case of nearest neighbor
spin/symbol interaction but the method of correspondence of Markov and Ising
chains becomes very cumbersome for longer range of correlations, $N\geqslant
3$. Note that while the first way of the temperature introduction is quite
natural, the second one can be interpreted as heuristic or axiomatic. We dive
here two methods of introducing information temperature in order to show the
validity and effectiveness of the method of introducing fictitious energy.

###  II.1 Temperature from equivalence

Consider an infinite random stationary ergodic sequence $\mathbb{S}$ of
symbols-numbers $a_{i}$,

$\mathbb{S}=...,a_{0},a_{1},a_{2},...$ (1)

taken from the binary alphabet $a_{i}\in\mathcal{A}$:

$\mathcal{A}=\\{0,1\\},\,\,\,\,i\in\mathbb{Z}=\\{...,-1,0,1,2...\\}.$ (2)

A higher number of states can be handled by binary coding. We use the notation
$a_{i}$ to indicate a position $i$ of the symbol $a$ in the chain and the
unified notation $\alpha^{k}$ to stress the value of the symbol
$a\in\mathcal{A}$.

We suppose that the symbolic sequence $\mathbb{S}$ is a high-order Markov
chain. The sequence $\mathbb{S}$ is a Markov chain if it possesses the
following property: the probability of symbol $a_{i}$ to have a certain value
$\alpha^{k}\in\mathcal{A}$ under the condition that _all_ previous symbols are
fixed depends only on $N$ previous symbols,

$\displaystyle P(a_{i}=\alpha^{k}|\ldots,a_{i-2},a_{i-1})$ (3)
$\displaystyle=P(a_{i}=\alpha^{k}|a_{i-N},\ldots,a_{i-2},a_{i-1}).$

For a _two-sided random chain_ the conditional probability that symbol $a_{i}$
is equal to unity, under condition that the _rest_ of symbols in the chain are
fixed, can be presented of the form AMUYa ,

$P(a_{i}=1|A_{i}^{-},A_{i}^{+})$ (4)
$=\displaystyle\frac{P(a_{i}=1,A_{i}^{+}|A_{i}^{-})}{P(a_{i}=1,A_{i}^{+}|A_{i}^{-})+P(a_{i}=0,A_{i}^{+}|A_{i}^{-})},$

where $A_{i}^{-}$ and $A_{i}^{+}$ are the semi-infinite words,
$(\ldots,a_{i-2},a_{i-1})$ and $(a_{i+1},a_{i+2},\ldots)$, surrounding symbol
$a_{i}$ respectively are,

$\underbrace{\ldots,a_{i-2},a_{i-1}}_{A_{i}^{-}},a_{i},\underbrace{a_{i+1},a_{i+2},\ldots}_{A_{i}^{+}}.$
(5)

Here the two-sided conditional probability $P(a_{i}=1|A_{i}^{-},A_{i}^{+})$ is
expressed by means of the Markov-like probability functions
$P(a_{i},A_{i}^{+}|A_{i}^{-})$. These probabilities containing semi-infinite
one-sided words can be expressed in terms of the conditional probability
function of the Markov chain.

For the stationary Markov chain, the probability $b(a_{1}a_{2}\dots a_{N})$ of
occurring a certain word $(a_{1},a_{2},\dots,a_{N})$ satisfies the condition
of compatibility for the Chapman-Kolmogorov equation (see, for example, Ref.
gar ):

$b(a_{1}\dots a_{N})$ $=\sum_{a=0,1}b(aa_{1}\dots a_{N-1})P(a_{N}\mid
a,a_{1},\dots,a_{N-1}).$ (6)

In works UYa  and UYaKM , we have introduced the model Markov chain for which
the conditional probability $p_{k}$ of occurring the symbol “0” after the
$N$-symbol words containing $k$ unities, e.g., after the word
$\underbrace{(11...1}_{k}\;\underbrace{00...0}_{N-k})$, is given by the
following expression:

$p_{k}=P(a_{N+1}=0\mid\underbrace{11\dots 1}_{k}\;\underbrace{00\dots
0}_{N-k})$ $=\frac{1}{2}+\mu(1-\frac{2k}{N}).$ (7)

For the ordinary one-step Markov chain with nearest-neighbor interaction using
Eq. (7) with $N=1$, we easily obtain the conditional probabilities

$\displaystyle
P(a_{i}=1,a_{i+1}=1|a_{i-1}=1)=\left(\frac{1}{2}+\mu\right)^{2},$ (8)
$\displaystyle
P(a_{i}=0,a_{i+1}=1|a_{i-1}=1)=\left(\frac{1}{2}-\mu\right)^{2}.$

Considering the equivalence of Markov chain to the two-sided random sequence,
we apply formula (4) to the word $(1,1,1)$:

$\displaystyle P(a_{i}=1|a_{i-1},a_{i+1})$ (9)
$\displaystyle=\frac{P(a_{i}=1,a_{i+1}|a_{i-1})}{P(a_{i}=1,a_{i+1}|a_{i-1})+P(a_{i}=0,a_{i+1}|a_{i-1})}.$

The Boltzmann distribution for the corresponding Ising model gives,

$\displaystyle P(a_{i}=1|a_{i-1}=1,a_{i+1}=1)=P(\uparrow\uparrow\uparrow)$
$\displaystyle=\frac{\exp(2\varepsilon/T)}{\exp(2\varepsilon/T)+\exp(-2\varepsilon/T)}.$
(10)

The energy of the spins interaction are
$\varepsilon_{\uparrow\uparrow}=\varepsilon_{\downarrow\downarrow}=-\varepsilon$,
$\varepsilon_{\uparrow\downarrow}=\varepsilon_{\downarrow\uparrow}=\varepsilon>0.$
Using Eqs. (8) - (II.1), we have

$\mu=\frac{1}{2}\tanh{\left(\frac{1}{\tau}\right)},\quad\quad\beta=\frac{1}{\tau}=\frac{1}{2}\ln\frac{1+2\mu}{1-2\mu}.$
(11)

Here we have introduced the informational temperature $\tau=T/\varepsilon$ of
the ordinary, one-step, Markov chain and $\beta=\varepsilon/T$ is the inverse
info-temperature. So, for $\tau\rightarrow{\pm\infty}$ we have
$\varepsilon/T\simeq 2\mu\rightarrow 0$, and $\tau\rightarrow\pm 0$ when
$\mu\rightarrow\pm{1/2}$. The negative values of $\tau$ describes an anti-
ferromagnetic ordering of spines or symbols ”0” and ”1”. We can say that
$\tau$ is the temperature $T$ measured in unites of energy $\varepsilon$,
$\tau={T}/\varepsilon$. Note that the informational temperature depends only
on the Markov chain parameter $\mu$, and not on the introduced fictitious
energy $\epsilon$.

###  II.2 Temperature from the entropy

In this section we present another method of introducing the information
temperature based on thermodynamic definition of temperature with direct
calculation of the block entropy and energy of Markov chain in the pair-
interaction approximation. The blocks are chosen to be $(N+1)$-length
subsequences of symbols occurring due to the fact that they encompass all the
information about the structure of the $N$-step Markov chain. The
probabilities of the blocks/words are defined by the Chapman-Kolmogorov
equation.

Using Eqs. (6) and (7) for $N=2$, we find the probabilities of 2-symbols words
and 3-points probabilities of symbols $000$ and $111$ occurring. Introducing
fictive energies of different spin configurations:

$\varepsilon(\downarrow\downarrow\downarrow)=\varepsilon(\uparrow\uparrow\uparrow)=-2\varepsilon_{1}-\varepsilon_{2},$
(12)
$\varepsilon(\downarrow\uparrow\downarrow)=\varepsilon(\uparrow\downarrow\uparrow)=2\varepsilon_{1}-\varepsilon_{2},$
(13)
$\varepsilon(\downarrow\downarrow\uparrow)=...=\varepsilon(\uparrow\uparrow\downarrow)=\varepsilon_{2},$
(14)

we calculate the energy

$E_{3}=-(2\varepsilon_{1}+\varepsilon_{2})\frac{\mu}{1-\mu},$ (15)

and the entropy per two bonds of random elements:

$H_{3}=-2P_{000}\ln P_{000}-6P_{001}\ln P_{001}.$ (16)

Here we use the well-known definition of entropy of subsequence of symbols of
length $L$, see, e.g., Refs. Cover ; Ebeling ,

$\displaystyle
H_{L}=-\sum_{a_{1},...,a_{L}\in\mathcal{A}}P(a_{1}^{L})\log_{2}P(a_{1}^{L}).$
(17)

Here $P(a_{1}^{L})=P(a_{1},\ldots,a_{L})$ is the probability to find the
$L$-word $a_{1}^{L}$ in the sequence and $L=3$ in the considered here case.

Calculating derivative $dH_{3}/dE$ as $(dH_{3}/d\mu)/(dE_{3}/d\mu)$, we obtain
the following result:

$\frac{1}{\tau}=\frac{1}{4}\ln\frac{1+2\mu}{1-2\mu},$ (18)

where

$\frac{1}{\tau}=\frac{\langle\,\varepsilon\rangle}{T},\quad\,\langle\,\varepsilon\rangle=\frac{2\,\varepsilon_{1}+\varepsilon_{2}}{3}.$
(19)

Emerged here average quantity $\langle\,\varepsilon\rangle$ can be treated
formally as the average (fictive) energy of symbols interaction and is
completely different from the energy $E_{3}$, Eq. (15). At the same time,
$\langle\,\varepsilon\rangle$ can be considered as the unity of measurement of
the temperature $T$. Putting in Eq. (19) $\varepsilon_{1}=\varepsilon_{2}=1$,
we obtain natural unity of the information temperature measure, $\tau=T$.

Let us note that the result presented by Eq. (18) is obtained without any
reference to the spin chain. Another important point of the above
consideration is independence of Eq. (18) from the introduced arbitrary
energies $\varepsilon_{1}$ and $\varepsilon_{2}$ of the symbols interactions.
Thus, despite the fact that the method of matching random sequences seems to
be more natural and consistent, it turns out to be more difficult and
restrictive in comparison with the entropy method. Note that using the method
of this section we can recover the result obtained above for the one-step
Markov chain, Eq. (11) and vis-versa the result Eq. (18) can be obtained by
the method of equivalence of the previous suction.

##  III Temperature characterizes entropy and complexity

Thus, we have above presented our earlier finding UsMPYa  on the informational
temperature which should be considered as macroscopic statistical
characteristic of stationary ergodic random sequences. The temperature is one
of the most important macroscopic parameters in statistical physics, and we
believe that the concept of informational temperature should also make deep
sense in relation to random sequences. In this section of the work, we present
two simplest examples of the application of the concept of information
temperature. Earlier, in the works of a number of authors Hogg ; Li ; Lopez ,
the question of the dependence of the entropy and complexity of the system on
the parameter of ordering of the system was discussed. This parameter did not
have a clear and unambiguous name and definition. It was called either “degree
of disorder” of the system Hogg , or the parameter “order – disorder” Li , or
a system parameter ranging “from ideal crystal to ideal gas” Lopez . If
earlier the answer to the question on what parameters the entropy depends
could be interpreted ambiguously, now we give the natural answer to this
question – it is the informational temperature.

The entropy per one bond $H_{2}$ for the ordinary one-step Markov chain
considered in Section II.1 can be found using the method of Section II.2 or
using the result of paper Li :

$H_{2}=-2P_{00}\ln P_{00}-2P_{01}\ln P_{01}=\ln 2$
$-(1/2+\mu)\ln(1/2+\mu)-(1/2-\mu)\ln(1/2-\mu).$ (20)

Here $P_{ij}$ are the probabilities of the words of the length two occurring:

$P_{00}=P_{11}=(1/2+\mu)/2,$ (21) $P_{01}=P_{10}=(1/2-\mu)/2,$ (22)

and the parameters $\mu$ is defined by Eq. (11). As a result we get the
entropy as a function of info-temperature,

$\displaystyle H_{2}$ $\displaystyle=$ $\displaystyle(\ln
2+\ln(1+\exp(-2\beta))/(1+\exp(-2\beta))$ (23) $\displaystyle+$
$\displaystyle\ln(1+\exp(2\beta))/(1+\exp(2\beta))).$

This is a very natural expression that approaches its minimum and maximum
values as $\tau=1/\beta$ approaches zero or infinity, respectively. This
result is presented in Fig. 1.

Figure 1: The entropy $H_{2}(\tau)$ for the 1-step Markov chain. The positive
values of $\tau$ describe persistent ferromagnetic ordering of the random
stationary ergodic sequence $\mathbb{S}$, the negative ones correspond to
anti-persistent anti-ferromagnetic case.

Earlier it was pointed out Hogg ; Li  that the algorithmic (entropic)
complexity is not corresponds to our intuitive understanding of complexity. In
this regard, the paper Lopez  proposes a definition that discriminates
complexity in the region of high disorder or, in our terms, when
$\tau\rightarrow\infty$. This is simply the interplay between the information
stored in the system and its disequilibrium. In our case, this definition has
the following form:

$\displaystyle C=hD,\quad\quad
D=\sum_{ij={0,1}}\left(P_{ij}-\frac{1}{4}\right).$ (24)

Here $h=H_{2}-H_{1}$ is the Shannon entropy, or the entropy per symbol,
$H_{1}=\ln 2$, $D$ is the so called disequilibrium factor, is equal in our
case to the value $D=\mu^{2}$, obtained with using Eqs. (21) and (22). The
final result for the complexity expressed via the informational temperature
$\tau$ is

$\displaystyle C(\tau)=\tanh^{2}(1/\tau)$ $\displaystyle\times$
$\displaystyle\left[\frac{\ln(1+\exp(-2/\tau))}{1+\exp(-2/\tau)}+\frac{\ln(1+\exp(2/\tau))}{1+\exp(2/\tau)}\right].$

This result is presented by the curve in Fig. 2.

Figure 2: The dependence of complexity $C(\tau)$ on the informational
temperature for the 1-step Markov chain.

##  IV Temperature is a parameter of a text complexity

Thus, in the previous section, we have presented two simplest examples of the
application of the concept of informational temperature. We have seen that
both entropy and information complexity can be considered as temperature
dependent functions. Consequently, the temperature itself characterizes the
measure of the complexity of the system. We believe that the importance of the
concept of informational temperature is not limited to these examples and has
a much wider scope.

We suppose that the informational temperature can be used to characterize the
informational and intellectual complexity of people and artificial
intelligence systems assuming that the latter, both human and AI, are
represented by written/generated texts. However, we note the insufficiency of
the obtained expressions for their use in applied purposes, since the memory
scale can reach several thousand units, while the results obtained are valid
only for a simple one-step Markov chain.

D. Tononi Tononi  indicates the possibility of characterizing a neural network
by some $\Phi$ function. This function is not clearly defined and difficult
for calculations. Can a neural network be characterized by the temperature of
the text that the neural network is able to generate? We suppose that the
answer to this question is positive.

###### Acknowledgements.

The author of this work apologizes to readers for the fact that the work is
published unfinished. The author of the work is not sure about the possibility
of its completion, as he is in a place under continuous artillery and bombing.

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††thanks: These two authors contributed equally††thanks: These two authors
contributed equally

# Sliding across a surface: particles with fixed and mobile ligands

Janna Lowensohn  Center for Nonlinear Phenomena and Complex Systems, Code
Postal 231, Université Libre de Bruxelles, Boulevard du Triomphe, 1050
Brussels, Belgium  Laurie Stevens  Interuniversity Institute of Bioinformatics
in Brussels, ULB-VUB, La Plaine Campus, 1050 Brussels, Belgium  Center for
Nonlinear Phenomena and Complex Systems, Code Postal 231, Université Libre de
Bruxelles, Boulevard du Triomphe, 1050 Brussels, Belgium  Applied Physics
Research Group, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels,
Belgium  Daniel Goldstein  Department of Physics and Astronomy, Tufts
University, 574 Boston Avenue, Medford, Massachusetts 02155, USA  Bortolo
Matteo Mognetti<EMAIL_ADDRESS>Center for Nonlinear Phenomena
and Complex Systems, Code Postal 231, Université Libre de Bruxelles, Boulevard
du Triomphe, 1050 Brussels, Belgium

###### Abstract

A quantitative model of the mobility of ligand presenting particles at the
interface is pivotal to understanding important systems in biology and
nanotechnology. In this work, we investigate the emerging dynamics of
particles featuring ligands that selectively bind receptors decorating an
interface. The formation of a ligand-receptor complex leads to a molecular
bridge anchoring the particle to the surface. We consider systems with
reversible bridges in which ligand-receptor pairs bind/unbind with finite
reaction rates. For a given set of bridges, the particle can explore a tiny
fraction of the surface as the extensivity of the bridges is finite. We show
how, at time scales longer than the bridges’ lifetime, the average position of
the particle diffuses away from its initial value. We distill our findings
into two analytic equations for the sliding diffusion constant of particles
carrying mobile and fixed ligands. We quantitatively validate our theoretical
predictions using reaction-diffusion simulations. We compare our findings with
results from recent literature and discuss the molecular parameters that
likely affect the particle’s mobility most. Our results, along with recent
literature, will allow inferring the microscopic parameters at play in complex
biological systems from experimental trajectories.

##  I Introduction

Ligand–receptor interactions underly most of the biological processes found at
the cell membrane, like adhesion and signaling [1]. Much work has quantified
the interaction strength between particles (e.g., colloids, viruses, or
vesicles) and surfaces (e.g., supported lipid bilayers or cell membranes)
mediated by ligand–receptor complexes [2, 3, 4, 5, 6, 7]. Different groups
have also studied the particle–surface first contact leading to the formation
of an interacting patch [8, 9, 10]. Despite the particle’s mobility being
pivotal in many biological processes, except for studies in a shear flow [11,
12, 13], little has been done to understand the rolling/sliding dynamics of a
functionalized particle at the interface [14, 15, 16, 17, 18]. For instance,
successful cellular invasions by Influenza A Viruses [19, 20] and Herpes
Viruses [21] require the invader to diffuse along the cell membrane while
remaining bound. Similarly, it is believed that the Malaria parasite uses
ligand gradients to reorient itself towards a configuration favoring
erythrocyte invasion [22]. In the self–assembly of functionalized colloids,
the mobility of bound particles is pivotal to relax disordered aggregates into
crystalline structures [23].

This study aims at improving our understanding of how particles bound to
receptor–expressing surfaces 111Without loss of generality, in this
contribution ligands and receptors refer to linker molecules tethered,
respectively, to the particles and the surface. can laterally diffuse (slide).
We develop a model in which the particle’s center of mass is constrained to
remain within a finite area, $\Omega$, by the presence of bridges. $\Omega$ is
entirely determined by a subset of bridges (in the following constraining
bridges). The emerging dynamics are then controlled by the timescale over
which a constraining bridge is added or removed along with the average
displacement of the particle following an update of $\Omega$. We summarize our
results using two simple equations (Eqs. 6, 10) expressing the diffusion
constant in terms of the single-bridge off–rate $k_{\mathrm{off}}$, the
average number of bridges $\langle n_{b}\rangle$, and the area of the surface
reachable by a single ligand. Our predictions hold in the limit in which the
evolution of $\Omega$ is slow as compared to the timescales taken by the
particle to explore $\Omega$. We clarify how this approximation holds based on
the finding that the number of constraining bridges remains constant in the
many-bridge limit ($\langle n_{\mathrm{cb}}\rangle\approx 5$). In a recent
contribution, Kowalewski et al. derived an expression similar to Eq. 6 (Eq. 7)
for the diffusion of a molecular walker (Fig. 1b) [25]. As compared to Ref.
25, the present contribution clarifies how Eqs. 6 and 7 hold only if $\langle
n_{\mathrm{cb}}\rangle$ does not increase with the number of bridges.

In Sec. II we discuss our model as compared to experimental systems. In Sec.
III, we derive the emerging diffusion constant for particles with mobile
ligands (Eq. 6). In Sec. IV, we validate the results of Sec. III using
reaction–diffusion simulations [16]. In Sec. V, we present our predictions for
fixed ligands (Eq. 10). Finally, in Sec. VI, we summarize and discuss the
findings of the paper in view of recent results.[17, 25]

Figure 1: (a), (top) Unilamellar Vesicles (UVs) carry ligands anchored to the
lipid bilayer. For liquid bilayers, ligands are mobile along the UV surface.
(bottom) The contact region between the vesicle and the surface is represented
by a 2D disk shown in black. Bridges form and break inside the contact region.
(b), (left) Colloidal particles carrying ligands with fixed tethering points.
(right) In molecular walkers, ligands are the legs of star-shaped polymers
tipped by reactive complexes (e.g., Ref. 26). (c), (left) Bridges form and
break with a rate constant equal to, respectively, $k_{\mathrm{on}}$ and
$k_{\mathrm{off}}$. (right) The bond potential $h$ controls the stretchability
of the bridges. (a, b, c) The different elements of the figure are not to
scale.

##  II The model system

Unilamellar vesicles are a class of particles usually employed to carry mobile
ligands (Fig. 1). Ligands (like membrane proteins or synthetic moieties) are
anchored to the membrane through trans-membrane domains, hydrophobic complexes
(inset of Fig. 1a, top), or covalent (e.g., glycosidic) bonds [2]. A schematic
model of functionalized vesicles at the interface would comprise a contact
region and an outer cap (Fig. 1a). Ligand-receptor bridges can form only
inside the contact region, in the following modeled by a disk (Fig. 1a,
bottom). The outer region acts as a finite reservoir of ligands. Neglecting
fluctuations in the direction orthogonal to the surface, we model the system
using a 2D representation in which the particle is identified with the disk
corresponding to the contact region (Sec. III.1). This model does not track
the position of the mobile, free ligands which are mapped into uniform,
depletable densities (as the total number of ligands is fixed).

Particles with fixed ligands (Fig. 1b, left) include colloids functionalized
by synthetic moieties or biological particles like viruses (e.g., Ref. 20).
Spherical particles tend to roll rather than slide [15, 16]. Sliding is
occasionally more prominent in non-spherical particles like, e.g., the
rod–shaped strand of the Influenza A virus. In Sec. V, we predict the sliding
diffusion constant using a 2D model (Fig. 8).

Ref. 25 employed a 2D model similar to the one of Fig. 1a, bottom to study
molecular walkers (Fig. 1b, left). Molecular walkers are made from star-shaped
polymers tipped by reactive moieties (e.g., Ref. 26). Each ligand can then
reach out to receptors found inside the circle centered over the projection of
the branching point onto the surface with a radius equal to the length of the
ligand.

Fig. 1c reports the important molecular parameters of the system. Bridges form
or break with a rate constant equal to $k_{\mathrm{on}}$ or
$k_{\mathrm{off}}$, respectively. We refer to Secs. IV and V for the
calculation of the rates, respectively, for fixed and mobile ligands. The
reaction rates have a major impact on the emerging dynamics. A system composed
of static bridges would result in a particle arrested on a surface. Another
important element is the bond potential $h$. Ligand/receptor backbones are
often constituted by flexible polymers. For ideal polymers, stretching the
tethering points far away is contrasted by a harmonic attraction,
$h(d)=-kd^{2}/2$ (where $d$ is the lateral distance between tethering points).
For non-ideal polymers, the bond potential is a multi–body function that also
depends on neighboring ligands/receptors. For more rigid backbones, the bond
potential is dominated by the finite stretchability of the bridges. In this
work, for fixed ligands, we use a square–well bond potential, $h(d)=0$ if
$d<\lambda$, $h(d)=\infty$ if $d>\lambda$, where $\lambda$ is the maximal
lateral extensibility of a bridge (e.g., [20]). This choice is mainly
motivated by the possibility of deriving universal analytic results which
could then be used (when compared with other systems) to assess the importance
of the bond potential $h$ (Sec. VI).

For mobile ligands (Fig. 1a), the bond potential $h$ plays a minor role as
averaging over the ligand position results in a null force exerted onto the
particle. This consideration holds only if the tethering point anchoring the
bridge to the vesicle can explore most of its available configurational space
before unbinding.

In this study, we only consider fixed, randomly distributed receptors. We
comment on the mobility of the receptors and their distribution in Secs. V and
VI.

##  III Theoretical predictions of the sliding diffusion constant

To model the emerging dynamics of particles carrying mobile ligands, in the
next section (Sec. III.1), we characterize the configurational space
($\Omega$) available to a particle featuring a static set of bridges. In Secs.
III.2 and III.3, we then investigate, respectively, the rate at which $\Omega$
evolves when the set of bridges changes because a bridge is either added or
removed and the average displacement of the particle following a change of
$\Omega$. Based on such an understanding, in Sec. III.4 we propose an analytic
expression for the emergent diffusion constant $D$ (Eq. 6). Eq. 6 holds in the
limit in which $D$ is not limited by the diffusion constant of the free
particle ($n_{b}=0$), $D_{0}$. The simulation protocol employed in Secs. IV, V
remains reliable at low values of $D_{0}$.

Figure 2: (a) CBs (thick, red points) in a system with $n_{b}=25$ bridges
uniformly distributed inside the particle’s perimeter (black line). The gray
circles represent configurations which simultaneously touch two CBs (such
configurations correspond to the vertices of $\Omega$ in Fig. 2c). (b) The
gray line is the curved polygon with vertices given by the set of CBs. Adding
a new bridge inside the curved polygon does not further constrain the
particle’s position. (c) The colored region highlights the configurational
area ($\Omega$) available to the center of mass for the set of CBs of panel a
and b. For comparison, $\Omega$ has been inserted in panel $a$ and $b$
(colored regions).

###  III.1 Constraining bridges (CBs)

Following on from Sec. II (Fig. 1a), we model particles functionalized by
mobile ligands using 2D disks of radius R moving over a surface decorated by
randomly distributed, fixed receptors (Fig. 2a). The model tracks the position
of the disk and the ensemble of receptors (dots in Fig. 2a) forming a bridge
with a ligand moving on the particle.

We assume that bridges cannot be formed with receptors outside the particle’s
perimeter. Therefore, for mobile ligands, we assume that the interaction range
is much smaller than the disk’s radius ($\lambda\ll R$). For a given particle
position, the bridges are taken as uniformly distributed inside the circle
(Fig. 2a). For a given set of bridges, the circle’s center can then explore a
finite fraction of the surface compatible with the fact that bridges remain
inside the perimeter of the particle rattling around the set of bridges (gray
circles in Fig. 2a).

Fig. 2c highlights the configurational space available to the disk’s center
($\Omega$) corresponding to the bridges of Fig. 2a. $\Omega$ comprises an
ensemble of vertices joined with curved edges (in the following we will refer
to such an object as a ’curved polygon’). The edges are arcs of circles of
radius $R$ centered on the bridges in contact with the perimeter (thick, red
points in Fig. 2a). Similarly, the vertices of $\Omega$ correspond to
configurations in which the perimeter of the particle (gray lines in Fig. 2a)
touches two bridges.

Figure 3: (bottom) Average number of constraining bridges (CBs) as a function
of the number of bridges. Gray bars represent the variance calculated using
$10^{3}$ independent samples with bridges unifomly distributed inside the
disk. The number of bridges does not fluctuate $n_{b}=\langle n_{b}\rangle$.
(top) Three samples with different number of bridges (from left, $n_{b}=20$,
$n_{b}=80$, and $n_{b}=2560$) featuring the same number of CBs (in red).

In general, only a finite fraction of bridges can come in contact with the
perimeter of the particle (Fig. 2a). In the following, we will refer to them
as ’Constraining Bridges’ (CBs). Intriguingly, the number of CBs
($n_{\mathrm{cb}}$) remains finite when increasing the number of bridges
$n_{b}$ (Fig. 3). Fig. 3 reports the average ($\langle
n_{\mathrm{cb}}\rangle$) and the variance of the number of CBs as a function
of $n_{b}$. The CBs can be defined as the smallest subset of bridges that,
when joined by circle arcs of radius $R$, contain all the remaining
$n_{b}-n_{\mathrm{cb}}$ points (Fig. 2b). The sets of CBs are then similar to
the vertices of a convex hull [27, 28] with the difference for convex hulls
being that the edges between the points on the border are straight lines (SI
Fig. 1) rather than the arcs shown in Fig. 2b. This key observation justifies
the fact that, contrary to what is observed in Fig. 3, the number of points
belonging to the convex hull diverges logarithmically with $n_{b}$ [27, 28].
Notice, for instance, that the stretching of two bridges far away from each
other towards the particle’s border pins the particle in a single
configuration and would impede any other bridge from touching the perimeter
(right panel of SI Fig. 1).

Figure 4: Average area of the surface outside the curved polygon ($\langle
A_{\mathrm{out}}\rangle$, Fig. 2b) and area of the configurational space
available to the center of mass of the particle $\langle|\Omega|\rangle$ (Fig.
2c). Dashed lines are the fitting functions reported in Eqs. 2 and 3. The
errorbars (calculated using 103 samples as in Fig. 3) are smaller than the
symbol size. The number of bridges does not fluctuate $n_{b}=\langle
n_{b}\rangle$. The $y$ scales are in units of $R^{2}$.

We define by $A_{\mathrm{out}}$ the area of the region outside the curved
polygon identified by the CBs (Fig. 2b). $A_{\mathrm{out}}$ will be used in
the next section to estimate the probability that $\Omega$ changes following
the formation of a new bridge.

We now consider the case in which bridges break and form (corresponding to
points that disappear and appear in Fig. 2a) with a rate constant equal to
$k_{\mathrm{on}}^{T}$ and $k_{\mathrm{off}}$, respectively. We decompose the
motion of the particle into two components. At short timescales, the particle
rattles around the current CBs. At larger timescales, comparable or larger
than the lifetime of a given set of CBs ($\tau_{\mathrm{cb}}$), the particle
is driven by the diffusive motion of the CBs 222Notice that the set of CBs is
expected to diffuse as the position of the added and deleted bridges is not
the same.. Hence, the emerging mobility of the particle is controlled by
$\tau_{\mathrm{cb}}$ and the typical displacement of the particle following an
update of the CBs, $\delta_{\mathrm{cb}}$. These two quantities are discussed
in the next two sections (Secs. III.2 and III.3). First, notice that these two
dynamics are not strictly independent as possible updates of the CBs depend on
the actual position of the particle. Second, this decomposition is valid only
in the limit in which the particle can explore an area comparable with
$\Omega$ (Fig. 2c and Fig. 4) before the next update of the CBs takes place.
This is often the case given that the system explores the polygon with a
diffusion constant $D_{0}$ with $D_{0}=10^{6}-10^{4}\,$nm${}^{2}/$s for
micrometer-nanometer particles.

###  III.2 CBs reconfiguration time ($\tau_{\mathrm{cb}}$)

Figure 5: Ditribution of the time to update the set of CBs,
$\tau_{\mathrm{cb}}$. Symbols are simulations featuring different average
numbers of bridges ($\langle n_{b}\rangle=20$–$2560$). For each value of
$\langle n_{b}\rangle$, we generate $5\cdot 10^{4}$ different sets of CBs (SI
Sec. II). The line is a Poisson distribution with average given by Eq. 2 with
$\langle n_{\mathrm{cb}}\rangle=4.9$ (Fig. 3).

There are two possible events leading to a change of the set of CBs (Fig. 2a)
and thus affect $\tau_{\mathrm{cb}}$. In the first one (event $a$), a free
ligand binds a bridge in the region outside the curved polygon
($A_{\mathrm{out}}$ in Fig. 2b). In event $b$, a receptor belonging to the CBs
unbinds. Notice how both events could, in principle, add/remove multiple
bridges to/from the set of CBs. For instance, in an event of type $a$, the new
bound receptor could prevent some of the existing CBs from touching the
perimeter. The rate at which an event of type $a$ happens is given by
$k_{a}=k_{\mathrm{on}}^{T}A_{\mathrm{out}}/A_{\mathrm{tot}}$. As defined
before, $k_{\mathrm{on}}^{T}$ is the rate of forming any bridge, contained or
not by the CBs. Events of type $b$ happen with a rate
$k_{b}=k_{\mathrm{off}}n_{\mathrm{cb}}$. $k_{\mathrm{off}}$ is the rate at
which a single bridge breaks (Fig. 1c). Instead, $k_{\mathrm{on}}^{T}$ reads
as $k_{\mathrm{on}}^{T}=k_{\mathrm{on}}\cdot n_{\ell}\cdot n_{r}$, where
$k_{\mathrm{on}}$ is the rate at which a bridge form from a given
ligand–receptor pair (Fig. 1c) and $n_{\ell}$ ($n_{r}$) is the number of free
ligands (receptors covered by the disk). If the number of bound receptors
$n_{b}$ is steady ($n_{b}=\langle n_{b}\rangle$), the rates of forming and
destroying a bridge are equal ($k_{\mathrm{on}}^{T}=k_{\mathrm{off}}\langle
n_{b}\rangle$). This equality allows expressing the average values of $k_{a}$
and $k_{b}$ as a function of $k_{\mathrm{off}}$ and geometric factors

$\displaystyle\langle k_{a}\rangle=k_{\mathrm{off}}\langle
n_{b}\rangle{\langle A_{\mathrm{out}}\rangle\over A_{\mathrm{tot}}}$
$\displaystyle\langle k_{b}\rangle=k_{\mathrm{off}}\langle
n_{\mathrm{cb}}\rangle\,\,,$ (1)

where $A_{\mathrm{tot}}=\pi R^{2}$ and we neglect correlations between
fluctuations in $n_{b}$ and $A_{\mathrm{out}}$. The average rate at which sets
of CBs are reconfigured is then given by
$\langle\tau_{\mathrm{cp}}\rangle^{-1}=\langle k_{a}\rangle+\langle
k_{b}\rangle$ (where we treat events $a$ and $b$ as independent). Notice that
events of type $a$ are the reverse of events of type $b$. These two types of
events decrease and increase $A_{\mathrm{out}}$ by the same quantity.
Therefore, in steady conditions, we have $\langle k_{a}\rangle=\langle
k_{b}\rangle$. From Eq. 1 we then derive

$\displaystyle\langle A_{\mathrm{out}}\rangle=\pi R^{2}{\langle
n_{\mathrm{cb}}\rangle\over\langle n_{b}\rangle}$
$\displaystyle\langle\tau_{\mathrm{cb}}\rangle={1\over
2k_{\mathrm{off}}\langle n_{\mathrm{cb}}\rangle}\,.$ (2)

In Fig. 4, we sample $\langle A_{\mathrm{out}}\rangle$ using the samples
generated in Fig. 3. Simulation results agree with the theoretical predictions
(Eq. 2). In Fig. 4 we also sample the average area ($\langle|\Omega|\rangle$)
of the configurational space $\Omega$ (Fig. 2c). Given that the distance
between the outer-most bridges and the disk’s border scales like $1/\langle
n_{b}\rangle$, we expect that the disk could move along a given direction by a
displacement $\delta\ell$, $\delta\ell\sim 1/\langle n_{b}\rangle$. This
suggests that the configurational area scales like
$\langle|\Omega|\rangle\sim\langle\delta\ell\rangle^{2}\sim\langle
n_{b}\rangle^{-2}$ in the large $\langle n_{b}\rangle$ limit. By fitting the
simulation results of Fig. 4, we find that $\langle|\Omega|\rangle$ is well
approximated by $\langle A_{\mathrm{out}}\rangle/\langle n_{b}\rangle$

$\displaystyle\langle|\Omega|\rangle=\pi R^{2}{\langle
n_{\mathrm{cb}}\rangle\over\langle n_{b}\rangle^{2}}\,\,.$ (3)

Figure 6: Distribution of the displacement, $\delta_{\mathrm{cb}}$, of the
particle following an update of the set of CBs calculated as described in the
text. Symbols correspond to the set of simulations presented in Fig. 5. The
black line is a double exponential function, $a_{1}\exp(-b_{1}\cdot
z)+a_{2}\exp(-b_{2}\cdot z)$, with $a_{1}=0.9964$, $b_{1}=1.1707$,
$a_{2}=2.67$, and $b_{2}=18.67$.

###  III.3 Single-step displacement ($\delta_{\mathrm{cb}}$)

We now estimate the average displacement of the center of mass of the particle
following a change in the set of CBs ($\delta_{\mathrm{cb}}$) by means of
numerical simulations. For a given set of CBs, we sample the position of the
disk center, ${\bf x}(\Omega)$, by generating points inside $\Omega$ as done
in Fig. 2c. While exploring $\Omega$, we attempt either to add or remove
bridges. The probability of adding and removing a bridge are given in SI Sec.
1. Once a reaction move changes the set of CBs, we estimate the average
position of the disk for the new set of CBs, ${\bf x}(\Omega^{\prime})$, and
calculate $\delta_{\mathrm{cb}}=|{\bf x}(\Omega^{\prime})-{\bf x}(\Omega)|$.
By repeating the procedure, we obtain the distribution of $\tau_{\mathrm{cb}}$
(Fig. 5) and of $\delta_{\mathrm{cb}}$ (Fig. 6).

In Figs. 5 & 6 symbols refer to systems with an average number of bridges
ranging from $\langle n_{b}\rangle=5$ to $\langle n_{b}\rangle=5120$ (as in
Figs. 3, 4). Fig. 5 shows how $\tau_{\mathrm{cb}}$ follows a Poisson
distribution consistent with Eq. 2. In particular,
$\langle\tau_{\mathrm{cb}}\rangle$ does not depend on $\langle n_{b}\rangle$.

Concerning the distributions of $\delta_{\mathrm{cb}}$, Fig. 6 shows how the
distributions for different $\langle n_{b}\rangle$ fall on the same master
curve when plotted as a function of $z=\langle
n_{b}\rangle\cdot\delta_{\mathrm{cb}}$. As a consequence
$\langle\delta_{\mathrm{cb}}\rangle$ scales like $1/\langle n_{b}\rangle$ with
a prefactor which is well-approximated by $R\sqrt{\pi/\langle
n_{\mathrm{cb}}\rangle}$ (inset of Fig. 6)

$\displaystyle\langle\delta_{\mathrm{cb}}\rangle=\sqrt{\pi\over\langle
n_{\mathrm{cb}}\rangle}{R\over\langle n_{b}\rangle}\,\,.$ (4)

Notice that, in our calculation of $\langle\delta_{\mathrm{cb}}\rangle$,
bridges form with uniform probability inside the particle’s perimeter. The
simulation setting employed in Secs. IV & V will consider explicit positions
of the receptors.

###  III.4 Emerging diffusion constant ($D$)

Having estimated $\tau_{\mathrm{cb}}$ and $\delta_{\mathrm{cb}}$, we are now
in a position to predict the emerging diffusion constant $D$. The dynamics is
limited by the reaction rates if
$D_{0}\langle\tau_{\mathrm{cb}}\rangle\geq\langle|\Omega|\rangle$. This
inequality certainly holds in the large $\langle n_{b}\rangle$ limit where
$\langle|\Omega|\rangle\sim\langle n_{b}\rangle^{-2}$ (Eq. 3) and
$\langle\tau_{\mathrm{cb}}\rangle\sim 1$ (Eq. 2). In a time interval $\Delta
T$, the average displacement of the particle will be (neglecting correlations
between sequential values of $\delta_{\mathrm{cb}}$)

$\displaystyle\langle\Delta{\bf x}^{2}\rangle={\Delta
T\over\langle\tau_{\mathrm{cp}}\rangle}\langle\delta_{\mathrm{cb}}\rangle^{2}=2D\Delta
T$ (5)

from which we extract

$\displaystyle\boxed{D={k_{\mathrm{off}}\pi R^{2}\over\langle
n_{b}\rangle^{2}}\,\,.}$ (6)

For a molecular walker (Fig. 1b), Ref. 25 recently reported the following
expression for $D$

$\displaystyle D={1\over 2}{k_{\mathrm{on}}k_{\mathrm{off}}\over
k_{\mathrm{on}}+k_{\mathrm{off}}}{R^{2}\over\langle n_{b}\rangle^{2}}\,,$ (7)

where $R$ is now the length of the walker’s branches. This contribution
identified the time-step ($\langle\tau\rangle$) of the coarse-grained dynamics
with the time taken to a single ligand to bind and sequentially unbind a
receptor ($\langle\tau\rangle=k_{\mathrm{on}}^{-1}+k_{\mathrm{off}}^{-1}$,
when neglecting avidity terms in the binding event). Ref. 25 then derived Eq.
7 by fitting simulation results (with $\langle n_{b}\rangle<16$) with
$D\sim\langle\delta\rangle^{2}/\langle\tau\rangle$ and
$\langle\delta\rangle^{2}\sim\langle n_{b}\rangle^{-2}$ (guessed from a
relation similar to Eq. 3). As compared to Ref. 25, we provide a microscopic
definition of $\langle\tau\rangle$ based on the identification of the set of
constraining bridges.

It may look surprising that in Eq. 6 $D$ increases with the radius of the disk
$R$. However, Eq. 6 holds only if the unconstrained diffusion constant $D_{0}$
is sufficiently high to allow the particle to sample the configurational space
$\Omega$ for timescales smaller or comparable with
$\langle\tau_{\mathrm{cb}}\rangle$. Such an approximation, for a given number
of bridges $n_{b}$, breaks down for sufficiently large values of $R$. In such
a limit, $D$ will be limited by $D_{0}$. Secondly, when comparing disks coated
with the same density of ligands, $\langle n_{b}\rangle$ would scale with the
area of the disk resulting in a diffusion coefficient scaling as $D\sim
1/R^{2}$.

Eq. 6 predicts how the ligand/receptor densities affect the mobility of the
system only through $\langle n_{b}\rangle$. This observation explains why
colloids functionalized with higher densities of ligands crystallize
better[23]. Indeed, higher coatings increase the multivalent effect and
therefore the melting temperature and $k_{\mathrm{off}}$ at the melting.

In Sec. V, we generalise Eq. 6 to the case of fixed ligands.

Figure 7: Simulation results: mobile ligands. (a) (left) Two configurational
spaces ($\Omega_{i}$, $i=1$, 2) visited by a $2\cdot 10^{-3}$s trajectory.
(center and right) Position of the disk center (dots) while constrained to
remain in $\Omega_{1}$ and $\Omega_{2}$, respectively. The disk’s center of
mass can explore a significant fraction of $\Omega_{i}$ before the latter
changes as a result of an update of the CBs (Fig. 2). (b) Full–lines: Mean
square displacements sampled using 100 trajectories (for each curve). The
shadow region represents the error bar. Dotted and dashed lines represent,
respectively, a linear fitting and the theoretical curve (Eq. 6). In (b) we
use i) $k_{\mathrm{off}}=636\,$s-1, $k_{\mathrm{on}}=0.318\,$s-1 (top curve,
corresponding to $L=10\,$nm, $\gamma k_{\mathrm{on},0}=10^{5}\,$M-1s-1 in Eq.
9) and ii) $k_{\mathrm{off}}=318\,$s-1, $k_{\mathrm{on}}=0.159\,$s-1 (bottom
curve, corresponding to $L=10\,$nm, $\gamma k_{\mathrm{on},0}=5\cdot
10^{4}\,$M-1s-1 in Eq. 9). In (a) we use system i).

##  IV Simulation validation

Following Ref. 16, we employ simulations merging Brownian diffusion and
reaction dynamics to validate Eq. 6. At each simulation step, we first employ
the Gillespie algorithm [30] to update the current set of bridges from time
$t$ to $t+\Delta t$, where $\Delta t$ is the simulation time step. The list of
possible reactions include the breakage of a bridge and the binding of a free
receptor to a free ligand. Consistently with the fact that we do not track the
explicit position of the ligands (Sec. II), the rate of forming a bridge is
the same for all ligand–receptor pairs (well–stirred approximation). Details
are reported in SI Sec. III.A. We then attempt to evolve the center of mass of
the particle, ${\bf x}_{\mathrm{CM}}$, using a Brownian dynamics update

$\displaystyle{\bf x}_{\mathrm{CM}}(t+\Delta t)={\bf
x}_{\mathrm{CM}}(t)+\sqrt{2D_{0}\Delta t}U\,\,,$ (8)

where $D_{0}$ is the diffusion constant of the particle at the surface without
bridges and $U$ is a normally distributed stochastic variable. If a bridge
exits the area covered by the particle, the new configuration is rejected.
High rejection rates may artificially slow down the dynamics. Therefore, as
compared to previous investigations [16], we slice the diffusion step into
$N_{B}$ consecutive Brownian updates in which we use Eq. 8 by replacing
$\Delta t$ with $\Delta t^{\prime}=\Delta t/N_{B}$. To limit the number of
rejets, we use the results of the previous section to gauge $N_{B}$ as a
function of $\Delta t$, $D_{0}$, and $\langle\delta_{\mathrm{cb}}\rangle$.
Understanding the impact of different reaction–diffusion algorithms [31] on
the mobility of the particle deserves future investigations.
We consider a disk of radius $R=100\,$nm placed at the center of a square with
side equal to $1\,\mu$m. We cover the surface with $N_{R}=17777$ randomly
distributed receptors corresponding to an average receptor–receptor distance
equal to $d=7.5\,$nm. This is a typical coverage density usually employed, for
instance, in experiments with ligands/receptors made of DNA oligomers [2, 32].
We do not expect that using regularly distributed receptors would have a major
impact on the results of this section [25]. The surface of the disk is covered
with $N_{L}=558$ mobile ligands, matching the receptor density. The on– and
off–rates, for flexible ligands, can be calculated as [33, 2]

$\displaystyle k_{\mathrm{on}}={\gamma k_{\mathrm{on,0}}\over\pi R^{2}L}$
$\displaystyle k_{\mathrm{off}}={\gamma k_{\mathrm{on,0}}\over
K_{\mathrm{eq}}}=\gamma k_{\mathrm{on,0}}K_{D}\,,$ (9)

where $K_{\mathrm{eq}}$, $K_{D}$, and $k_{\mathrm{on},0}$ are, respectively,
the equilibrium costant, the dissociation constant, and the on rate measured
for ligands and receptors free in solution (i.e. not anchored to the particle
and surface). $L$ is a length comparable with the size of the ligands.
$\gamma$ ($\gamma<1$) is a non-dimensional term accounting for the fact that
the diffusion constants of anchored ligands/receptors are reduced as a result,
e.g., of the drag exherted by the bilayer onto the anchoring point [34, 35,
32] or by the solvent onto the polymeric backbone supporting the reactive
complex [36]. $k_{\mathrm{on},0}$ can span multiple orders of magnitudes
(e.g., Refs. 37, 38, 21) and is always smaller than the diffusion–limited
value $k^{(\mathrm{dl})}_{\mathrm{on},0}\approx 10^{8}\,M^{-1}s^{-1}$ (where
$M$ is the molarity unit, $1\cdot M=0.602\,$nm-3), calculated using the
Smoluchowski’s equation for a nanometer–size probe. Importantly, our
simulations model bridge formation/denaturation as Poissonian events and
therefore neglect rebinding events [39]. This is not expected to be a serious
drawback as the emergent diffusion is order of magnitudes smaller than the
diffusions constant of the ligands (which could be $\approx\mu m^{2}/s$). The
calculation of the on/off rates of ligands tethered to a surface is currently
debated [40].

Fig. 7 reports our simulation results for the mobile ligand system with
$\langle n_{b}\rangle=111$. Fig. 7a (left) shows the two configurational
spaces ($\Omega_{1}$ and $\Omega_{2}$) sampled by an $\approx 2\cdot 10^{-3}$s
trajectory. Figs. 7a (center and right) show the center of mass trajectory
(points) while constrained to remain in $\Omega_{1}$ and $\Omega_{2}$,
respectively. Notice that the trajectory exploring $\Omega_{1}$ is much
shorter than the one constrained by $\Omega_{2}$. The variability of the
trajectory length is related to the stochastic nature of $\tau_{\mathrm{cb}}$.
Importantly, in both cases, the trajectories can travel a distance comparable
to the size of $\Omega_{i}$. This justifies the assumptions employed in the
previous section and clarifies how the long-time diffusion is controlled by
the evolution of $\Omega$ arising from the formation/rupture of bridges.

In Fig. 7b, we report the mean squared displacement for two pairs of on and
off rates with $k_{\mathrm{on}}/k_{\mathrm{off}}$ constant (resulting in the
same $\langle n_{b}\rangle$). In both cases, we show how Eq. 6 predictions
agree with simulation results. A validation employing different $\langle
n_{b}\rangle$ is presented in the next section. The simulation programs
employed to obtain Fig. 7b is available online 333URL:
https://github.com/bmognetti/SlidingDiffusionConstant_Mobile.git.

Figure 8: Particles with fixed ligands. (a) A ligand–receptor pair can form a
bridge if the distance between the tethering points (dots and squares) is
smaller than $\lambda$. We report only ligands/receptors forming a bridge. (b)
By representing the relative position of the bound receptors as compared to
the conjugated ligands, fixed ligand systems are mapped into mobile ligand
systems with $R=\lambda$ (Fig. 1).  Figure 9: Simulation results: fixed
ligands. (a and b) Translational diffusion constant at a given
$k_{\mathrm{off}}$ and $k_{\mathrm{on}}$, respectively, and different numbers
of bridges (as obtained by changing, respectively, $k_{\mathrm{on}}$ and
$k_{\mathrm{off}}$). (c) Rotational diffusion constant at a given
$k_{\mathrm{off}}$ and different $\langle n_{b}\rangle$. The lines in (a) and
(b) are the theoretical predictions (Eq. 10) while in (c) are the fits to the
data points. We generate 100, $1\,$s trajectories for each data point. We
calculate the error bars using 5 independent estimates of the diffusion
constant obtained by dividing the batch of trajectories into 5 subgroups. The
error bars on the number of bridges are smaller than the symbol.

##  V Fixed ligands

We now discuss the case in which ligands are not mobile but fixed to the
surface of the particle. We define by ${\bf r}^{\mathrm{(L)}}_{i}$ the
coordinates of the ligands’ tethering points (square symbols in Fig. 8a). A
receptor, grafted in ${\bf r}^{\mathrm{(R)}}_{i}$, can then form a bridge with
ligand $i$ if $|{\bf r}^{\mathrm{(R)}}_{i}-{\bf
r}^{\mathrm{(L)}}_{i}|<\lambda$, where $\lambda$ is the maximal lateral
distance between the tethering points of a ligand–receptor pair forming a
bridge. For a given set of bridges, the possible configurations of the
particle are then the ones in which none bound receptor exits the circle of
radius $\lambda$ centered over the conjugated ligand (dashed circles in Fig.
8a). By shifting all the bound receptor’s position by the position of the
corresponding conjugated ligand (${\bf r}^{\mathrm{(R)}}_{i}\to{\bf
r}^{\mathrm{(R)}}_{i}-{\bf r}^{\mathrm{(L)}}_{i}$), as done in Fig. 8b, we
show how the possible configurations of the particle (for a given orientation)
correspond to the possible configurations of a disk of radius $\lambda$ with
mobile ligands binding uniformly distributed receptors (Fig. 2). In
particular, the area available to the center of the particle with fixed
ligands (for a given orientation) will follow from the expression of
$\langle|\Omega|\rangle$ (Eq. 3) by replacing $R$ with $\lambda$. Similarly,
$\lambda$ replaces $R$ in the average displacement
$\langle\delta_{\mathrm{cb}}\rangle$ (Eq. 4), while the average time to
reconfigure a set of constraining bridges $\langle\tau_{\mathrm{cb}}\rangle$
(Eq. 2) does not change. Finally, the sliding diffusion constant will mimic
Eq. 6 as follows

$\displaystyle\boxed{D_{\mathrm{fixed}}={k_{\mathrm{off}}\pi\lambda^{2}\over\langle
n_{b}\rangle^{2}}\,.}$ (10)

The previous equation implies that, if $\lambda\ll R$, the emerging diffusion
for fixed ligands will be much smaller than for mobile ligands. Moreover, the
configurational area available to the particle’s center of mass will also
decrease. The latter observation implies that the assumptions underlying Eq. 6
are less severe in the present case: the trajectory will have the same time
($\tau_{\mathrm{cb}}$) to explore configurational spaces ($\Omega_{i}$)
smaller than the ones considered in Fig. 7a.

To validate Eq. 10, we simulate a $d=$100 nm diameter disk carrying
$N_{L}=200$ randomly distributed ligands and a surface featuring 0.1
receptors$/$nm-2. As for mobile ligands, we do not expect that major
differences would arise from using surfaces with regular patterns of
ligands/receptors (except for settings in which bridges could form only in
particular alignment conditions). Ligands can bind receptors placed at a
lateral distance smaller than $\lambda=10.0\,$nm (Fig. 8). For fixed ligands,
the relations between $k_{\mathrm{on}}$, $k_{\mathrm{off}}$ and
$k_{\mathrm{on},0}$, $k_{\mathrm{off},0}$ are different from what stated by
Eq. 9. In general, $k_{\mathrm{on}}$ and/or $k_{\mathrm{off}}$ are also a
function of the relative distance between the tethering points (e.g. Ref. 2).
In this section, consistent with the fact that bridges do not exert any force
onto the particle when the distance between tethering points is smaller than
$\lambda$, we consider configuration–independent rates. This choice is
consistent with previous literature (e.g. 20). In Sec. VI, we discuss how
different bond potentials between bridged tethering points could affect our
findings.

Given that ligands are fixed, the particle’s orientation, $\theta$, is a
non–degenerate variable. In particular, the system features an emerging
rotational diffusion constant, $D_{\theta}$. The (rotational) diffusion
constant of the free particle is $D_{0}=10^{5}\,$nm2/s
($D_{\theta,0}=100\,$rad/s). As found in Sec. IV, in the limit of validity of
the theory, $D_{0}$ and $D_{\theta,0}$ do not affect the emerging diffusion
constants ($D_{\mathrm{fixed}}$ and $D_{\theta}$). In all simulations, we use
$\Delta t=10^{-5}\,$s and $N_{B}=10$ (Sec. IV).

Figs. 9a and 9b report simulation results for $D_{\mathrm{fixed}}$ as a
function of $\langle n_{b}\rangle$. We consider sets of runs with
$k_{\mathrm{off}}$ constant and different $k_{\mathrm{on}}$ (Fig. 9a) and sets
with $k_{\mathrm{on}}$ constant and different $k_{\mathrm{off}}$ (Fig. 9b).
The resulting average number of bridges ranges between $\langle
n_{b}\rangle\approx 8$ and $\langle n_{b}\rangle\approx 100$. Systems with
$k_{\mathrm{off}}$ constant can be directly compared with Eq. 10 (dashed lines
in Fig. 9a). For the simulations at constant $k_{\mathrm{on}}$, the
predictions of Eq. 10 are given by the horizontal dashed lines calculated
using the different $k_{\mathrm{off}}$ employed by the simulations. In the
large $N_{L}$ limit (at constant $\langle n_{b}\rangle$), it can be shown
that, for constant $k_{\mathrm{on}}$, $D_{\mathrm{fixed}}$ decreases as
$1/\langle n_{b}\rangle^{3}$ (SI Sec. IV). Overall, the simulation results of
Fig. 9a are in excellent agreement with Eq. 10 for all values of
$k_{\mathrm{on}}$ and $k_{\mathrm{off}}$. $D_{\mathrm{fixed}}$ is not a
function of the particle’s shape, as highlighted by the geometric construction
in Fig. 8 and SI Fig. 3a. In particular, contrarily to $D_{0}$,
$D_{\mathrm{fixed}}$ is always isotropic.

From the simulation results, it is not immediate to speculate about systematic
errors in Eqs. 6 and 10. Our theory neglects that, for low values of $\langle
n_{b}\rangle$, the emerging diffusion constant could be limited by $D_{0}$.
However, in Figs. 7b, 9a, and 9b, the theory overestimates simulations rarely.
Possible systematic errors could arise from treating successive unit
displacements $\delta_{\mathrm{cb}}$ as independent variables while some
degree of correlation is expected given that successive $\Omega_{i}$ share a
subset of constraining bridges. A detailed study of the statistical properties
of the coarse-grained dynamics underlying our model deserves future
investigations.

Fig. 9c and SI Fig. 2 report the results for the rotational diffusion
constant, $D_{\theta}$ vs $n_{b}$, at constant $k_{\mathrm{off}}$ (Fig. 9c)
and $k_{\mathrm{on}}$ (SI Fig. 2). $D_{\theta}$ mimics the trends observed for
$D_{\mathrm{fixed}}$. By fitting the data points at constant
$k_{\mathrm{off}}$ (Fig. 9c) we find

$\displaystyle D_{\mathrm{\theta}}=c{k_{\mathrm{off}}\pi\over\langle
n_{b}\rangle^{2}}$ (11)

with $c\approx 0.13$. Contrarily to what observed for $D_{\mathrm{fixed}}$,
$D_{\theta}$ is affected by the shape of the particle. In particular,
$D_{\theta}$ decreases when considering more elongated particles (SI Fig. 3b).
The simulation programs employed to obtain Fig. 9 is available online 444URL:
https://github.com/StevensLaurie/Disk_Diffusion.

##  VI Discussions and conclusions

In this paper we have derived two simple equations (Eqs. 6, 10) for the
emergent sliding diffusion constant of particles bound to a surface through
reversible linkages. We validated the theoretical predictions by employing
reaction diffusion simulations. The diffusion constant is controlled by the
rate at which ligand–receptor pairs unbind ($k_{\mathrm{off}}$), the average
number of bridges ($\langle n_{b}\rangle$), and the area of the surface
accessible to a single ligand ($\pi R^{2}$ in Eq. 6 or $\pi\lambda^{2}$ in Eq.
10). Our results strictly apply to the case of fixed receptors. For mobile
receptors, particles anchored to a surface through static bridges can
laterally diffuse[35, 34].

Ref. 17 recently presented numerical and analytic predictions of the emerging
sliding diffusion of a 1D system. As compared to our setting, this
contribution employs harmonic (rather than square-well) bond potentials $h$
(Fig. 1c) to model the forces exerted by the bridges onto the particle. The
prediction of Ref. 17 for $D$ (in the large $\langle n_{b}\rangle$ limit)
reads as follows

$\displaystyle D={k_{B}T\over\Gamma+\langle n_{b}\rangle\gamma+\langle
n_{b}\rangle{k\over k_{\mathrm{off}}}+\langle n_{b}\rangle{\gamma
k_{\mathrm{on}}\over k_{\mathrm{off}}}}\,,$ (12)

where $\Gamma$ and $\gamma$ are the friction coefficients controlling the
diffusion of the free particle ($D_{0}=k_{B}T/\Gamma$) and of the reactive
tips of the ligands, while $k$ is the spring constant of the bond potential.
Ref. 17 predicts how, for $\langle n_{b}\rangle\to 0$, the emerging sliding
diffusion is limited by $D_{0}$. In the many–bridge limit, it predicts a
$1/\langle n_{b}\rangle$ scaling law which is in contrast with Eqs. 6 & 10.
Such a disagreement unlikely arises from the fact that Ref. 17 employs a 1D
system. Indeed, adapting our model to a 1D system would not impact the
$1/\langle n_{b}\rangle^{2}$ scaling found in Eqs. 6, 10. It sounds more
reasonable that the two different types of bond potentials could play a
crucial role. This observation points to the fact that polymeric details of
the ligand/receptor molecules, not directly linked to the rate coefficients
($k_{\mathrm{on},0}$ and $k_{\mathrm{off}}$), may have a major impact on the
mobility of the particle. The bond potential is largely tunable. For instance,
ligands resembling fully flexible ideal polymers or thin rods would result,
respectively, in a quadratic and logarithmic bond potential [43].

Existing literature in biophysics, employing models with cross-linked objects,
already reported emerging friction terms $\zeta$ ($\zeta\sim 1/D$) which are
non–linear in the number of crosslinkers. For instance, Wierenga et al. [44]
show how the friction between two sliding cytoskeletal filaments is super
exponential in the number of crosslinkers, $\langle n_{b}\rangle$. In this
system, the highly non-linear increase of $\zeta$ in $\langle n_{b}\rangle$
arises from the necessity of having a specific sequence of $\langle
n_{b}\rangle$ unbinding-rebinding events to observe a net ($O(1)$) relative
displacement of the filaments. Intriguingly, cooperative effects may also
reduce the friction constant. For instance, Fogelson et al. [45] study
monovalent particles diffusing over a substrate decorated by receptors. In
this system, multiple particles diffuse faster (as compared to a diluted
system) as a result of an effective repulsive force mediated by the
functionalized surface.

When comparing the predictions of Eqs. 6, 10 with experiments employing
spherical particles, one should consider that the latter tend to roll rather
than slide [15, 16]. The prediction of the emergent rotational diffusion
constant deserves future investigations. As compared to the present
investigation, Ref. 16 studied large rates constants ($k_{\mathrm{on},0}\geq
10^{6}\,$M-1s-1). Due to the short simulation times employed (milliseconds
rather than seconds as in this study), Ref. 16 did not detect sliding motion
for small $k_{\mathrm{on},0}$ and large $n_{b}$. In the regime of validity of
Eqs. 6 & 10, we predict how the diffusion constant is not a function of the
drag exerted by the medium onto the particle. This result could be tested in
experiments, for instance, by adding inert polymers to increase the viscosity
of the medium without affecting the biochemistry of the system. On the other
hand, sliding is likely the major component of the mobility of non-spherical
particles, like the elongated form of Influenza A Virus (IAV) [20]. However,
it is believed that IAV particles achieve their mobility because of the
catalytic effect of neuramidase ligands cleaving the sialic acid present on
the receptors [19]. The outcomes of the present study (as well as Refs. 17,
25) will help quantitatively assess the contribution to the mobility of the
particle due to the presence of activity in IAV and other biological systems.
Finally, for fixed ligands, we predict that elongated particles would diffuse
homogeneously. This observation will allow for the detection of catalytic
activity which has been shown to be responsible for persistent motions [19].

## Supplementary Material

See supplementary material for figures supporting the claims made in the
manuscript and details about the simulation alghoritms.

## Acknowledgements

We thank M. Holmes-Cerfon (New-York University, USA) for bringing Refs. 28,
25, and 45 to our attention. We thank M. Holmes-Cerfon (New-York University,
USA), S. Korosec (Simon Fraser University, Canada), and N. R. Forde (Simon
Fraser University, Canada) for useful discussions. JL is supported by a BAEF
grant. LS is supported by a seed grant from the Interuniversity Institute of
Bioinformatics in Brussels (IB2). BMM is supported by a PDR grant of the
FRS-F.N.R.S. (grant n∘T015821F). Computational resources have been provided by
the Consortium des Équipements de Calcul Intensif (CÉCI), funded by the
F.R.S.-FNRS under Grant No. 2.5020.11 and by the Walloon Region.

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