Present address: ] Computational Biology Unit, Departments of Chemistry and
Informatics, University of Bergen, 5007 Bergen, Norway

# Quantifying the efficiency of principal signal transmission modes in
proteins

Anil Kumar Sahoo  Fachbereich Physik, Freie Universität Berlin, Arnimallee 14,
14195 Berlin, Germany  Max Planck Institute of Colloids and Interfaces, Am
Mühlenberg 1, 14476 Potsdam, Germany  Richard Schwarzl  Fachbereich Physik,
Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany  Markus S.
Miettinen  [  Fachbereich Physik, Freie Universität Berlin, Arnimallee 14,
14195 Berlin, Germany  Roland R. Netz<EMAIL_ADDRESS>Fachbereich
Physik, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany  Centre
for Condensed Matter Theory, Department of Physics, Indian Institute of
Science, Bangalore 560012, India

###### Abstract

On the microscopic level, biological signal transmission relies on coordinated
structural changes in allosteric proteins that involve sensor and effector
modules. The timescales and microscopic details of signal transmission in
proteins are often unclear, despite a plethora of structural information on
signaling proteins. Based on linear-response theory, we develop a theoretical
framework to define frequency-dependent force and displacement transmit
functions through proteins and, more generally, viscoelastic media. Transmit
functions quantify the fraction of a local time-dependent perturbation at one
site, be it a deformation, a force or a combination thereof, that survives at
a second site. They are defined in terms of equilibrium fluctuations from
simulations or experimental observations. We apply the framework to our all-
atom molecular dynamics simulation data of a parallel, homodimeric coiled-coil
(CC) motif that connects the sensor and effector modules of a blue-light-
regulated histidine kinase from bacterial signaling systems extensively
studied in experiments. Our analysis reveals that signal transmission through
the CC is possible via shift, splay, and twist deformation modes. Based on
results of mutation experiments, we infer that the most relevant mode for the
biological function of the histidine kinase protein is the splay deformation.

protein, allostery, coiled coil, signal transfer, response function, molecular
simulation

## Introduction

Signal transmission within a cell, between cells, and from the exterior to the
interior of a cell is necessary for any form of life, be it bacteria, plants,
or animals. Signal-transducing units, at the molecular level, involve signal
receptors (sensors) that sense intracellular or environmental changes (such as
photons, hormones or pH) and corresponding effectors that are responsible for
sparking a response Hunter (2000); Smock and Gierasch (2009). Sensor and
effector modules are usually distinct domains of membrane-bound or cytosolic
proteins Rosenbaum  _et al._ (2009); Bhate  _et al._ (2015); Gleichmann  _et
al._ (2013). How information, at the molecular level, is transmitted through
proteins has been subject of intense research in the last few decades Wodak
_et al._ (2019); Cooper and Dryden (1984); Süel  _et al._ (2003); Changeux and
Edelstein (2005); Cui and Karplus (2008); Motlagh  _et al._ (2014); Nussinov
_et al._ (2014); Feher  _et al._ (2014); Thirumalai  _et al._ (2019).
Experimental techniques, such as NMR spectroscopy Brüschweiler  _et al._
(2009); Guo and Zhou (2016), time-resolved crystallography Berntsson  _et al._
(2017a); Mehrabi  _et al._ (2019), cryo-electron microscopy Wahlgren  _et al._
(2022), and single-molecule experiments Lerner  _et al._ (2018); Schoeler  _et
al._ (2015); Bauer  _et al._ (2019) have provided information about protein
structure, dynamics, and mechanical signaling pathways. Computational
approaches combining molecular simulation techniques Miyashita  _et al._
(2003); Zheng  _et al._ (2006); Van Wynsberghe and Cui (2006); Sharp and
Skinner (2006); Young  _et al._ (2013) and tools from information theory
Lenaerts  _et al._ (2008) and graph theory Chennubhotla and Bahar (2006);
Sethi  _et al._ (2009); Negre  _et al._ (2018), along with various linear or
non-linear correlation analyses Lange and Grubmüller (2006); Kong and Karplus
(2009), have provided molecular-level insights into protein allosteric
communication pathways Mitchell  _et al._ (2016); Wang  _et al._ (2020). A
combination of spectroscopy techniques and molecular dynamics (MD) simulation
description has led to recent insights into time-resolved processes of
allosteric regulation Buchli  _et al._ (2013); Viappiani  _et al._ (2014);
Buchenberg  _et al._ (2017); Bozovic  _et al._ (2020, 2021); Ali  _et al._
(2022). However, the quantitative relation between the dynamics of a protein
and its signal transfer efficacy, which ultimately governs physiological
response, is missing.

We here show how to quantify the signal-transfer efficiency through proteins
in terms of frequency-dependent force and displacement transmit functions. A
force transmit function describes the fraction of the frequency-dependent
force applied at the protein sensor position that survives at the effector
position. The displacement transmit function is defined similarly but is based
on spatial displacements. In fact, the presence of strongly correlated
fluctuations at the sensor and effector positions, as quantified by two-point
correlation functions, is not sufficient for efficient signal transmission,
because equilibrium fluctuations produce a background that the signal has to
compete with. Transmit functions weigh the correlation between the sensor and
effector positions by the fluctuation magnitude and therefore quantify the
fractional signal transmission, they are thus very different from ordinary
two-point correlation functions. Our theoretical framework builds on our
previously developed convolution theory Hinczewski  _et al._ (2010) and is
exact on the linear-response level Kubo (1966). From the time-domain transmit
function, the protein response to any temporal perturbation signal can be
calculated by convolution, all one needs as input to our theory are time
series of positions or displacements that can be obtained from MD simulations
or single-molecule experiments, e.g., by fluorescence resonance energy
transfer Lerner  _et al._ (2018).

Here, we focus on coiled-coil motifs, frequently found in various plant and
bacterial signal transduction systems, that connect signal receptor and
response or effector protein Lupas and Gruber (2005); Casino  _et al._ (2010);
Anantharaman  _et al._ (2006); Matthews  _et al._ (2006); Hulko  _et al._
(2006); Inouye (2006); Bhate  _et al._ (2015); Möglich (2019). Signal
receptors are commonly oligomeric domains and modular in architecture,
produced by recombination of sensor and effector modules. Signaling through
coiled coils leads to conformational rearrangements within effectors that
trigger interaction with regulator proteins, starting signaling cascades.
Studies investigating the role of coiled coils in signal propagation are,
however, limited to quantifying their structural changes in the presence of an
external signal, e.g., pivoting or rotation of the helices Wahlgren  _et al._
(2022); Möglich (2019).

We perform all-atom MD simulations of a coiled-coil (CC) motif consisting of
two identical $\alpha$-helices arranged in parallel that is part of an
engineered blue-light-regulated histidine kinase Möglich  _et al._ (2009). In
this synthetic enzyme, the CC connects the light-oxygen-voltage sensor module
from Bacillus subtilis YtvA and the histidine kinase complex from the
bacterium Bradyrhizobium japonicum, see Fig. 1A. We study the dynamics of the
isolated CC in terms of its shift, splay, and twist deformation modes.
Computed transmit functions reveal that all modes are able to transmit
signals. From the time-domain transmit functions we derive transmit properties
of different time-dependent signals. For step signals we find that
transmission via the splay mode is drastically reduced by single-point
mutations in the CC, whereas the twist mode is least affected. Together with
the experimental observation of light-induced splaying of the CC Berntsson
_et al._ (2017a, b) and experimental mutation studies Gleichmann  _et al._
(2013), this suggests that splay is the most relevant deformation mode of the
CC for the biological function of this sensor histidine kinase Möglich  _et
al._ (2009).

Figure 1:  A) The full-length structure of the dark-adapted blue-light-
regulated histidine kinase YF1 (PDB ID: 4GCZ) Diensthuber  _et al._ (2013). In
YF1 the sensor, linker, and effector modules are homodimeric at the molecular
level. B) Schematic representation of signal transmission from sensor to
effector sites. C) Simulation unit cell containing the coiled-coil (CC)
linker. Water and ions are not shown. D) Time series and corresponding
probability density functions (PDFs) for the N and C-termini distances of the
CC, $|\vec{n}|$ and $|\vec{c}|$. E) Schematic representation of shift, splay,
and twist modes of the CC. F) Schematics for defining the shift vectors
$\vec{n}_{\mathrm{sh}}$, $\vec{c}_{\mathrm{sh}}$ and the splay vectors
$\vec{n}_{\mathrm{sp}}$, $\vec{c}_{\mathrm{sp}}$ as the parallel and
perpendicular components of the distance vectors $\vec{n}$, $\vec{c}$ with
respect to the symmetrized CC axis
$\vec{a}=(\vec{a}_{\mathrm{1}}+\vec{a}_{\mathrm{2}})/2$. G) Schematic for
describing twist as rotation of splay unit vectors $\hat{n}_{\mathrm{sp}}$ and
$\hat{c}_{\mathrm{sp}}$ around the long-axis $\vec{a}$. The angle between
$\hat{n}_{\mathrm{sp}}$ and $\hat{c}_{\mathrm{sp}}$ is denoted as the twist
angle $\phi$.

## Results

###  .1 Theory of transmit functions

The input–output relation for a general responsive system can be quantified by
two transmit functions: force and displacement transmission. Consider we apply
to a system’s sensor (input) position a frequency-dependent force
$\tilde{F}_{\mathrm{s}}(\omega)$ and to its effector (output) position a force
$\tilde{F}_{\mathrm{e}}(\omega)$, see Fig. 1B. In Fourier space, the sensor
and effector positions change as Hinczewski  _et al._ (2010)

$\displaystyle\begin{split}\tilde{X}_{\mathrm{s}}(\omega)=\tilde{J}^{\mathrm{s}}_{\mathrm{self}}(\omega)\tilde{F}_{\mathrm{s}}(\omega)+\tilde{J}_{\mathrm{cross}}(\omega)\tilde{F}_{\mathrm{e}}(\omega)\\\
\tilde{X}_{\mathrm{e}}(\omega)=\tilde{J}^{\mathrm{e}}_{\mathrm{self}}(\omega)\tilde{F}_{\mathrm{e}}(\omega)+\tilde{J}_{\mathrm{cross}}(\omega)\tilde{F}_{\mathrm{s}}(\omega),\end{split}$
(1)

where $\tilde{J}_{\mathrm{cross}}(\omega)$ and
$\tilde{J}^{\mathrm{s/e}}_{\mathrm{self}}(\omega)$ are the frequency-dependent
cross and self (sensor/effector-side) linear response functions, respectively.
Note that there is only a single cross-response. However, the self-responses
$\tilde{J}^{\mathrm{s}}_{\mathrm{self}}(\omega)$ and
$\tilde{J}^{\mathrm{e}}_{\mathrm{self}}(\omega)$ are different for a general
asymmetric system. It should be further noted that Eq. 1 is exact on the
linear response level and is applicable for any two positions in a protein.
The force transmit function $\tilde{T}^{\mathrm{s\rightarrow
e}}_{\mathrm{F}}(\omega)$ is defined for the boundary condition of a
stationary effector position $\tilde{X}_{\mathrm{e}}=0$, from Eq. 1, we obtain

$\displaystyle\tilde{T}^{\mathrm{s\rightarrow
e}}_{\mathrm{F}}(\omega)\equiv\frac{\tilde{F}_{\mathrm{e}}(\omega)}{-\tilde{F}_{\mathrm{s}}(\omega)}=\frac{\tilde{J}_{\mathrm{cross}}(\omega)}{\tilde{J}^{\mathrm{e}}_{\mathrm{self}}(\omega)}.$
(2)

Similarly, setting $\tilde{X}_{\mathrm{s}}=0$ in Eq. 1, we obtain
$\tilde{T}^{\mathrm{e\rightarrow
s}}_{\mathrm{F}}\equiv-\tilde{F}_{\mathrm{s}}/\tilde{F}_{\mathrm{e}}=\tilde{J}_{\mathrm{cross}}/\tilde{J}^{\mathrm{s}}_{\mathrm{self}}$.
Response functions are causal, i.e., there is no positional response before a
force is applied, which implies $\tilde{J}_{\mathrm{cross}}(\omega)$ and
$\tilde{J}_{\mathrm{self}}(\omega)$ are analytic in the upper-half complex
plane. If furthermore $\tilde{J}_{\mathrm{self}}(\omega)$ is non-zero in the
upper-half complex plane, from Eq. 2 it follows that transmit functions are
also causal.

The displacement transmit function $\tilde{T}^{\mathrm{s\rightarrow
e}}_{\mathrm{X}}(\omega)$ is defined as the ratio of the displacement of the
effector site $\tilde{X}_{\mathrm{e}}$ divided by the displacement at the
sensor site $\tilde{X}_{\mathrm{s}}$ under force-free boundary condition
$\tilde{F}_{\mathrm{e}}=0$. Inverting Eq. 1 yields

$\displaystyle\begin{split}\begin{pmatrix}\tilde{F}_{\mathrm{s}}\\\
\tilde{F}_{\mathrm{e}}\end{pmatrix}&=\frac{1}{\tilde{J}^{\mathrm{s}}_{\mathrm{self}}\tilde{J}^{\mathrm{e}}_{\mathrm{self}}-{\tilde{J}_{\mathrm{cross}}}^{2}}\begin{pmatrix}\tilde{J}^{\mathrm{e}}_{\mathrm{self}}&-\tilde{J}_{\mathrm{cross}}\\\
-\tilde{J}_{\mathrm{cross}}&\tilde{J}^{\mathrm{s}}_{\mathrm{self}}\end{pmatrix}\begin{pmatrix}\tilde{X}_{\mathrm{s}}\\\
\tilde{X}_{\mathrm{e}}\end{pmatrix}\\\
&=\begin{pmatrix}\tilde{G}^{\mathrm{s}}_{\mathrm{self}}&\tilde{G}_{\mathrm{cross}}\\\
\tilde{G}_{\mathrm{cross}}&\tilde{G}^{\mathrm{e}}_{\mathrm{self}}\end{pmatrix}\begin{pmatrix}\tilde{X}_{\mathrm{s}}\\\
\tilde{X}_{\mathrm{e}}\end{pmatrix},\end{split}$ (3)

where $\tilde{G}$’s are the moduli determined by inverting the response matrix
$\begin{pmatrix}\tilde{J}^{\mathrm{s}}_{\mathrm{self}}&\tilde{J}_{\mathrm{cross}}\\\
\tilde{J}_{\mathrm{cross}}&\tilde{J}^{\mathrm{e}}_{\mathrm{self}}\end{pmatrix}$.
From Eq. 3 we obtain

$\displaystyle\tilde{T}^{\mathrm{s\rightarrow
e}}_{\mathrm{X}}\equiv\frac{\tilde{X}_{\mathrm{e}}}{\tilde{X}_{\mathrm{s}}}=\frac{-\tilde{G}_{\mathrm{cross}}}{\tilde{G}^{\mathrm{e}}_{\mathrm{self}}}=\frac{\tilde{J}_{\mathrm{cross}}}{\tilde{J}^{\mathrm{s}}_{\mathrm{self}}}=\tilde{T}^{\mathrm{e\rightarrow
s}}_{\mathrm{F}}$ (4)

and by setting $\tilde{F}_{\mathrm{s}}=0$ similarly
$\tilde{T}^{\mathrm{e\rightarrow
s}}_{\mathrm{X}}\equiv\tilde{X}_{\mathrm{s}}/\tilde{X}_{\mathrm{e}}=\tilde{T}^{\mathrm{s\rightarrow
e}}_{\mathrm{F}}$. Thus, force and inverse displacement transmit functions are
the same. Note the striking resemblance between transmit functions defined
here and the transfer function which characterizes the output of a linear
time-invariant system (e.g., an electric circuit consisting of resistors,
inductors and capacitors) in the context of signal processing Oppenheim  _et
al._ (1996).

In practice, one need not apply external forces to determine linear response
functions. $J(t)$ can be obtained from equilibrium time-correlation functions
$C(t)$ using the fluctuation–dissipation theorem Kubo (1966):

$\displaystyle J(t)=-\frac{1}{k_{\mathrm{B}}T}\theta(t)\frac{d}{dt}C(t),$ (5)

where $k_{\mathrm{B}}$ is the Boltzmann’s constant, $T$ represents
temperature, and $\theta(t)$ is the Heaviside step function. The needed cross
and self correlation functions are defined as $C_{\mathrm{cross}}(t)=\langle
X_{\mathrm{s}}(0)X_{\mathrm{e}}(t)\rangle$ and
$C_{\mathrm{self}}^{\mathrm{s}}(t)=\langle
X_{\mathrm{s}}(0)X_{\mathrm{s}}(t)\rangle$,
$C_{\mathrm{self}}^{\mathrm{e}}(t)=\langle
X_{\mathrm{e}}(0)X_{\mathrm{e}}(t)\rangle$, respectively (see Materials and
Methods for details). Note that the trajectories of the positions of two sites
in a protein needed for the calculation of $C(t)$ can be obtained from MD
simulations but also from single-molecule experiments Lerner  _et al._ (2018).

###  .2 MD simulations of the CC linker

We consider the bacterial signaling protein introduced above to demonstrate
the applicability of our theoretical framework to real systems. The water-
soluble structure of the whole protein (Fig. 1A) is too large to carry out
simulations for a sufficiently long time. Also, we are primarily interested in
the CC because of its relevance in diverse signal receptors Lupas and Gruber
(2005); Anantharaman  _et al._ (2006); Bhate  _et al._ (2015); Möglich (2019)
and of the availability of structural and experimental data Möglich  _et al._
(2009); Diensthuber  _et al._ (2013); Gleichmann  _et al._ (2013); Berntsson
_et al._ (2017a, b); Ohlendorf  _et al._ (2016). We therefore perform explicit
solvent all-atom MD simulations of only the CC linker excluding the sensor and
effector modules. A summary of simulation details is given in Materials and
Methods and the simulation unit cell is shown in Fig. 1C. We find that the
distribution of the N-termini distance at the sensor side is broader compared
to the C-termini distance at the effector side, both distributions exhibit
tails that reflect intermittent splaying of the $\alpha$-helix termini (see
Fig. 1D,E). To check the long-time stability of secondary and tertiary
structures of the CC, we calculate three different order parameters: the
fraction of native contacts $Q$ between the two $\alpha$-helices, the root-
mean-square deviation (RMSD) of distances between the native and simulated CC
structures, the former taken from the crystal structure of the full-length
protein shown in Fig. 1A, and the secondary structure (SS) content. These
order parameters are defined in SI Appendix, section 1 and their time-averaged
values are given in SI Appendix, Fig. S1. The overall configuration of the CC
remains stable within 20 $\mu$s of simulation as indicated by an average RMSD
of 2.5 Å. The two $\alpha$-helices remain bound to each other ($Q=0.92$) due
to salt bridges (involving residues Arg135 and Glu138,142) and hydrophobic
interactions (involving residues Leu136,139,143 and Val146) Gleichmann  _et
al._ (2013); Diensthuber  _et al._ (2013). Individual $\alpha$-helices do not
melt as their fraction of SS content values is above 0.85.

###  .3 The CC linker transmits signals via shift, splay and twist modes

We consider the CC linker deformations that correspond to forces of equal
magnitude and opposite direction acting on the two N-termini and on the two
C-termini. Thus, these deformations conserve linear momentum. We define three
distinct deformation modes by the orientation of the terminal displacement
vector with respect to the distance vector between the terminal groups: splay,
where the displacement is parallel to the terminal separation, and shift and
twist, where the displacement is perpendicular to the terminal separation, see
schematics in Fig. 1E. Splay conserves angular momentum, whereas shift and
twist do not and are counteracted by a rotation of the sensor module. Since
the typical rotational diffusion time of the sensor module, estimated to be of
the order of 1 $\mu$s (see Materials and Methods), is much longer than shift
and twist relaxation times, as will be shown below, shift and twist modes are
nevertheless possible signal transmission modes.

Figure 2:  A) Self and cross-correlation functions ($C_{\mathrm{self}}$ and
$C_{\mathrm{cross}}$) for the twist mode of the CC. B) Response functions $J$,
obtained according to Eq. 5 from numerical derivatives of the correlation
functions in A (solid lines). The vertical dash-dotted line represents the
cutoff beyond which the response functions are set to zero to prevent noise
artifacts when calculating Fourier transforms. C) Real and D) imaginary parts
of $\tilde{J}(\omega)$ obtained from discrete Fourier transform of $J(t)$ are
shown as solid lines. Dashed lines represent simultaneous fits to the real and
imaginary parts by a sum of 10 Debye relaxation functions (for further details
see SI Appendix, section 3). Dashed lines in panel B represent the inverse
Fourier transform of the fits in C and D.

The three signal transmission modes are obtained as described below. The
position vectors of the N-termini and C-termini are $\vec{N}_{\mathrm{i}}$ and
$\vec{C}_{\mathrm{i}}$, respectively, where $i=1,2$ refers to the first and
second $\alpha$-helix. These vectors are used to construct the separation
vectors $\vec{c}=\vec{C}_{\mathrm{2}}-\vec{C}_{\mathrm{1}}$,
$\vec{n}=\vec{N}_{\mathrm{2}}-\vec{N}_{\mathrm{1}}$, and
$\vec{a}_{\mathrm{i}}=\vec{C}_{\mathrm{i}}-\vec{N}_{\mathrm{i}}$ (Fig. 1F).
From the end-to-end vectors $\vec{a}_{\mathrm{i}}$, we define the long axis of
the CC as $\vec{a}=(\vec{a}_{\mathrm{1}}+\vec{a}_{\mathrm{2}})/2$. With
respect to $\vec{a}$, we separate $\vec{n}$ into the parallel component, shift
($n_{\mathrm{sh}}=|\vec{n}_{\mathrm{sh}}|=\vec{n}\cdot\hat{a}$) and the
perpendicular component, splay
($n_{\mathrm{sp}}=|\vec{n}_{\mathrm{sp}}|=|\vec{n}-\vec{n}_{\mathrm{sh}}|$).
Similarly, for the C-termini we obtain $c_{\mathrm{sh}}$ and
$c_{\mathrm{sp}}$. The twist angle $\phi$, depicted in Fig. 1G, is obtained
from the scalar product of the two splay unit vectors $\hat{n}_{\mathrm{sp}}$
and $\hat{c}_{\mathrm{sp}}$ as
$\phi=\cos^{-1}(\hat{n}_{\mathrm{sp}}\cdot\hat{c}_{\mathrm{sp}})$. We show
time series of these deformation modes and the corresponding probability
distribution functions in SI Appendix, Fig. S2. Importantly, we find a strong
bias towards clockwise rotation of the CC around its long axis $\vec{a}$, from
the handedness plot obtained from the simulation (see SI Appendix, Fig. S2),
in agreement with the experimental observation of the left-handed supercoiling
of the CC Berntsson  _et al._ (2017a).

The N and C-termini of the CC correspond to the sensor (s) and effector (e)
side, respectively. For the shift mode, the two self-correlation functions are
defined as $C_{\mathrm{self}}^{\mathrm{sh,s}}=\langle
n_{\mathrm{sh}}(0)n_{\mathrm{sh}}(t)\rangle$ and
$C_{\mathrm{self}}^{\mathrm{sh,e}}=\langle
c_{\mathrm{sh}}(0)c_{\mathrm{sh}}(t)\rangle$ and the cross-correlation
function is defined as $C_{\mathrm{cross}}^{\mathrm{sh}}=\langle
n_{\mathrm{sh}}(0)c_{\mathrm{sh}}(t)\rangle$ $=\langle
c_{\mathrm{sh}}(0)n_{\mathrm{sh}}(t)\rangle$. Correlation functions for the
splay mode are defined by interchanging the shift quantities with the related
splay quantities, e.g., $C_{\mathrm{self}}^{\mathrm{sp,s}}=\langle
n_{\mathrm{sp}}(0)n_{\mathrm{sp}}(t)\rangle$. For the twist mode, the cross
and two self-correlation functions are defined by the scalar product of the
splay unit vectors as
$C_{\mathrm{cross}}^{\mathrm{tw}}=\langle\hat{n}_{\mathrm{sp}}(0)\cdot\hat{c}_{\mathrm{sp}}(t)\rangle=\langle\hat{c}_{\mathrm{sp}}(0)\cdot\hat{n}_{\mathrm{sp}}(t)\rangle$
and
$C_{\mathrm{self}}^{\mathrm{tw,s}}=\langle\hat{n}_{\mathrm{sp}}(0)\cdot\hat{n}_{\mathrm{sp}}(t)\rangle$
and
$C_{\mathrm{self}}^{\mathrm{tw,e}}=\langle\hat{c}_{\mathrm{sp}}(0)\cdot\hat{c}_{\mathrm{sp}}(t)\rangle$,
respectively. To disentangle twist from overall CC rotation, we calculate
twist correlation functions in the molecular coordinate frame obtained by
removing the CC center-of-mass translation and rigid-body rotation around its
principal axes at each time step. We present results for the twist mode in
Fig. 2 and for shift and splay modes in SI Appendix, Fig. S3. Self and cross-
correlation functions are shown in Fig. 2A, the corresponding response
functions $J(t)$, obtained according to Eq. 5, are shown in Fig. 2B. All self
and cross-response functions smoothly decay to zero. The relaxation time,
defined as the largest decay time $\tau_{\mathrm{max}}$ of a multi-exponential
fit of $J$ (SI Appendix, section 3), is found to be the fastest for twist
($\tau_{\mathrm{max}}=2.6$ ns), followed by shift ($\tau_{\mathrm{max}}=7.1$
ns) and splay ($\tau_{\mathrm{max}}=9.4$ ns). As expected, for each mode the
two self response functions are greater than the cross response at all times.
The real and imaginary parts of the Fourier-transformed response functions, Re
$\tilde{J}(\omega)$ and Im $\tilde{J}(\omega)$, are shown in Fig. 2C,D. There
is a distinct low-frequency plateau/peak in the real/imaginary part of the
twist response. To obtain analytical representations, we fit multi-Debye
functions (dashed lines) to the Fourier-transformed response functions
$\tilde{J}(\omega)$ in Fig. 2C,D (details are provided in SI Appendix,
sections 2 and 3). Inverse Fourier transforms of the fit functions (dashed
lines) reproduce the time-domain response functions, $J(t)$, data shown in
Fig. 2B very well.

Force transmit functions, $\tilde{T}_{\mathrm{F}}(\omega)$, obtained from the
fitted $\tilde{J}_{\mathrm{self}}(\omega)$ and
$\tilde{J}_{\mathrm{cross}}(\omega)$ using Eq. 2 are shown for all three
different signaling modes in Fig. 3. $\tilde{T}_{\mathrm{F}}(\omega)$
quantifies the system’s response to all possible excitation frequencies. We
find that for the entire frequency range, force transmission through the twist
mode (blue) is the highest, followed by the shift mode (red) and the splay
mode (green). From Re $\tilde{T}_{\mathrm{F}}(\omega)$ in Fig. 3, it is also
evident that no force transmission is possible via the shift and splay mode
for an input signal of frequency $>400$ GHz ($\times 10^{-3}\mathrm{ps}^{-1}$)
and $>10$ GHz, respectively. These cutoff frequencies are similar to the water
Debye mode at a frequency of about $\simeq 20$ GHz Rinne  _et al._ (2015),
which suggests that the damping is partially due to the coupling to the
hydration water. Except for the twist mode, the transmit functions are
generally asymmetric, i.e., the sensor-to-effector side transmit function
$\tilde{T}_{\mathrm{F}}^{\mathrm{s\rightarrow e}}(\omega)$ and the effector-
to-sensor side transmit function $\tilde{T}_{\mathrm{F}}^{\mathrm{e\rightarrow
s}}(\omega)$ are different. It should be noted that
$\tilde{T}_{\mathrm{F}}(\omega)$ presented here characterizes the transmit
properties of an isolated parallel CC. The effects of added sensor and
effector protein modules can straightforwardly be obtained from the sensor and
effector response functions using our previously developed convolution theory
von Hansen  _et al._ (2013).

Figure 3:  (top) Real and (bottom) imaginary part of force transmit functions
$\tilde{T}_{\mathrm{F}}$ for the shift (sh), splay (sp), and twist (tw) modes
of the CC obtained using analytical representations for the self and cross-
response functions (details in SI Appendix, section 3) according to Eq. 2 .
The sensor-to-effector (s $\rightarrow$ e) and effector-to-sensor (e
$\rightarrow$ s) transmit functions are shown as solid and dashed lines,
respectively.

###  .4 Signal transmission in the time domain

For practical purposes, signal transmission in the time domain needs to be
characterized. The force transmit function in the time domain,
$T^{\mathrm{s\rightarrow e}}_{\mathrm{F}}(t)$, is given by the inverse Fourier
transform of $\tilde{T}^{\mathrm{s\rightarrow e}}_{\mathrm{F}}(\omega)$ and
describes the transmission of a $\delta$-function input force signal
$F_{\mathrm{s}}(t)$ at the sensor side. Once
$T_{\mathrm{F}}^{\mathrm{s\rightarrow e}}(t)$ is known, the transmitted force
$F_{\mathrm{e}}(t)$ at the effector side due to an arbitrary input force
signal $F_{\mathrm{s}}(t)$ can be obtained via convolution

$\displaystyle
F_{\mathrm{e}}(t)=\int_{-\infty}^{t}T_{\mathrm{F}}^{\mathrm{s\rightarrow
e}}(\tau)F_{\mathrm{s}}(t-\tau)d\tau.$ (6)

$T_{\mathrm{F}}(t)$ for the different signaling modes are shown in Fig. 4A,
the method by which we obtain analytical representations for
$T_{\mathrm{F}}(t)$ from $\tilde{T}_{\mathrm{F}}(\omega)$ is explained in SI
Appendix, section 4. It is seen that the transmitted force signals for all the
different modes decay rather quickly within 100 ps.

Figure 4:  Transmitted force profiles for the different modes of the CC for
different input force signals: A) $\delta$ pulse, B) step function, and C)
rectangular pulse of width $\tau=1$ ns. Insets in panel A and C represent
zoomed-in force profiles. The force profiles in B and C are rescaled by the
input force strength.

In reality, signals are not transmitted via infinitely short $\delta$-pulses
but rather by pulses of finite duration. How efficiently the CC linker
transmits such an input force signal depends on how quickly it responds to a
suddenly imposed force and how quickly it relaxes back to its equilibrium
state after the removal of force. To understand theses complex dynamics, we
discuss the transmitted output forces for input force step and rectangular
force pulse, $F_{\mathrm{e}}^{\mathrm{stp}}(t)$ and
$F_{\mathrm{e}}^{\mathrm{rec}}(t)$, both obtained from the convolution
integral Eq. 6, for further details see SI Appendix, section 4. For a force
switched on at $t=0$, $F_{\mathrm{e}}^{\mathrm{stp}}(t)$ is shown in Fig. 4B
for the different modes. The transmitted force plateau values are reached for
the different modes after 30–50 ns, which is around 2 orders of magnitude
faster than the reported experimental timescale of 2 $\mu$s associated with
light-induced conformational transitions within the sensor module Berntsson
_et al._ (2017a). Signal transmission by all three modes would hence be
sufficiently fast to not become time-limiting. The plateau value of
$F_{\mathrm{e}}^{\mathrm{stp}}$ is the highest for the twist mode, followed by
shift and splay modes. In Fig. 4C we present the transmitted force
$F_{\mathrm{e}}^{\mathrm{rec}}(t)$ for a rectangular force pulse signal of
duration $\tau=1$ ns that is switched on at $t=0$. It is seen that the
transmitted force through each mode decay to zero within 40 ps after the
removal of applied force. Results for different durations of the rectangular
force pulse of $\tau=10^{-2},10^{-1},10^{1},10^{2}$ ns are presented in SI
Appendix Fig. S4.

###  .5 Robustness with respect to mutations

It has been experimentally demonstrated that single-point mutations within the
CC reduce the signal response of the blue-light-regulated histidine kinase YF1
Diensthuber  _et al._ (2013); Gleichmann  _et al._ (2013). To study this in
our framework, we perform MD simulations of two different experimentally
studied mutants, Q133L, where Gln133 is replaced by Leu, and R135L, where
Arg135 is replaced by Leu. We find that these single-point mutations do not
affect the overall coiled-coil conformation, as seen from their different
structural order parameter values compared with that of the wild-type CC in SI
Appendix, Fig. S1. However, the dynamics and signaling response of these two
mutants are completely different from each other and from the wild-type CC.
For the mutant Q133L in Fig. 5A, the step force transmission for the shift and
splay modes are negligible. Interestingly, the plateau value of the twist-mode
in Fig. 5A is larger than that of the wild-type CC in Fig. 4B. Based on the
absence of signaling for Q133L in experiments Gleichmann  _et al._ (2013),
this suggests that the signaling in the histidine kinase is not connected to
the twist mode. For the mutant R135L in Fig. 5B, the plateau value of the
shift-mode transmission is finite but negative, the splay-mode step
transmission is much reduced, and the plateau value of the twist-mode
transmission is almost half in comparison to the wild-type CC, in experiments
the signaling activity is reduced but not absent Gleichmann  _et al._ (2013).
In addition, structural characterization by electron paramagnetic resonance
spectroscopy and X-ray solution scattering has revealed that light induces a
splaying apart of the sensor domains and hence of the N-termini of the CC
Berntsson  _et al._ (2017a, b). Thus, by comparison with the experimental
findings, we conclude that the signaling mode in the histidine kinase is
predominantly of the splay type. Berntsson et al. Berntsson  _et al._ (2017a)
have experimentally observed superhelical coiling of the CC subsequent to its
light-induced splaying, which suggests coupling between splay and twist
deformations Möglich (2019). We quantify the coupling between different
deformation modes by calculating the Pearson correlation coefficients
(Materials and Methods) as summarized in Table 1 and indeed find significant
splay-twist coupling which could explain the experimentally seen twisting by
coupling to splay. We note that our simulations reveal that the twist mode is
most stable with respect to mutations, so it is conceivable that the CC linker
might in a different biological context also function as a twist transmitter.

Figure 5:  Comparison of transmitted force profiles for an input step force for modified CC systems: A,B) CC with different single-point mutations (Q133L or R135L) and C) CC with charged N-termini. All force profiles are rescaled by the input force strength. For a comparison with the wild-type CC with charge-neutral N and C-termini, see Fig. 4B. Table 1: Correlation coefficients between different deformation modes for the same ends (sensor and effector) and for different ends (cross). type | sh-sp | sh-tw | sp-tw
---|---|---|---
sensor | -0.021 | -0.012 | -0.012
effector | 0.26 | -0.011 | 0.004
cross | -0.057 | -0.006 | 0.006

###  .6 Transmission asymmetry

Asymmetry could possibly be important for the efficient information transfer
from the sensor to the effector side. To look into this, we introduce the
rectification factor $\gamma$ as the ratio of sensor-to-effector
($\mathrm{s\rightarrow e}$) and effector-to-sensor ($\mathrm{e\rightarrow s}$)
step-force transmission profile plateau values:

$\displaystyle\gamma=\lim_{t\to\infty}\frac{F_{\mathrm{e}}^{\mathrm{stp}}(t)}{F_{\mathrm{s}}^{\mathrm{stp}}(t)}.$

From the results in Fig. 4B we conclude that rectification for the splay mode
is the highest, $\gamma^{\mathrm{sp}}=3.2$, followed by the shift mode,
$\gamma^{\mathrm{sh}}=2.4$. In contrast, no rectification is observed for the
twist mode, i.e., $\gamma^{\mathrm{tw}}=1$. These results can be rationalized
by the fact that the rectification factor is given by the ratio of the real
parts of the zero-frequency self responses for the sensor and the effector
side, see SI Appendix, Fig. S3C. To study the relation between the
rectification factor and the structural asymmetry in more detail, we introduce
an additional asymmetry between the sensor and effector ends of the CC by
uncapping the sensor-side $\alpha$-helix termini, which thereby become
positively charged at neutral pH, resulting in the structure
NH${}_{3}^{+}$-CC-CONH2 (note that the results presented in Figs. 1–4 are
obtained for the CC linker with charge-neutral end groups: ACE-CC-CONH2).
Though the step-force transmission profiles for ACE-CC-CONH2 and
NH${}_{3}^{+}$-CC-CONH2 are qualitatively the same, as follows by comparing
Figs. 4B and 5C, we observe a pronounced difference of the plateau values for
the $\mathrm{s\rightarrow e}$ and $\mathrm{e\rightarrow s}$ transmission for
the shift and splay modes of NH${}_{3}^{+}$-CC-CONH2. However, its
$\mathrm{s\rightarrow e}$ and $\mathrm{e\rightarrow s}$ plateau values for the
twist transmission are the same and remain unaffected in comparison to the
charge neutral-termini system ACE-CC-CONH2. We thus find that the
rectification factor can be tuned by changing the chemical structure of the
sensor and effector terminal groups. It is tempting to relate the $\gamma$
values for the different modes to their functional relevance, e.g., the
highest value for $\gamma^{\mathrm{sp}}$ found here and the light-induced
splaying of the CC linker observed in experiments Berntsson  _et al._ (2017a,
b).

## Discussion and Conclusion

We introduce the theoretical framework to quantify the signal transmission
between two distinct sites of a protein in terms of the associated self and
cross response functions. The response functions are via the
fluctuation–dissipation theorem related to equilibrium time-correlation
functions, which can be generated from MD simulations but also from single-
molecule experiments. Note that in experiments, trajectories of separation
coordinates typically include effects due to the coupling to measurement
devices, which can be filtered out by using dynamic deconvolution theory
Hinczewski  _et al._ (2010); von Hansen  _et al._ (2012). The displacement
transmit function relates the correlations between two sites and the
fluctuations at the site at which the input signal is applied, it thus
quantifies the ratio of the output to the input signal and thus conveys more
useful information than the often considered dynamic cross-correlation.

We apply our theoretical framework to the CC linker from the bacterial
signaling protein YF1 and show that all three deformation modes, twist, shift
and splay, achieve signal transfer from the sensor to the effector end of the
CC. Although twist is in principle a better signal transmission mode, it does
not conserve angular momentum and will therefore lead to a rotation of the
sensor domain, this is probably why nature is not using it, at least for this
protein construct. The experimentally observed splaying followed by
superhelical coiling of the CC upon triggering Berntsson  _et al._ (2017a) are
expected due to a coupling between splay and twist deformation modes. Our
analysis of simulation data for the wild-type CC and two important single-
point mutants Diensthuber  _et al._ (2013) suggests that splay is actually the
signaling mode realized in the experimentally studied histidine kinase
Berntsson  _et al._ (2017a, b).

Previous experiments have indicated that the length of the CC linker, not only
the actual linker sequence, is instrumental in determining the response to
light signals Ohlendorf  _et al._ (2016). Our framework is directly applicable
to study the CC length-dependent signaling. Moreover, our method will be
useful to understand the dynamics of activation pathways of other cell
signaling proteins with complex topology, e.g., G-protein-coupled receptors
Thal  _et al._ (2018); Hilger  _et al._ (2018); Rodríguez-Espigares  _et al._
(2020), which are the most frequent targets of drugs due to their involvement
in diverse physiological processes. In this context it should be kept in mind
that signal transmission through general protein networks can be predicted
from the response functions of individual components by repeated application
of convolution relations for serial and parallel connections Hinczewski  _et
al._ (2010); von Hansen  _et al._ (2013). Our study, thus, provides a way
forward to relate atom-level protein dynamics to large-scale intermolecular
communications of biological signaling networks.

Our theory is formulated at the linear-response level and thus is scale-
invariant with respect to the input signal amplitude. In order to obtain the
signal threshold beyond which the signal strength surpasses the noise
background, one needs to compare the signal strength with the root-mean-square
of the fluctuating force or displacement, similar to the definition of the
signal-to-noise ratio in information theory MacKay (2003). We have in this
paper only considered signaling between identical deformation modes at the two
ends, i.e. twist to twist, splay to splay and shift to shift, off-diagonal
signaling modes might be relevant experimentally and will be considered in
future work.

## Materials and Methods

### Models and force-field parameters

From the crystal structure of the whole signal transducing protein unit (PDB
ID: 4GCZ), sensor (N-terminal) and effector (C-terminal) modules are deleted
to obtain the structure of the CC linker Diensthuber  _et al._ (2013). The CC
is composed of two parallel $\alpha$-helices each containing the same 23
residues ([126]Ile-Thr-Glu-His-Gln-Gln-Thr-Gln-Ala-Arg-Leu-Gln-Glu-Leu-Gln-
Ser-Glu-Leu-Val-His-Val-Ser-Arg[148]). The CC is simulated in a rhombic
dodecahedron box of volume 227 nm3 filled with 7135 water molecules (and
counterions needed to neutralize the system). CHARMM36m protein force field
parameters Huang  _et al._ (2017), the TIP3P water model Jorgensen  _et al._
(1983); MacKerell Jr  _et al._ (1998) and ion parameters from Ref. Venable
_et al._ (2013) are used. Four different systems with changes of N- and
C-termini capping groups and/or a mutated residue are considered:
NH${}_{3}^{+}$-CC-CONH2, ACE-CC-CONH2, ACE-CC[Q133L]-CONH2, ACE-
CC[R135L]-CONH2. The acetyl (ACE) cap, CH3-CO-, is used at the N-terminal and
the “-CONH2" group is used at the C-terminal Arg, to simulate charge neutral
termini. The two mutated systems are selected from the study by Gleichmann et
al. Gleichmann  _et al._ (2013).

### MD simulation details

For each system, a simulation is performed for 20 $\mu$s in the $NpT$ ensemble
at temperature $T=300$ K and pressure $p=1$ bar with periodic boundary
conditions using Gromacs 2020.1 Abraham  _et al._ (2015). The stochastic
velocity rescaling thermostat Bussi  _et al._ (2007) with a time constant
$\tau_{T}=0.1$ ps is used to control temperature, while for pressure control
an isotropic Parrinello–Rahman barostat Parrinello and Rahman (1981) is used
with a time constant $\tau_{p}=2$ ps and compressibility $\kappa=4.5\times
10^{-5}$ bar-1. The LINCS algorithm Hess  _et al._ (1997) is used to constrain
the bonds involving hydrogen atoms, allowing a timestep $\Delta t=2$ fs.
Electrostatic interactions are computed using the particle mesh Ewald method
Darden  _et al._ (1993) with a real-space cutoff distance of 1.2 nm, while van
der Waals interactions are modeled using Lennard-Jones potentials with a
cutoff distance of 1.2 nm where the resulting forces smoothly switch to zero
between 1 nm to 1.2 nm. For the data analysis, simulation trajectories are
saved every 1 ps. Images are rendered using the visual molecular dynamics
(VMD) software Humphrey  _et al._ (1996). Analysis is performed using in-house
developed codes and Gromacs analysis modules Abraham  _et al._ (2015).
Correlation functions for observable $A(t)$ and $B(t)$ are calculated as

$\displaystyle
C(\tau)=\frac{1}{L-\tau}\int_{0}^{L-\tau}A(t)B(t+\tau)dt=\langle
A(t)B(t+\tau)\rangle,$

where $L$ is the trajectory length.

### Protein rotational relaxation time

The rotational relaxation time $\tau_{\mathrm{r}}$ of an object is related to
the rotational diffusion coefficient as $D_{\mathrm{r}}=1/2\tau_{\mathrm{r}}$
and is estimated from the Stokes’ rotational diffusion coefficient
$D_{\mathrm{r}}=k_{\mathrm{B}}T/8\pi\eta R_{\mathrm{h}}^{3}$ Landau and
Lifshitz (2013). Using the viscosity of the medium as that of water,
$\eta=8.9\times 10^{-1}$ Pa$\cdot$s, and the hydrodynamic radius as half of
the largest length scale of the full-length protein YF1, $R_{\mathrm{h}}=7$
nm, we obtain $\tau_{\mathrm{r}}=2\ \mu$s which exceeds the transmission
relaxation times by far.

### Coupling between deformation modes

Pearson correlation coefficients between two different deformation modes are
obtained as

$\displaystyle R_{ij}=\frac{\langle(x_{i}-\langle x_{i}\rangle)(x_{j}-\langle
x_{j}\rangle)\rangle}{\sqrt{\langle(x_{i}-\langle
x_{i}\rangle)^{2}\rangle}\sqrt{\langle(x_{j}-\langle
x_{j}\rangle)^{2}\rangle}},$

where $\langle\cdot\rangle$ denotes the time average and $i,j$ refers to
shift, splay or twist at the same ends or at different ends of the CC. The
correlation coefficients $R_{ij}$ for $i\neq j$ (as $R_{ii}=1$) are summarized
in Table 1.

### Data, Materials, and Software Availability

All study data are included in the main text and/or SI Appendix. Derivations
and additional figures that support the findings of this study are included in
SI Appendix.

## Acknowledgements

We acknowledge funding by the European Research Council (ERC) Advanced Grant
NoMaMemo Grant No. 835117, the Infosys foundation and computing time on the
HPC cluster at ZEDAT, FU Berlin.

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