# Enhanced Atmospheric Turbulence Resiliency with Successive Interference
Cancellation DSP in Mode Division Multiplexing Free-Space Optical Links

Yiming Li, Zhaozhong Chen, Zhouyi Hu, David M. Benton, Abdallah A. I. Ali,
Mohammed Patel, Martin P. J. Lavery, and Andrew D. Ellis, Manuscript received
June 00, 2022; revised June 00, 2022. Research supported by EPSRC under grant
number EP/T009047/1, EP/T009012/1, EP/S003436/1, EP/S016171/1, and the
European Union’s Horizon 2020 research and innovation programme under the
Marie Skłodowska-Curie grant agreement No. 713694.Y. Li, Z. Hu, D. Benton, A.
Ali, M. Patel, and A. Ellis are with Aston Institute of Photonic Technology,
Aston University, Birmingham, B4 7ET, UK (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>andrew.ellis@aston.ac.uk).Z. Chen, and M. Lavery are
with James Watt school of engineering, University of Glasgow, Glasgow, G12
8QQ, UK (e-mail<EMAIL_ADDRESS>Martin.Lavery@glasgow.ac.uk).

###### Abstract

We experimentally demonstrate the enhanced atmospheric turbulence resiliency
in a 137.8 Gbit/s/mode mode-division multiplexing free-space optical
communication link through the application of a successive interference
cancellation digital signal processing algorithm. The turbulence resiliency is
further enhanced through redundant receive channels in the mode-division
multiplexing link. The proof of concept demonstration is performed using
commercially available mode-selective photonic lanterns, a commercial
transponder, and a spatial light modulator based turbulence emulator. In this
link, 5 spatial modes with each mode carrying 34.46 GBaud dual-polarization
quadrature phase shift keying signals are successfully transmitted with an
average bit error rate lower than the hard-decision forward error correction
limit. As a result, we achieved a record-high mode- and polarization-division
multiplexing channel number of 10, a record-high line rate of 689.23 Gbit/s,
and a record-high net spectral efficiency of 13.9 bit/s/Hz in emulated
turbulent links in a mode-division multiplexing free-space optical system.

###### Index Terms:

Mode division multiplexing (MDM), free-space optics (FSO), multiple-input
multiple-output (MIMO), successive interference cancellation (SIC),
turbulence.

##  I Introduction

Free space optical (FSO) communication is a promising wireless communication
technique for a wide range of applications. FSO systems can provide a very
high unlicensed bandwidth for data transmission at more than one hundred
Gigabits per second [1], and can also provide extremely long transmission
distance, inherent security, and robustness to electromagnetic interference
[2, 3].

Mode-division multiplexing (MDM) can be used in FSO communication systems to
increase the throughput or mitigate the impact of atmospheric turbulence [4,
5]. A popular basis for MDM is orbital angular momentum (OAM) modes, which can
be decomposed into the rotational symmetric complete orthogonal
Laguerre–Gaussian (LG) modal basis set [6, 7, 8]. To further increase the
transmission capacity, it is preferable to employ a complete orthogonal mode
set, such as LG modes [9], where orthogonality guarantees negligible
crosstalk, and the rotational symmetry guarantees good compatibility with
conventional optical lenses and fiber based delivery with relatively low modal
crosstalk [10].

Atmospheric turbulence can significantly reduce the system performance by
introducing scintillation, beam wandering, beam spreading, angle-of-arrival
fluctuations, and phase fluctuations [11]. The performance degradation is even
more pronounced in a MDM FSO system as it can introduce the inter-mode
crosstalk [7, 12].

Adaptive optics (AO) may be used to mitigate atmospheric turbulence effects
[8, 13, 14], and may be assisted by a coaxial Gaussian beam as reference for
MDM FSO systems [8]. However, the AO systems can not fully compensate for the
turbulence effects and residual inter-mode crosstalk remains [8, 13],
requiring a tradeoff between control bandwidth and compensation errors [14].

Multiple-input multiple-output (MIMO) digital signal processing (DSP) is
another approach to mitigate the negative effects of atmospheric turbulence in
the MDM FSO systems [15, 16]. MIMO DSP shifts the complexity from the optical
to the digital domain, enabling rapid feed-forward channel state information
(CSI) extraction. Most reported MIMO algorithms in optical communications are
based on a minimum mean square error (MMSE) MIMO equalizer [15, 16, 17, 18,
19]. This works well in few-mode fibre (FMF) systems where the channel matrix
is almost unitary [17, 19, 18], but introduces significant performance
degradation in fading channels [20, 15, 16].

It is widely accepted in the radio frequency wireless MIMO communication
systems that adding redundant receive channels can provide extra diversity,
which is beneficial to the bit error rate (BER) performance [20]. To date, in
MDM FSO systems, diversity gain has only been considered in special cases such
as single-input multiple-output (SIMO) systems or systems with multiple
spatially separated apertures [5, 16]. The generalized MIMO algorithms with
redundant receive channels have not been applied to the MDM FSO systems to
combat turbulence induced fading yet.

In this paper, we experimentally demonstrated enhanced atmospheric turbulence
resiliency in MDM FSO communication systems. In particular: (1) We validate
for the first time that successive interference cancellation (SIC) MIMO
decoding provides enhanced turbulence resiliency; (2) We provide the first
demonstration that a generalized MIMO algorithm with redundant receive
channels provides enhanced turbulence resiliency by comparing different MIMO
systems (e.g. $10\times 10$, $10\times 12$, and $6\times 6$ to $6\times 12$);
(3) We employ commercial components, including mode selective photonic
lanterns (MSPLs) and coherent transponders to demonstrate practical
applicability. As a result, we have demonstrated a record high independent
channel number of 10, line rate of 689.23 Gbit/s, and net spectral efficiency
of 13.9 bit/s/Hz in emulated turbulent MDM FSO links.

The remainder of this paper is organized as follows: the experimental setup is
given in Section II, Section III proposes the selection of phase screens for
turbulence generation, Section IV proposes the SIC MIMO DSP with redundant
receive channels for enhanced turbulence resiliency, Section V provides the
experimental results, and Section VI summarizes the key advantages of the
proposed MDM FSO system.

##  II Experimental Setup

The experimental setup is shown in Fig. 1.

Figure 1: The experimental setup. DP-QPSK: dual-polarization quadrature phase
shift keying; TS: training sequence; P: pilot; D: data; SM-EDFA: single-mode
erbium-doped fiber amplifier; AOM: acousto-optic modulator; FDLs: fibre delay
lines; VOAs: variable optical attenuators; MSPL: mode-selective photonic
lantern; SLM: spatial light modulator; HWP: half-wave plate; PCs: polarization
controllers; ASE: amplified spontaneous emission; OBPF: optical bandpass
filter; OSA: optical spectrum analyzer; DSP: digital signal processing; CD:
chromatic dispersion; LMS: least mean squares; SIC: successive interference
cancellation; MIMO: multiple-input multiple-output. (a) Frame structure
generated by Ciena transponder; (b) signal burst after AOM; (c) delayed frame
structure after transmitter FDLs; (d) time-division multiplexing (TDM) signal
after receiver FDLs; (e) TDM signal after receiver coupler; (f) offline DSP.

At the transmitter, a Ciena WaveLogic 3 transponder with a 39.385 GSa/s
onboard arbitrary waveform generator (AWG) was used to generate the 34.46
GBaud dual-polarization quadrature phase shift keying (DP-QPSK) signals at
1550.12 nm. As shown in Fig. 1(a), the signals had a pilot-aided frame
structure with a frame length of 20,000 symbols. Each frame had a training
sequence of $1680$ symbols, and a pilot symbol for every 9 data symbols in the
payload. The training sequence and the pilots were generated by a random
permutation of balanced QPSK symbols. The data symbols were generated from a
PRBS-15 pseudo random binary sequence (PRBS). The generated signals were
shaped by a root-raised cosine (RRC) filter with a roll-off factor of 0.1. The
signals were then amplified by a single-mode erbium-doped fibre amplifier (SM-
EDFA). In order to emulate the use of independent transmitters and receivers,
at the transmitter, the resultant optical signal was passed through an
acousto-optical modulator (AOM), which was operated at a period of 160 µs and
a duty cycle of 12.5%, generating a 20 µs signal burst (Fig. 1(b), more than
34 frames). The signals were then split into 5 copies by a fibre coupler and
then each copy was delayed by a variable fibre delay line (FDL) with lengths
of 0, 280, 560, 840, and 1120 symbols, respectively to generate decorrelated
signals for different modes (Fig. 1(c)), and the decorrelation length between
adjacent modes is 1/6th of the training sequence length. The decorrelated
signals were then connected to 5 variable optical attenuators (VOAs) to
compensate for the mode-dependent loss at the transmitter side to guarantee
that each transmitted mode contributed approximately the same power at the
receiver side. The attenuated signals were finally connected to the five
lowest order linearly polarized (LP) modes of a 6-mode MSPL for mode
multiplexing and then coupled into free space using a collimator with a focal
length of 18.4 mm and a lens diameter of $D=8.4$ mm.

In the FSO channel, we employed a spatial light modulator (SLM) with
$1920\times 1152$ pixels to emulate indefinite number of independent
turbulence realizations. The size of each pixel is $9.2\times 9.2$
µ$\text{m}^{2}$. In order to modulate the both polarization components of the
signal, we passed the collimated light through the SLM twice (red paths in
Fig. 1). Simulated phase screen were displayed on the left and right half of
the SLM. The distance between the collimators and the SLM, the mirrors and the
SLM, and the two mirrors were ~50 cm, ~50 cm, and ~20 cm, respectively. During
the first pass, the horizontal polarization component was modulated by
turbulence phase screen. A half-wave plate (HWP) located in-between the first
and second pass swapped the horizontal and vertical components. And then the
unaffected component in the first pass was modulated by the same turbulence
phase screen during the second pass. The selection of phase screens for
turbulence generation will be detailed in Section III.

At the receiver side, the free-space beam was coupled into a FMF by a second
identical $D=8.4$ mm diameter FSO coupler with a focal length of 18.4 mm. The
received signals were then demultiplexed by a second 6-mode MSPL at the
receiver side. 6 polarization controllers were connected to the output of the
MSPL to balance the received power in different polarization states.
Afterwards, the 6 signals were delayed by the 25 km, 20 km, 15 km, 10 km, 5
km, and 0 km FDLs to enable the time-division multiplexing (TDM) receiver [21,
22]. As shown in Fig. 1(d), the FDLs generated an approximately 24.5 µs delay
between adjacent modes, which was slightly longer than the signal burst. 6
VOAs were then connected to the output of the FDLs to compensate for the mode-
dependent loss of the devices at the receiver side. The TDM signals were then
coupled into one single-mode fibre (Fig. 1(e)) and amplified by another SM-
EDFA. Afterwards, the variable optical noise was loaded by a combination of an
amplified spontaneous emission source, an optical bandpass filter, a SM-EDFA,
a VOA, and a 50:50 coupler. The average optical signal-to-noise ratio (OSNR)
was then measured by an optical spectrum analyzer to enable OSNR scanning. The
resultant signal was received by a standard coherent receiver with a 50 GSa/s
sampling rate and 23 GHz bandwidth oscilloscope. Finally, the received digital
traces were demodulated and decoded by the offline DSP (Fig. 1(f)).

##  III Turbulence Generation

In this work, we emulated the beam propagation in a thin phase turbulent
channel representing a channel approximately 400-500 m in length, using a
single aberration inducing phase screen implemented with a SLM to emulate each
instantiation of the turbulent channel. The screens used were generated by
implementing a power spectrum inversion method, in which phase screens are
randomly generated by performing a Fast Fourier Transform (FFT) of associated
spatial frequency power spectrum. The power spectrum that is commonly used for
atmospheric turbulence is the von Kármán spectrum [11], that has Fourier
series coefficients, $c_{n,m}$, in the form,

$\displaystyle{c_{n,m}}=$ $\displaystyle
w(n,m)\frac{{\sqrt{0.023}}}{{{L}}}{\left({\frac{2}{{{r_{0}}}}}\right)^{{}^{\frac{5}{6}}}}{\left({f_{{x_{n}}}^{2}+f_{{y_{m}}}^{2}+\frac{1}{{L_{0}^{2}}}}\right)^{-\frac{{11}}{{12}}}}$
(1)
$\displaystyle\times{\left\\{{\exp\left[{-\left({f_{{x_{n}}}^{2}+f_{{y_{m}}}^{2}}\right){{\left({\frac{{2\pi{l_{0}}}}{{5.92}}}\right)}^{2}}}\right]}\right\\}^{\frac{1}{2}}},$

where $w(n,m)$ is the random array that obey complex circular Gaussian
statistics with zero-mean and unit-variance. $L=8.832$ mm is the length of the
$960\times 960$ pattern. $r_{0}$ is the Fried’s parameter. The spatial
frequency grid is determined by parameters $f_{x_{n}}=n/L$ and
$f_{y_{m}}=m/L$. Further, this power spectrum is influenced by the physical
channel parameters that determine the largest, $L_{0}$ to smallest, $l_{0}$
eddies sizes present in a turbulent channel.

Turbulence-induced phase $\phi(x,y)$ can be generated by considering discrete
x- and y-directed spatial frequencies denoted as $f_{x_{n}}$ and $f_{y_{m}}$,
respectively. To accurately represent modes like tilt aberration, the sub-
harmonic can be employed[23, 24]. A $3\times 3$ grid of 2D arrays of spatial
frequencies is used, where the frequency grid spacing for each value of $p$ is
${1\mathord{\left/{\vphantom{1{\left({3^{p}L}\right)}}}\right.\kern-1.2pt}{\left({3^{p}L}\right)}}$,
and the corresponding Fourier series coefficient is $c_{n,m,p}$. A specific
phase screen, $\phi(x,y)$ can then be computed as a sum of $N_{p}+1$
independent screens using the formula

$\displaystyle\phi(x,y)=$
$\displaystyle\sum\limits_{n=-\infty}^{+\infty}{\sum\limits_{m=-\infty}^{+\infty}{{c_{n,m}}\exp\left[{j2\pi\left({{f_{{x_{n}}}}x+{f_{{y_{m}}}}y}\right)}\right]}}\hfill$
(2) $\displaystyle+$
$\displaystyle\sum\limits_{p=1}^{{N_{p}}}{\sum\limits_{n=-1}^{1}{\sum\limits_{m=-1}^{1}{{c_{n,m,p}}\exp\left[{j2\pi\left({{f_{{x_{n}}}}x+{f_{{y_{m}}}}y}\right)}\right]}}}\hfill.$

Each computed screen provides an individual instance of turbulence and can
vary in strength, where the full impact of a particular turbulence strength
can only be realized by considering an appropriate number of instances over
time. It is generally considered that turbulence is approximately frozen at a
time base close to the coherence time, commonly called the Greenwood frequency
[24], of the atmosphere and can vary between 60 Hz to 1 kHz depending on the
particular environment condition. Therefore, if a channel measurement is
performed suitably faster than this Greenwood frequency, which is the typical
case for the channel estimation algorithm in our system, then one single
instance of turbulence needs to be considered when the channel mitigation
strategies are applied. We considered two sets of 120 independent patterns,
both with $L_{0}=10\,\text{m}$ and $l_{0}=0.1\,\text{mm}$, for stronger
($r_{0}=0.8\,\text{mm}$, $D/r_{0}=10.5$) and weaker ($r_{0}=3.0\,\text{mm}$,
$D/r_{0}=2.8$) turbulence, respectively. Example patterns are shown in Fig.
2(a) and Fig. 2(c), whilst Fig. 2(b) and Fig. 2(d) show the normalized power
probability density function (PDF) at the receiver (before MSPL), which agree
well with normalized lognormal curve fits corresponding to scintillation
indexes ($\sigma_{I}^{2}$) of 0.079 and 0.0050, respectively [11].

Figure 2: The typical turbulence realizations ($L_{0}=10\,\text{m}$,
$l_{0}=0.1\,\text{mm}$). PDF: probability density function. (a) The typical
pattern of the stronger turbulence ($r_{0}=0.8\,\text{mm}$); (b) The power
distribution of the stronger turbulence ($\sigma_{I}^{2}=0.079$); (c) The
typical pattern of the weaker turbulence ($r_{0}=3.0\,\text{mm}$); (d) The
power distribution of the weaker turbulence ($\sigma_{I}^{2}=0.0050$).

##  IV MIMO DSP Algorithm

###  IV-A Preliminaries

As shown in Fig. 3,

Figure 3: Schematic diagram for TDM MDM optical communication systems.

a generalized mathematical model for the TDM MDM optical communication system
with $N_{t}$ transmit channels and $N_{r}$ receive channels can be given as

${{\mathbf{y}}_{r}}={\mathbf{\Phi H\Psi s}}+{{\mathbf{n}}_{r}},$ (3)

where ${\mathbf{y}}_{r}$ is the $N_{r}\times 1$ received signal vector,
$\mathbf{H}$ is the $N_{r}\times N_{t}$ channel matrix, ${\mathbf{s}}$ is the
$N_{t}\times 1$ transmitted signal vector, ${\mathbf{n}}_{r}$ is the
$N_{r}\times 1$ independent and identically distributed (i.i.d.) circularly-
symmetric complex additive white Gaussian noise (AWGN) vector, where the
expectation and the variance of each element are 0 and $N_{0}$, respectively.
${\mathbf{\Psi}}={\text{diag}}\left({{e^{j{\psi_{1}}}},\cdots,{e^{j{\psi_{N_{t}}}}}}\right)$
and
${{\mathbf{\Phi}}}={\text{diag}}\left({{e^{j{\varphi_{1}}}},\cdots,{e^{j{\varphi_{{N_{r}}}}}}}\right)$
are the diagonal phase matrices of the transmit and receive channels, where
$\psi_{l}$ and $\varphi_{k}$ are the phase noise at the $l^{th}$ transmit
channel and the $k^{th}$ receive channel, respectively.

If the diagonal elements of $\mathbf{\Phi}$ and $\mathbf{\Psi}$ are
approximately identical, corresponding to a single transmit laser and a local
oscillator, both with phase matched fiber paths, then MIMO equalization may be
carried out prior to phase estimation [17]. However, the approximately
identical phase should be conserved by carefully splitting and delaying the LO
beam or using multiple oscilloscopes at the receiver [18]. This extra
complexity can be reduced by employing carrier-asynchronous algorithms to
combat decorrelated LOs, but prior attempts have suffered from iterative phase
and channel estimation structure [19]. Moreover, inherent difficulties exist
in the above mentioned DSPs when employing advanced MIMO decoding algorithms
such as SIC.

To accommodate these practical realities, we propose a new carrier-
asynchronous DSP structure, which is shown in Fig. 1(f) (green text), to
separate the MIMO decoder from the phase estimation, the channel estimation,
and the ISI equalization. By doing so, redundant receive channels and advanced
MIMO decoding algorithms such as SIC can be supported.

The phase and channel estimation algorithm is fully detailed in one of our
previous work [25]. Noting the fact that the phases are well conserved at the
transmitter side, we can set $\mathbf{\Psi}$ to a $N_{t}\times N_{t}$ identity
matrix as reference. Therefore, (3) can be simplified to

${\mathbf{y}}_{r}={\mathbf{\Phi Hs}}+{\mathbf{n}}_{r}.$ (4)

After the phase and channel estimation, we can obtain $\mathbf{\hat{\Phi}}$
and ${\mathbf{\hat{H}}}$, which are the estimations of $\mathbf{\Phi}$ and
$\mathbf{H}$, respectively. And the phase information can be cancelled out as

${\mathbf{y}}={{\mathbf{\hat{\Phi}}}^{H}}{{\mathbf{y}}_{r}}\approx{\mathbf{Hs}}+{{\mathbf{\hat{\Phi}}}^{H}}{{\mathbf{n}}_{r}}={\mathbf{Hs}}+{\mathbf{n}},$
(5)

where ${\left(\cdot\right)^{H}}$ is the Hermitian adjoint operator and
${\mathbf{n}}={{\mathbf{\hat{\Phi}}}^{H}}{\mathbf{n}}_{r}$. Moreover,
${\mathbf{n}}$ has the same statistical property as $\mathbf{n}_{r}$.

Afterwards, a MIMO equalizer should be introduced to mitigate the negative
influence from ISI. We only update the coefficients at the known pilot symbols
$\mathbf{s}_{p}$. Moreover, we modified the reference information symbols of
our MIMO equalizer to

${\mathbf{I}}_{eq}={\mathbf{\hat{H}s}_{p}},$ (6)

which is different from the reference information symbols
${\mathbf{I}}_{eq}={\mathbf{s}_{p}}$ in the conventional MIMO equalizers. By
doing so, the ISI can be mitigated without decoding the symbols, and an
arbitrary MIMO decoding algorithm, such as SIC, can be applied after the MIMO
equalization.

###  IV-B Successive Interference Cancellation

To focus on the $k^{th}$ transmit channel, we can rewrite (5) as

${\mathbf{y}}={\mathbf{Hs}}+{\mathbf{n}}=\sum\limits_{i=1}^{{N_{t}}}{{{\mathbf{h}}_{i}}{s_{i}}}+{\mathbf{n}}={{\mathbf{h}}_{k}}{s_{k}}+\sum\limits_{\begin{subarray}{l}i=1\\\
i\neq k\end{subarray}}^{{N_{t}}}{{{\mathbf{h}}_{i}}{s_{i}}}+{\mathbf{n}},$ (7)

where ${\mathbf{h}}_{i}$ is the $i^{th}$ column vector of the channel matrix
$\mathbf{H}$, and $s_{i}$ is the $i^{th}$ element of the transmitted signal
vector $\mathbf{s}$. For the $k^{th}$ channel, the second term in the last
equation is referred to as the inter-channel interference (ICI). We may notice
that the ICI can be reduced if we can properly cancel out some of the
interfering channels.

The schematic diagram for the SIC algorithm is shown in Fig. 4.

Figure 4: Schematic diagram for the SIC algorithm.

The strategy of the SIC algorithm is to successively cancel out the
interference of channels which have already been decoded [26]. For the
$k^{th}$ channel, the subtraction process can be written as

${{\mathbf{\tilde{y}}}_{k}}={\mathbf{y}}-\sum\limits_{i=1}^{k-1}{{{{\mathbf{\hat{h}}}}_{i}}{{\tilde{s}}_{i}}},$
(8)

where ${{{\mathbf{\hat{h}}}}_{i}}$ is the $i^{th}$ column vector of the
estimated channel matrix $\mathbf{\hat{H}}$, and ${{\tilde{s}}_{i}}$ is the
decoded symbol in the $i^{th}$ channel. If the channel matrix is accurately
estimated and the first $k-1$ symbols are properly decoded, (8) can be
rewritten as

${{\mathbf{\tilde{y}}}_{k}}\approx{{\mathbf{h}}_{k}}{s_{k}}+\sum\limits_{i=k+1}^{{N_{t}}}{{{\mathbf{h}}_{i}}{s_{i}}}+{\mathbf{n}},$
(9)

where the approximation indicates that the nonlinear error propagation may
occur in this process if estimation or decoding error occurs. However, the
impact of this phenomenon can be minimized by using an optimal decoding order
for each frame of the signal [27]. Moreover, it will be shown in Section V
that we can still achieve a significantly better decoding performance by using
the SIC algorithm.

After the subtraction process, the first $k-1$ channels are cancelled out.
Therefore, the $k^{th}$ MMSE receiver should be modified to

${\hat{s}_{k}}={\left[{{{\left({{{{\mathbf{\hat{H}}}}_{k}}{\mathbf{\hat{H}}}_{k}^{H}+{N_{0}}{{\mathbf{I}}_{{N_{r}}\times{N_{r}}}}}\right)}^{-1}}{{{\mathbf{\hat{h}}}}_{k}}}\right]^{H}}{{\mathbf{y}}_{k}},$
(10)

where
${{\mathbf{\hat{H}}}_{k}}=\left[{{{{\mathbf{\hat{h}}}}_{k}},{{{\mathbf{\hat{h}}}}_{k+1}},\cdots,{{{\mathbf{\hat{h}}}}_{{N_{t}}}}}\right]$,
and $\mathbf{I}_{{N_{r}}\times{N_{r}}}$ represents the ${N_{r}}\times{N_{r}}$
identity matrix.

Finally, decoding in the $k^{th}$ channel can be calculated using the
conventional decision rule as [20]

${\tilde{s}_{k}}=\mathop{\arg\min}\limits_{{{\tilde{s}}_{k}}\in\mathcal{S}}\left({{{\left|{{{\tilde{s}}_{k}}-{{\hat{s}}_{k}}}\right|}}}\right),$
(11)

where $\mathcal{S}$ represents the set of all possible transmitted symbols,
and $\left|\cdot\right|$ is the modulus operator for complex numbers.

###  IV-C Vector Space Explanation

In order to better understand why the proposed MIMO DSP algorithm can obtain
enhanced atmospheric turbulence resiliency, Fig. 5 gives an vector space
explanation for a system with 3 receive channels. In order to simplify the
analysis, we only consider the special situation of negligible AWGN, where the
MMSE algorithm degenerates to multiplying by a Moore–Penrose inverse matrix of
$\mathbf{H}$, which can also be regarded as a projection and a scaling of the
column vectors in $\mathbf{H}$. The AWGN in a practical system should be
considered as another dimension in the random vector space and analogous
analysis can then apply to the generalized situation with AWGN.

Figure 5: The vector space explanation for the enhanced atmospheric turbulence
resiliency. ${{\mathbf{h}}_{k}}$ is the $k^{th}$ column vector in the channel
matrix ${\mathbf{H}}$, and ${{\mathbf{\tilde{h}}}_{k}}$ is the effective
projection of ${{\mathbf{h}}_{k}}$. The vectors cancelled out by the SIC are
represented by dotted vectors. (a) the $3\times 3$ MMSE MIMO decoder; (b) the
$3\times 3$ SIC MIMO decoder; (c) the $2\times 3$ MMSE MIMO decoder with
redundant receive channels.

Fig. 5(a) illustrates a $3\times 3$ linear MMSE MIMO decoder in vector space.
Considering the fact that the scaling effect is just for normalization, we are
only interested in the projection effect. For arbitrary ${{\mathbf{h}}_{k}}$,
the effective projection ${{\mathbf{\tilde{h}}}_{k}}$ should be in the 1
dimensional orthogonal complement of the subspace spanned by
$\left\\{{{{\mathbf{h}}_{i}}\left|{1\leqslant i\leqslant{N_{t}},i\neq
k}\right.}\right\\}$ to eliminate the negative influence of ICI.

Fig. 5(b) illustrates a $3\times 3$ SIC MIMO decoder. In the first channel, no
interference cancellation is performed and the same performance as the MMSE
decoder should be expected. However, when decoding the second channel, the
first channel can be cancelled out, and ${{\mathbf{\tilde{h}}}_{2}}$ is a
projection in the orthogonal complement of ${{\mathbf{h}}_{3}}$, which is a 2
dimensional vector subspace. Therefore, the degree of freedom is increased to
2 for the second channel. Similarly, the last channel needs no projection and
all the 3 degrees of freedom can be exploited. As a result, the effective
projections in the SIC system will be longer than or equal to the effective
projections in the MMSE system, and a larger effective signal-to-interference-
plus-noise ratio (SINR) can be achieved in the SIC system.

Fig. 5(c) illustrates a $2\times 3$ MMSE MIMO decoder which has redundant
receive channels. The third channel is directly cancelled out because we are
not transmitting any symbols in that channel. As a result, arbitrary
${{\mathbf{h}}_{k}}$ can always be projected onto a 2 dimensional orthogonal
complement subspace and a better turbulence resiliency can be obtained.

It is also worth noting that in an ideal system with mutually orthogonal
column vectors in the channel matrix, the equation
${{\mathbf{\tilde{h}}}_{k}}={{\mathbf{h}}_{k}}$ holds. Therefore, the system
performance can not be improved by either applying the SIC algorithm or adding
redundant receive channels. Unfortunately, although the orthogonality can be
well conserved in certain kinds of channels (e.g. short FMF link or back-to-
back FSO link without turbulence), this condition doesn’t hold in a turbulent
channel.

##  V Experimental Results

Table V lists the experimental parameters for our MDM FSO communication
system.

TABLE I: Experimental Parameters

Parameter | Value
---|---
Wavelength | 1550.12 nm
Laser linewidth | 100 kHz
Symbol format | DP-QPSK
Baud rate | 34.46 GBaud
Roll-off factor | 0.1
Receiver sampling rate | 50 GSa/s
Hard-decision forward error correction |
(HD-FEC) limit with 6.25% overhead [28] |  $4.7\times 10^{-3}$
Diameter of the receiver lens (D) | 8.4 mm
Fried’s parameter ($r_{0}$) for weaker turbulence | 3.0 mm
Fried’s parameter ($r_{0}$) for stronger turbulence | 0.8 mm
Inner scale of turbulence ($l_{0}$) | 0.1 mm
Outer scale of turbulence ($L_{0}$) | 10 m

First, we examine the BER performance against average OSNR of the $10\times
12$ MIMO system under different turbulence conditions, which is shown in Fig.
6.

Figure 6: Average BER curves of the $10\times 12$ MIMO system under different
turbulence intensities. Reference: Theoretical reference curve in an ideal
system; Turb free: turbulence free system with a blank pattern. Weaker:
typical weaker turbulence pattern when $r_{0}=3.0~{}\text{mm}$; Stronger:
typical stronger turbulence pattern when $r_{0}=0.8~{}\text{mm}$.

The theoretical reference curve assumes that: (1) The channel matrix is a
$N_{r}\times N_{t}$ submatrix of a $N_{r}\times N_{r}$ unitary matrix; (2) No
other imperfection except the AWGN exists. For the turbulence free situation,
a blank pattern with factory pre-calibration is used in the system, here
observe a ~2.2 dB implementation penalty for the SIC system at the hard-
decision forward error correction (HD-FEC) limit, which we believe is
dominated by the inter-mode crosstalk and residual mode-dependent losses in
the MSPL pair. As expected, these imperfections have a slightly higher impact
on the MMSE system (~2.7 dB implementation penalty). In the weaker turbulence
case, the SIC system has a ~3 dB penalty but the penalty for the MMSE system
is ~4.2 dB, which indicates a moderate reduction in channel matrix
orthogonality. For the stronger turbulence, a larger penalty of ~6.9 dB is
observed in the SIC system, but the MMSE system does not even achieve the HD-
FEC limit, and the loss of channel matrix orthogonality is severe. The result
in the stronger turbulence case also indicates that the proposed DSP algorithm
has only mitigated rather than eliminated the negative influence of
atmospheric turbulence.

In order to better understand the enhanced turbulence resiliency by the
proposed SIC system, Fig. 7 depicts the BER of the best channel (the channel
with minimum BER at each average OSNR), the worst channel (the channel with
maximum BER at each average OSNR), and the average BER of all channels (also
shown in Fig. 6) in the stronger turbulence.

Figure 7: BER curves of the $10\times 12$ SIC (red lines) and MMSE (blue
dotted lines) MIMO systems under the stronger turbulence when
$r_{0}=0.8\,\text{mm}$. Avg (circles): average BER; Min (lower triangles): the
best channel with minimum BER at each average OSNR; Max (upper triangles): the
worst channel with maximum BER at each average OSNR.

The best channels in both the SIC system and the MMSE system have similar BER
performance, this is because we don’t execute any interference cancellation
for the first decoded channel in the SIC system. At the high OSNR region, the
best channel in the SIC system performs slightly better than the MMSE system.
This is because the second decoded channel may obtain a better BER performance
by cancelling out the first decoded channel. On the other hand, the worst
channel in the SIC system has a much better performance than it in the MMSE
system. This is because a higher effective SINR can be achieved in the SIC
system by cancelling out the previously decoded channels. Considering the fact
that the worst channel is the dominant factor in the average BER performance,
the SIC system can obtain a significantly lower average BER in turbulent
channels.

The constellation diagrams for the $10\times 10$ and $10\times 12$ MIMO
systems under the stronger turbulence case are depicted in Fig. 8 without
noise loading.

Figure 8: Constellation diagrams of different transmitted channels for the
$10\times 10$ MMSE (the first row), $10\times 12$ MMSE (the second row),
$10\times 10$ SIC (the third row), and $10\times 12$ SIC (the last row) MIMO
configurations without noise loading. The SIC order is given at the bottom
left of each constellation. EVM: error vector magnitude. Red crosses:
reference QPSK constellation points.

The SIC order is given at the bottom left of each constellation. Because the
channel matrices are different between the $10\times 10$ and $10\times 12$
MIMO systems, the SIC order may also be different. As shown in Fig. 8, the
$10\times 10$ and $10\times 12$ SIC systems have a significantly lower error
vector magnitude (EVM) than the $10\times 10$ and $10\times 12$ MMSE systems,
respectively. This is because certain ICI is cancelled out by applying the SIC
algorithm. However, no ICI is cancelled out in the first decoded channel
(${\text{LP}_{\text{21bY}}}$ in both cases). In this channel, identical EVM
performance is observed in both SIC and MMSE decoders. On the other hand, a
comparison between the $10\times 10$ and $10\times 12$ MMSE MIMO systems shows
that 1 more redundant receive mode (2 more redundant receive channels) can
significantly reduce the EVM of the constellation in all the 10 channels.
Compared with the $10\times 10$ SIC MIMO system, We can also obtain a better
average EVM performance in the $10\times 12$ SIC MIMO system because of the
information obtained from the 2 redundant receive channels (EVM = 21.23%  for
the $10\times 12$ SIC MIMO system and EVM = 25.78%  for the $10\times 10$ SIC
MIMO system, respectively), although a higher EVM in specific channels (e.g.
${\text{LP}_{\text{21aY}}}$) may be observed due to the different SIC order.
Moreover, less nonlinear propagation (apparently discrete points in the
diagram, see for example ${\text{LP}_{\text{01X}}}$) is also observed in the
$10\times 12$ SIC MIMO system. This is also because of the information
obtained from the 2 redundant receive channels and less symbol error occurs in
the decoded channels.

We tested 120 independent turbulence patterns under the stronger turbulence
case without noise loading. As shown in Fig. 9, the BER of the SIC system is
consistently lower than the MMSE system.

Figure 9: The BER performance of the $10\times 12$ MIMO system under 120
stronger turbulence realizations.

As a result, the average BER among different turbulence realizations is
decreased by an order of magnitude, from $8.02\times 10^{-3}$ in the MMSE
system to $4.76\times 10^{-4}$ in the SIC system (well below the HD-FEC
limit). The probability distribution of the BER in Fig. 9 is depicted in Fig.
10 for better clarity.

Figure 10: The probability distribution of the BER performance in the
$10\times 12$ system under 120 stronger turbulence realizations.

Because of the limited length of our recorded data, the BER should be taken as
being below $10^{-6}$ for the 18 turbulence realizations where no errors we
recorded. If we consider an outage when the BER is larger than the HD-FEC
limit, we have an outage probability of 48.3% in the MMSE system and only 2.5%
in the SIC system, suggesting a more robust transmission using SIC.

To further illustrate the enhanced turbulence resiliency obtained from the
redundant receive channels, we increased the ratio of received to transmitted
channels by disabling the two highest order mode inputs of the transmitter.
Fig. 11 depicts the BER performance against average OSNR of a $6\times 12$
MIMO system under the same turbulence conditions as Fig. 6.

Figure 11: Average BER curves of the $6\times 12$ MIMO system under different
turbulence intensities. Reference: Theoretical reference curve in an ideal
system; Turb free: turbulence free system with a blank pattern. Weaker:
typical weaker turbulence pattern when $r_{0}=3.0\,\text{mm}$; Stronger:
typical stronger turbulence pattern when $r_{0}=0.8\,\text{mm}$.

For the turbulence free situation, a blank SLM pattern with factory pre-
calibration is used in the system. In this turbulence free scenario, we
observe a ~1.4 dB implementation penalty between the reference and the SIC
system at the HD-FEC limit, while the MMSE system has a slightly higher
implementation penalty of ~1.5 dB. In the weaker turbulence case, the
performance of the SIC system has a ~1.7 dB implementation penalty, compared
to ~2.0 dB for the MMSE system. In the stronger turbulence case, a larger
implementation penalty of ~4.0 dB is observed in the SIC system, while the
implementation penalty of the MMSE system is ~8.2 dB. When compared with the
corresponding results in Fig. 6, we can conclude that a significantly better
turbulence and device imperfection resiliency can be obtained under different
turbulence conditions by increasing the number of redundant receive channels
in the MDM FSO system.

Fig. 12 depicts the BER performance of a $6\times 12$ system using 120
independent patterns, which are the same as the patterns used in Fig. 9, under
the stronger turbulence case without noise loading.

Figure 12: The BER performance of the $6\times 12$ MIMO system under 120
stronger turbulence realizations.

Similar to Fig. 9, the BER of the SIC system is consistently lower than the
MMSE system. As a result, the average BER among different turbulence
realizations is decreased from $1.56\times 10^{-4}$ to $2.86\times 10^{-6}$.
If we consider an outage when the BER is larger than the HD-FEC limit, the
outage probability of the MMSE and the SIC system will be decreased to 1.67%
and 0%, respectively. When compared with the corresponding results in Fig. 9,
we can conclude that a significantly better turbulence resiliency can be
obtained in different turbulence realizations by increasing the number of
redundant receive channels.

In order to comprehensively compare different transmit and receive channel
numbers as well as different MIMO decoding algorithms, Fig. 13 depicts the
average BER performance of different MIMO systems using 120 independent
patterns under the strong turbulence case without noise loading.

Figure 13: The average BER performance of different MIMO systems under 120
stronger turbulence realizations.

As shown in Fig. 13, when the number of transmit channels ($N_{t}$) is fixed,
the average BER will decrease as the number of receive channels ($N_{r}$)
increases. On the other hand, when $N_{r}$ is fixed, the average BER will
increase as $N_{t}$ increases. Moreover, in the same system setup when $N_{t}$
and $N_{r}$ are both fixed, the SIC algorithm always performs better than the
MMSE algorithm.

##  VI Conclusion

In this paper, we experimentally demonstrate the enhanced turbulence
resiliency in a MDM FSO communication link with 5 transmitted spatial modes,
each carrying a 34.46 GBaud DP-QPSK signal. By employing the SIC MIMO
algorithm and redundant receive channels, the average BER of the $10\times 12$
MIMO system goes down to $4.76\times 10^{-4}$, which is well below the HD-FEC
limit, among 120 independent turbulence realizations where $D=8.4$ mm and
$r_{0}=0.8\,\text{mm}$ ($D/r_{0}=10.5$). Moreover, the outage probability is
significantly decreased when using the SIC algorithm, which indicates the
possibility of a much more robust transmission in the MDM FSO system through
turbulent channels.

We also compare the system performance among systems with different transmit
and receive channel numbers. The results indicate that we can obtain an even
better BER performance by increasing the number of redundant receive channels.
These results may also provide useful guidelines when deploying the MDM FSO
system in an even stronger turbulence environment.

As a result, the system demonstrates a record-high independent channel number
of 10 and line rate of 689.23 Gbit/s. Considering the 8.4% training sequence,
10% pilot rate, 6.25% HD-FEC cost, and 0.1 rolloff factor, a record-high net
spectral efficiency of 13.9 bit/s/Hz was also achieved in emulated turbulent
MDM FSO links.

## Acknowledgment

We thank Dr. Chao Gao for fruitful discussions. We also thank Ciena and Dr.
Charles Laperle for kindly providing the WaveLogic 3 transponder used in our
experiments.

## References

  * [1] Y. Ren, Z. Wang, P. Liao, L. Li, G. Xie, H. Huang, Z. Zhao, Y. Yan, N. Ahmed, A. Willner _et al._ , “Experimental characterization of a 400 Gbit/s orbital angular momentum multiplexed free-space optical link over 120 m,” _Optics letters_ , vol. 41, no. 3, pp. 622–625, 2016.
  * [2] R. Lange, B. Smutny, B. Wandernoth, R. Czichy, and D. Giggenbach, “142 km, 5.625 Gbps free-space optical link based on homodyne BPSK modulation,” in _Free-space laser communication technologies XVIII_ , vol. 6105. International Society for Optics and Photonics, 2006, p. 61050A.
  * [3] M. A. Khalighi and M. Uysal, “Survey on free space optical communication: A communication theory perspective,” _IEEE communications surveys & tutorials_, vol. 16, no. 4, pp. 2231–2258, 2014.
  * [4] J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur _et al._ , “Terabit free-space data transmission employing orbital angular momentum multiplexing,” _Nature photonics_ , vol. 6, no. 7, pp. 488–496, 2012.
  * [5] N. K. Fontaine, R. Ryf, Y. Zhang, J. C. Alvarado-Zacarias, S. van der Heide, M. Mazur, H. Huang, H. Chen, R. Amezcua-Correa, G. Li, M. Capuzzo, R. Kopf, A. Tate, H. Safar, C. Bolle, D. T. Neilson, E. Burrows, K. Kim, M. Bigot-Astruc, F. Achten, P. Sillard, A. Amezcua-Correa, and J. Carpenter, “Digital turbulence compensation of free space optical link with multimode optical amplifier,” in _45th European Conference on Optical Communication (ECOC 2019)_ , 2019, pp. 1–4.
  * [6] A. E. Willner, Z. Zhao, C. Liu, R. Zhang, H. Song, K. Pang, K. Manukyan, H. Song, X. Su, G. Xie _et al._ , “Perspectives on advances in high-capacity, free-space communications using multiplexing of orbital-angular-momentum beams,” _Apl Photonics_ , vol. 6, no. 3, p. 030901, 2021.
  * [7] Z. Qu and I. B. Djordjevic, “500 Gb/s free-space optical transmission over strong atmospheric turbulence channels,” _Optics Letters_ , vol. 41, no. 14, pp. 3285–3288, 2016.
  * [8] Y. Ren, G. Xie, H. Huang, N. Ahmed, Y. Yan, L. Li, C. Bao, M. P. Lavery, M. Tur, M. A. Neifeld _et al._ , “Adaptive-optics-based simultaneous pre-and post-turbulence compensation of multiple orbital-angular-momentum beams in a bidirectional free-space optical link,” _Optica_ , vol. 1, no. 6, pp. 376–382, 2014.
  * [9] A. Trichili, K.-H. Park, M. Zghal, B. S. Ooi, and M.-S. Alouini, “Communicating using spatial mode multiplexing: Potentials, challenges, and perspectives,” _IEEE Communications Surveys & Tutorials_, vol. 21, no. 4, pp. 3175–3203, 2019.
  * [10] R. Brüning, Y. Zhang, M. McLaren, M. Duparré, and A. Forbes, “Overlap relation between free-space Laguerre Gaussian modes and step-index fiber modes,” _JOSA A_ , vol. 32, no. 9, pp. 1678–1682, 2015.
  * [11] L. C. Andrews and R. L. Phillips, _Laser Beam Propagation through Random Media, Second Edition_. SPIE Press, 2005\.
  * [12] L. Li, R. Zhang, P. Liao, Y. Cao, H. Song, Y. Zhao, J. Du, Z. Zhao, C. Liu, K. Pang _et al._ , “Mitigation for turbulence effects in a 40-Gbit/s orbital-angular-momentum-multiplexed free-space optical link between a ground station and a retro-reflecting UAV using MIMO equalization,” _Optics letters_ , vol. 44, no. 21, pp. 5181–5184, 2019.
  * [13] S. Chen, S. Li, Y. Zhao, J. Liu, L. Zhu, A. Wang, J. Du, L. Shen, and J. Wang, “Demonstration of 20-Gbit/s high-speed Bessel beam encoding/decoding link with adaptive turbulence compensation,” _Optics letters_ , vol. 41, no. 20, pp. 4680–4683, 2016.
  * [14] T. Weyrauch, M. A. Vorontsov, T. G. Bifano, J. A. Hammer, M. Cohen, and G. Cauwenberghs, “Microscale adaptive optics: wave-front control with a $\mu$-mirror array and a VLSI stochastic gradient descent controller,” _Applied Optics_ , vol. 40, no. 24, pp. 4243–4253, 2001.
  * [15] H. Huang, Y. Cao, G. Xie, Y. Ren, Y. Yan, C. Bao, N. Ahmed, M. A. Neifeld, S. J. Dolinar, and A. E. Willner, “Crosstalk mitigation in a free-space orbital angular momentum multiplexed communication link using 4$\times$ 4 MIMO equalization,” _Optics letters_ , vol. 39, no. 15, pp. 4360–4363, 2014\.
  * [16] Y. Ren, Z. Wang, G. Xie, L. Li, A. J. Willner, Y. Cao, Z. Zhao, Y. Yan, N. Ahmed, N. Ashrafi _et al._ , “Atmospheric turbulence mitigation in an OAM-based MIMO free-space optical link using spatial diversity combined with MIMO equalization,” _Optics letters_ , vol. 41, no. 11, pp. 2406–2409, 2016.
  * [17] R. G. Van Uden, C. M. Okonkwo, H. Chen, H. de Waardt, and A. M. Koonen, “Time domain multiplexed spatial division multiplexing receiver,” _Optics express_ , vol. 22, no. 10, pp. 12 668–12 677, 2014.
  * [18] G. Rademacher, B. J. Puttnam, R. S. Luís, T. A. Eriksson, N. K. Fontaine, M. Mazur, H. Chen, R. Ryf, D. T. Neilson, P. Sillard _et al._ , “Peta-bit-per-second optical communications system using a standard cladding diameter 15-mode fiber,” _Nature Communications_ , vol. 12, no. 1, pp. 1–7, 2021.
  * [19] K. Shibahara, T. Mizuno, and Y. Miyamoto, “MIMO carrier phase recovery for carrier-asynchronous SDM-MIMO reception based on the extended Kalman filter,” _Optics Express_ , vol. 29, no. 11, pp. 17 111–17 124, 2021\.
  * [20] J. G. Proakis and M. Salehi, _Digital communications_. McGraw-hill New York, 2001, vol. 4.
  * [21] Y. Li, Z. Hu, D. M. Benton, A. Ali, M. Patel, and A. D. Ellis, “Demonstration of 10-channel mode-and polarization-division multiplexed free-space optical transmission with successive interference cancellation DSP,” _Optics Letters_ , vol. 47, no. 11, pp. 2742–2745, 2022.
  * [22] Z. Hu, Y. Li, D. Benton, A. Ali, M. Patel, and A. Ellis, “Single-wavelength transmission at 1.1-Tbit/s net data rate over a multi-modal free-space optical link using commercial devices,” _Optics Letters_ , vol. 47, no. 13, 2022.
  * [23] R. G. Lane, A. Glindemann, and J. C. Dainty, “Simulation of a Kolmogorov phase screen,” _Waves in Random Media_ , 1992.
  * [24] M. Chen and M. Lavery, “Optical angular momentum interaction with turbulent and scattering media,” in _Structured Light for Optical Communication_. Elsevier, 2021, pp. 237–258.
  * [25] Y. Li, Z. Hu, and A. Ellis, “CRLB Approaching Pilot-aided Phase and Channel Estimation Algorithm in MIMO Systems with Phase Noise and Quasi-Static Channel Fading,” _arXiv preprint arXiv:2112.02583_ , 2021.
  * [26] D. Tse and P. Viswanath, _Fundamentals of wireless communication_. Cambridge university press, 2005.
  * [27] J. Benesty, Y. Huang, and J. Chen, “A fast recursive algorithm for optimum sequential signal detection in a BLAST system,” _IEEE Transactions on Signal Processing_ , vol. 51, no. 7, pp. 1722–1730, 2003.
  * [28] L. M. Zhang and F. R. Kschischang, “Staircase codes with 6% to 33% overhead,” _Journal of Lightwave Technology_ , vol. 32, no. 10, pp. 1999–2002, 2014.


# Modulating internal transition kinetics in responsive macromolecules by
collective crowding

Upayan Baul  Applied Theoretical Physics - Computational Physics,
Physikalisches Institut, Albert-Ludwigs-Universität Freiburg, D-79104
Freiburg, Germany  Nils Göth  Applied Theoretical Physics - Computational
Physics, Physikalisches Institut, Albert-Ludwigs-Universität Freiburg, D-79104
Freiburg, Germany  Michael Bley  Applied Theoretical Physics - Computational
Physics, Physikalisches Institut, Albert-Ludwigs-Universität Freiburg, D-79104
Freiburg, Germany  Joachim Dzubiella  Applied Theoretical Physics -
Computational Physics, Physikalisches Institut, Albert-Ludwigs-Universität
Freiburg, D-79104 Freiburg, Germany  Cluster of Excellence livMatS @ FIT -
Freiburg Center for Interactive Materials and Bioinspired Technologies,
Albert-Ludwigs-Universität Freiburg, D-79110 Freiburg, Germany

###### Abstract

Packing and crowding are used in biology as mechanisms to (self-)regulate
internal molecular or cellular processes based on collective signalling. Here,
we study how the transition kinetics of an internal ’switch’ of responsive
macromolecules is modified collectively by their spatial packing. We employ
Brownian dynamics simulations of a model of ’responsive colloids’ (RCs), in
which an explicit internal degree of freedom – here, the particle size –
moving in a bimodal energy landscape responds self-consistently to the density
fluctuations of the crowded environment. We demonstrate that populations and
transition times for the two-state switching kinetics can be tuned over one
order of magnitude by ’self-crowding’. An exponential scaling law derived from
a combination of Kramers’ and liquid state perturbation theory is in very good
agreement with the simulations.

Over the last two decades the phenomenon of macromolecular crowding has
attracted great attention because of its large effects on biomolecular
mobility, reactions, structure and dynamics [1, 2, 3, 4, 5]. In particular,
crowding leads to variations of (two-state) folding energy landscapes and
kinetics, typically leading to more compact molecular states and speeding up
folding times [6, 7, 8, 9]. In fact, it was proposed that cells can regulate
and maintain a relatively constant level of packing of macromolecules to tune
molecular processes, which was termed ’homeocrowding’ as an acronym of
macromolecular crowding homeostasis [10].

Similar mechanisms play decisive roles also on larger scales, for example, in
bacterial ’quorum sensing’, which is the self-regulation of internal gene
expression rates in response to fluctuations in overall cell-population
density [11]. These fascinating collective effects are inspiring for the
design of ’intelligent’ synthetic materials, where soft and responsive
particles, e.g., polymer-based hydrogels, potentially mimic these effects for
discovering new functionalities, such as advanced time responses, feedback and
adaptivity [12, 13]. In particular, the colloid/polymer duality of soft
microgels [14, 15] and their stimuli-triggered conformational switching [16]
open new avenues for creating programmable, self-regulating materials.

Despite the progress in this field, our understanding is still based on
relatively simple soft matter concepts such as excluded volume, depletion
interactions, or scaled particle theory [17, 10]. In other words, in
conventional theoretical studies of macromolecular crowding the crowders are
hard and inert, while in reality they are soft and responsive. Like in
bacteria, control by crowding is collective and self-consistent, affecting the
crowder and the crowded simultaneously [11]. Physical models that self-
consistently address crowding effects and their consequence on internal
processes, such as internal transitions or reactions, are still absent. A
promising approach in this regard is the explicit consideration of internal
degrees of freedom (DOFs) which can respond to environmental changes [18, 19,
20, 21, 22, 23], including local density fluctuations [24].

In this letter, we study collective crowding effects for a single
macromolecular species (’self-crowding’) employing a model of Responsive
Colloids (RCs) [19, 24]. The latter explicitly resolves an internal ’property’
DOF that responds self-consistently to changes in the local packing. As a case
study, we focus on the two-state switching kinetics of the size of the
macromolecule, modeling, for example, conformational transitions in globular
protein folding [6, 7, 8], switching of intrinsically disordered proteins
(IDPs) [25] or synthetic supramolecular polymers [26], or bimodal hydrogel
volume transitions [12, 27, 28, 29]. We find that populations and the
transition times between the two sizes (small and large) modified by packing
density spread over one order of magnitude. Importantly, we develop a scaling
theory based on a combination of Kramers’ escape [30] and liquid state
perturbation theory [31] extended to RCs. The theory is in very good agreement
with the calculated kinetics and provides useful, experimentally testable
scaling laws for a general class of Hamiltonians.

We consider $N$ interacting particles in a volume $V$ with number density
$\rho=N/V$ in absence of any external fields. The coarse-grained RC
Hamiltonian is defined as [19]

$H(\bm{\mathbf{r}}^{N},\sigma^{N})=\sum_{i}^{N}\psi(\sigma_{i})+\frac{1}{2}\sum_{i\neq
j}^{N}\phi(\mathbf{r}_{i},\mathbf{r}_{j};\sigma_{i},\sigma_{j}),$ (1)

and describes the energy of the microstate of the configuration
{$\mathbf{r}_{i},\sigma_{i}$}$=:\\{\mathbf{x}_{i}\\}$, where the
$\mathbf{r}_{i}$ and $\sigma_{i}$ are positions and sizes of particle
$i=1..N$, respectively. The first term on the right-hand-side carries the one-
body contribution to the energy associated with changes in the size $\sigma$.
Thus, it represents the energy landscape $\beta\psi(\sigma)=-\ln\,p(\sigma)$,
where $p(\sigma)$ is the corresponding probability distribution, and
$\beta=1/k_{B}T$ the inverse thermal energy. We refer to $p(\sigma)$ as the
parent distribution since it describes the intrinsic internal polydispersity
of an isolated particle. To model two-state kinetics, we choose a bimodal form
of the explicit, double-Gaussian form

$\displaystyle
p(\sigma)=\frac{1}{\sqrt{2\pi}}\biggl{[}\frac{c}{\delta_{1}}e^{\left(-\frac{1}{2}\left(\frac{\sigma-\mu_{1}}{\delta_{1}}\right)^{2}\right)}+\frac{1-c}{\delta_{2}}e^{\left(-\frac{1}{2}\left(\frac{\sigma-\mu_{2}}{\delta_{2}}\right)^{2}\right)}\biggr{]}.$
(2)

This and its corresponding energy landscape $\psi(\sigma)$ are shown in Fig.
1(a), highlighting the energy minima $\mu_{1}$, $\mu_{2}$, and the local
maximum, defining the transition state (TS). In going from continuous
distributions to the two-state (small (S) versus large (L)) picture, we
consistently use the TS as the state boundary. In our study, $c=0.5$,
$\mu_{1}=0.63\sigma_{0}$, $\mu_{2}=1.0\sigma_{0}$, and
$\delta_{1}=\delta_{2}\equiv\delta=0.1\sigma_{0}$, with unit length
$\sigma_{0}$ set to 1.0. The choice leads to an unperturbed activation energy
barrier $E_{0}^{a}\simeq 1k_{B}T$ (Fig. 1(a)).

The second term in Eq. (1) represents the two-body, effective pair potential,
$\phi(\mathbf{r}_{i},\mathbf{r}_{j};\sigma_{i},\sigma_{j})$, which is
explicitly dependent on both the positions and the property of the interacting
RCs. Here, we choose the Hertzian potential

$\phi(r;\sigma_{i},\sigma_{j})=\epsilon\left(1-\frac{r}{\sigma_{ij}}\right)^{5/2}\Theta\left(1-\frac{r}{\sigma_{ij}}\right),$
(3)

where $\Theta$ is the Heaviside step function, $\epsilon$ dictates the
magnitude of the interaction, $r=|\mathbf{r}_{j}-\mathbf{r}_{i}|$ is the
interparticle distance, and $\sigma_{ij}=(\sigma_{i}+\sigma_{j})/2$. The
Hertzian potential reflects the soft, purely repulsive interactions between
spherical elastic colloids with diameter $\sigma_{i}$, widely used to describe
swollen microgels [32, 33, 34, 35]. For our RC system we choose
$\beta\epsilon=500$, which is in the typical range for modeling real
experimental systems [34].

Figure 1: (a) Parent distribution $p(\sigma)$ of size $\sigma$ (solid) and
corresponding energy landscape $\psi(\sigma)$ (dashed) at dilute (single-
particle) conditions. Locations of energy minima ($\mu_{1},\mu_{2}$) and the
corresponding transition state (TS) are shown, separating small (S, blue) and
large (L, red) states. (b) Example emergent distribution $N(\sigma;\rho)$
(solid) and corresponding energy landscape $\tilde{\psi}(\sigma;\rho)$
(dashed). The crowding changes the activation energy from $E_{0}^{a}$ in (a)
to $E_{\rm SL}^{a}$ in (b). (c, d) Representative snapshots of system
configurations under dilute and crowded conditions, respectively. The color
gradient from blue (small sizes) to red (large sizes) visualizes the dynamical
sizes according to the distributions in the top panels (note the scale bars).

The equilibrium probability distribution of sizes in the dispersion is defined
as
$N(\sigma;\rho)=\frac{1}{N}\langle\sum_{i=1}^{N}\delta(\sigma-\sigma_{i})\rangle$
by the canonical ensemble average. It represents the emergent property
distribution accounting for the effects of interactions among RCs. It is a
central in our study, because

$\beta\tilde{\psi}(\sigma;\rho)=-\ln N(\sigma;\rho)$ (4)

defines the crowding-modified energy landscape, which tunes the internal two-
state switching. Fig. 1(b) exemplifies the modified energy barrier
$E_{\mathrm{SL}}^{a}$ for S to L transitions. Since we have a Hamilton system
with explicit response of the sizes, the modification of the energy landscape
is collective and self-consistent.

We assume both the translation and the size dynamics to be overdamped and
propagated by Brownian dynamics (BD) [36], as

$\mathbf{x}_{i}(t+\Delta
t)=\mathbf{x}_{i}(t)+\frac{D_{l,i}}{k_{B}T}\mathbf{F}_{i}(t)\Delta
t+\bm{\xi}_{i},$ (5)

where $\Delta t$ is the simulation timestep. $\mathbf{F}_{i}=-\nabla_{i}H$ is
the generalized force experienced by $i$-th particle from pairwise
interactions given Hamiltonian $H$, Eq. (1), and $\nabla=(\partial/\partial
x,\partial/\partial y,\partial/\partial z,\partial/\partial\sigma)$ is a
4-gradient. The term $\bm{\xi}_{i}$ represents stochastic forces stemming from
the solvent fluctuations. For the evolution along a coordinate
$l=x,y,z,\sigma$, the corresponding component $\xi_{l}$ is drawn from a
Gaussian distribution with mean $\langle\xi_{l}\rangle=0$ and variance
$\langle\xi_{l}^{2}\rangle=2D_{l}\Delta t$, which is strictly
$\delta$-correlated in time and obeys the standard fluctuation-dissipation
theorem [37]. For translation ($l=x,y,z$), the diffusivity $D_{l,i}$ in Eq.
(5) is defined in units $\sigma_{0}^{2}/\tau_{\rm BD}$, where $\tau_{\rm BD}$
sets our unit (Brownian) time scale. The diffusion is related to the friction
coefficient $\zeta_{l,i}$ through $D_{l,i}=k_{B}T/\zeta_{l,i}$. We assume the
latter to follow the Stokes-Einstein relation $\zeta_{l,i}\propto\sigma_{i}$,
as recently verified for conformationally fluctuating proteins [38]. Setting
the time scale of size ($\sigma$) fluctuations, we define
$D_{\sigma}=\alpha\,\sigma_{0}^{2}/\tau_{\rm BD}$ and use the parameter
$\alpha$ (same for all particles) to tune the intrinsic time scale of internal
switching.

We simulate fully periodic systems in the canonical ($NVT$) ensemble
comprising $N=512$ RCs for $\alpha=(0.001,0.01,0.1,1.0,10.0)$ and particle
densities in the range from the low density limit (LDL),
$\rho\sigma_{0}^{3}\to 0$, up to $\rho\sigma_{0}^{3}=1.91$. The LDL is
approximated by simulating an isolated particle. More simulation details are
given elsewhere [39]. Simulation snapshots with particle size distributions
resolved by color are illustrated in Figs. 1(c) and (d).

Resulting emergent size distributions $N(\sigma;\rho)$ are presented in Fig.
2(a). At dense packing, the particles are compressed and the bimodal
distribution shifts to stabilize smaller sizes. For all densities, the
$N(\sigma;\rho)$ distributions also fit well to bimodal Gaussians (Eq. (2))
[39]. Fig. 2(b) shows the crowding-modified landscapes
$\beta\tilde{\psi}(\sigma;\rho)=-\ln N(\sigma;\rho)$ computed using the fitted
$N(\sigma;\rho)$ distributions. Small sizes are substantially stabilized over
the larger states with increasing density, with small-to-large population
ratios exceeding a factor of 10, see the inset of Fig. 2(b). This is in
agreement with earlier computational studies which reported crowding induced
coil-to-globule transitions of two-state biopolymer systems consistent with
experiments [5]. Importantly, the transition barriers are drastically modified
by packing, which is the key to characterize the changes in the transition
kinetics.

Figure 2: (a) Emergent size distributions $N(\sigma;\rho)$, (b) energy
landscapes $\tilde{\psi}(\sigma;\rho)$, and (c) RDF $g(r)$ for varying number
densities $\rho$. The distributions in (a) can be well fitted to bimodal
distributions of form of Eq. (2) [39], which we take to compute
$\tilde{\psi}(\sigma;\rho)$ in (b) using Eq. (4). The inset in (b) shows the
population ratio between small (S) and large (L) states versus $\rho$,
calculated from integrations over the distribution,
$n_{S}/n_{L}=\int_{-\infty}^{\rm TS}N(\sigma){\rm d}\sigma/\int_{\rm
TS}^{\infty}N(\sigma){\rm d}\sigma$. The dashed gray line in (c) shows the
theoretical estimate in the LDL given by
$g(r)\approx\int\mathrm{d}\sigma^{\prime}\int\mathrm{d}\sigma
p(\sigma^{\prime})p(\sigma)\exp{\left[-\beta\phi(r;\sigma,\sigma^{\prime})\right]}$.

For further analysis, we introduce the two-body density-size distribution
function [19], $g(\mathbf{x},\mathbf{x}^{\prime})\equiv
g(r;\sigma,\sigma^{\prime})$. By integrating out $\sigma$ and
$\sigma^{\prime}$ [19], we recover the usual radial distribution function
(RDF) $g(r)$, which represents the average over the polydisperse
distributions. Due to the - on average - smaller size of the particles, the
first peak of the RDF, shown in Fig. 2(c), shifts to smaller interparticle
distances with increasing $\rho$. The first peak grows for higher density as
expected from enhanced packing correlations, although the decreasing particle
size counterbalances this effect. This can be rationalized by the effective
packing fraction, defined as $\eta=\pi\rho\langle\sigma^{3}\rangle/6$ [24],
which approaches sublinear behavior with increasing $\rho$ [39]. Such a weak
scaling with density may indeed facilitate homeocrowding as hypothesized for
biological cells [10].

From our simulations we directly access the mean first passage times (MFPTs)
$\tau^{\rm FP}$ for small to large (SL) or large to small (LS) transitions
from time series analysis [40, 39]. For our overdamped systems, Kramers’
escape theory applies [30], which essentially reads

$\tau^{\rm FP}(\alpha,\rho)\sim\alpha^{-1}\exp(-\beta E^{a}(\rho)),$ (6)

where $E^{a}(\rho)$ is the energy barrier obtained from
$\tilde{\psi}(\sigma;\rho)$, cf. Fig. 2(b). To check the prefactor scaling
with $\alpha$, we plot first the SL-MFPT, $\tau^{\rm FP}_{\rm SL}$, as a
function of $\alpha$ in Fig. 3(a). Indeed, we observe a clear $\alpha^{-1}$
dependence for all densities, that is, the intrinsic time scale of $\sigma$
simply scales the MFPT linearly.

The density-dependence of the MFPTs, though, is non-trivial and dictated by
self-crowding induced changes to $\tilde{\psi}(\sigma;\rho)$, and consequently
the changes in the activation energies $E^{a}(\rho)$ (see Figs. 1(a,b) and
2(b)). In particular, the SL transition slows because of an increasing energy
barrier with density, while the LS transition accelerates. The calculated
energy barriers are presented in Fig. 3(b). They depend apparently linearly on
the density, changing by up to a factor 2-3, due to the compression of the
energy landscapes. This change of activation energy results in a drastic,
almost one-order-of magnitude spread of all MFPTs versus density, as presented
in Fig. 3(c) as a log-lin plot. We observe that the MFPTs depend exponentially
on the crowding density. [We note here that we find, as expected, the
exponential relationship between times $\tau^{\rm FP}$ and energy $E^{a}$
holds across all $\rho$ [39], as predicted by Kramers’, Eq. (6).] At a
critical density of about $\rho\sigma_{0}^{3}\simeq 1.7$ the transition
vanishes in our systems because the L-state becomes unstable, as indicated by
the saddle point in the distribution. Here, $\tau^{\rm FP}_{\rm LS}$
accordingly approaches zero, which can be interpreted as a completely
stabilized (folded or globule) S-state.

Figure 3: (a) Simulation MFPTs of small to large (SL) size switching (in units
of the Brownian time scale $\tau_{\rm BD}$) versus $\alpha$ at various
densities shown using log-log scale. The dashed gray line denotes the
$\alpha^{-1}$ Kramers’ scaling behavior, Eq. (6). (b) Comparison of theory
(lines) and simulations (sim., red squares) of the activation energy $E^{a}$
versus density $\rho$. (c) Comparison of theory and simulations of the MFPT
(scaled by the LDL) as a function of $\rho$, using a log-lin scale.

The LS-transition can be identified with a folding transition from coil to
globule states [7, 8]; qualitatively consistent with our results, it was
demonstrated experimentally and through simulations that crowding enhances
biomolecular folding rates by a multiple, while unfolding rates decrease [6,
8]. Similarly, large crowders speed up two-state polymer looping kinetics in
DNA models [9]. Note, however, that the latter systems dealt with inert, non-
responsive crowders while in our homo-crowding case the system responds
collective and self-consistent, i.e., the internal kinetics responds in quorum
for all particles in the same way.

We finally present a perturbation theory, leading to scaling laws which
describe the density-dependence of the switching kinetics quantitatively. For
this, we consider the mean force on the particle size coordinate due to the
interacting environment

$F(\sigma)=-\rho\int_{V}\mathrm{d}^{3}r\int_{-\infty}^{\infty}\mathrm{d}\sigma^{\prime}N(\sigma^{\prime})\frac{\partial\phi(r;\sigma,\sigma^{\prime})}{\partial\sigma}g(r;\sigma,\sigma^{\prime}).$
(7)

This force is the source for the compression and tilt of the energy landscape
$\psi$. In a first order perturbation approach to the spatial structure we
apply the LDL
$g(r;\sigma,\sigma^{\prime})\approx\exp{\left[-\beta\phi(r;\sigma,\sigma^{\prime})\right]}$.
By integrating the force up to the particle’s size,
$\mathcal{F}(\sigma)=-\int_{0}^{\sigma}\mathrm{d}\sigma^{\prime}F(\sigma^{\prime})$,
this results in a free energy of the analytical form linear in $\rho$ (with
details in [39])

$\displaystyle\mathcal{F}(\sigma)=\frac{5}{12}\pi\rho\epsilon\kappa\left(\sigma^{3}+3\sigma^{2}\left<\sigma\right>+3\sigma\left<\sigma^{2}\right>\right),$
(8)

where $\kappa\approx 6.377\cdot 10^{-4}$ from numerical evaluation for
$\beta\epsilon=500$ in the Hertzian pair potential, and $\mathcal{F}(\sigma)$
can be conveniently expressed by the expectation values $\langle\sigma\rangle$
and $\langle\sigma^{2}\rangle$ with respect to the emergent distribution. From
another first order perturbation approach, $N(\sigma;\rho)\simeq p(\sigma)$ in
Eq. (7), we obtain analytically $\langle\sigma\rangle=(\mu_{1}+\mu_{2})/2$ and
$\langle\sigma^{2}\rangle=\delta^{2}+(\mu_{1}^{2}+\mu_{2}^{2})/2$.

Employing the modified free energy landscape
$\tilde{\psi}(\sigma)\simeq\psi(\sigma)+\mathcal{F}(\sigma)$ in Kramers’
theory, Eq. (6), we obtain following linear forms in $\rho$ for the activation
energies,
$E_{\mathrm{SL}}^{a}\approx\tilde{\psi}(\mu_{3})-\tilde{\psi}(\mu_{1})=E_{0}^{a}+\frac{\nu_{\mathrm{SL}}}{\beta}\rho\sigma_{0}^{3}$
and
$E_{\mathrm{LS}}^{a}\approx\tilde{\psi}(\mu_{3})-\tilde{\psi}(\mu_{2})=E_{0}^{a}-\frac{\nu_{\mathrm{LS}}}{\beta}\rho\sigma_{0}^{3}$
with $\beta E_{0}^{a}=1.019$, and $\nu_{\mathrm{SL}}=0.557$ and
$\nu_{\mathrm{LS}}=0.697$ from evaluation of Eq. (8) [39]. The theoretical
results are shown through lines in Fig. 3(b) and compare very well with the
simulations.

Applying Kramer’s rule, $\tau^{\mathrm{FP}}\propto\exp(\beta E^{a})$, now
leads to MFPT predictions

$\displaystyle\tau^{\rm FP}_{\rm SL}(\rho)\propto\left(\exp(\tilde{\nu}_{\rm
SL})\right)^{\rho}\;\;{\rm and}\;\;\tau^{\rm FP}_{\rm
LS}(\rho)\propto\left(\exp(\tilde{\nu}_{\rm LS})\right)^{-\rho},$ (9)

with $\tilde{\nu}_{\rm SL}=\nu_{\rm SL}\sigma_{0}^{3}/\beta$ and
$\tilde{\nu}_{\rm LS}=\nu_{\rm LS}\sigma_{0}^{3}/\beta$, establishing
exponential scaling laws for the dependence of the MFPT on the packing
density. As with the activation energies, the theoretical predictions for
$\tau^{\rm FP}$ are in very good agreement with the computational results, cf.
Fig. 3(c). From the nature of the perturbation theory, the scaling laws are
quite universal for any pair potential in the LDL. In our case, however, it
works up to surprisingly high densities. The reason lies on one hand in the
softness of the Hertzian pair potential, featuring only weakly correlated
RDFs, cf. Fig. 2. On the other hand, our parent distribution is relatively
broad, perturbing the energy landscape of the internal DOF only linearly in
leading order, cf. Eq. (8).

Our work provides a general methodical framework for studying the effects of
(homeo)-crowding on the internal kinetics in complex responsive systems; this
may include multi-component mixtures, complex (polymodal) energy landscapes,
and various internal DOFs. It will be fascinating to add internal activity
[41] into the RC model to study actively self-regulated crowding effects. For
example, internally fueled switching of hydrogel properties [12, 27, 28] are
expected to lead to complex collective nonequilibrium effects [42, 43] and may
be programmable to mimic the quorum sensing displayed by bacterial populations
[11, 13].

###### Acknowledgements.

The authors acknowledge support for computing facilities provided by the High
Performance Cluster framework by the state of Baden-Württemberg (bwHPC) and
the German Research Foundation (DFG) through grant no. INST 39/963-1 FUGG
(bwForCluster NEMO).

## References

  * Ellis [2001] R. Ellis, Macromolecular crowding: obvious but underappreciated, Trends Biochem. Sci. 26, 597 (2001).
  * Zhou  _et al._ [2008] H.-X. Zhou, G. Rivas, and A. P. Minton, Macromolecular crowding and confinement: Biochemical, biophysical, and potential physiological consequences, Annu. Rev. Biophys. 37, 375 (2008).
  * Morelli  _et al._ [2011] M. Morelli, R. Allen, and P. Rein ten Wolde, Effects of macromolecular crowding on genetic networks, Biophysical Journal 101, 2882 (2011).
  * Kim and Mittal [2013] Y. C. Kim and J. Mittal, Crowding induced entropy-enthalpy compensation in protein association equilibria, Phys. Rev. Lett. 110, 208102 (2013).
  * Kang  _et al._ [2015] H. Kang, P. A. Pincus, C. Hyeon, and D. Thirumalai, Effects of macromolecular crowding on the collapse of biopolymers, Phys. Rev. Lett. 114, 068303 (2015).
  * Cheung  _et al._ [2005] M. S. Cheung, D. Klimov, and D. Thirumalai, Molecular crowding enhances native state stability and refolding rates of globular proteins, Proc. Natl. Acad. Sci. U.S.A. 102, 4753 (2005).
  * Hong and Gierasch [2010] J. Hong and L. M. Gierasch, Macromolecular crowding remodels the energy landscape of a protein by favoring a more compact unfolded state, J. Am. Chem. Soc. 132, 10445 (2010).
  * Dupuis  _et al._ [2014] N. F. Dupuis, E. D. Holmstrom, and D. J. Nesbitt, Molecular-crowding effects on single-molecule rna folding/unfolding thermodynamics and kinetics, Proc. Natl. Acad. Sci. U.S.A. 111, 8464 (2014).
  * Shin  _et al._ [2015] J. Shin, A. G. Cherstvy, and R. Metzler, Kinetics of polymer looping with macromolecular crowding: effects of volume fraction and crowder size, Soft Matter 11, 472 (2015).
  * van den Berg  _et al._ [2017] J. van den Berg, A. J. Boersma, and B. Poolman, Microorganisms maintain crowding homeostasis, Nat. Rev. Microbiol. 15, 309 (2017).
  * Miller and Bassler [2001] M. B. Miller and B. L. Bassler, Quorum sensing in bacteria, Annu. Rev. Microbiol. 55, 165 (2001).
  * Heuser  _et al._ [2015] T. Heuser, E. Weyandt, and A. Walther, Biocatalytic feedback-driven temporal programming of self-regulating peptide hydrogels, Angew. Chem., Int. Ed. 54, 13258 (2015).
  * Walther [2019] A. Walther, Viewpoint: From responsive to adaptive and interactive materials and materials systems: A roadmap, Adv. Mater. 32, 1905111 (2019).
  * Lyon and Fernandez-Nieves [2012] L. A. Lyon and A. Fernandez-Nieves, The polymer/colloid duality of microgel suspensions, Annu. Rev. Phys. Chem. 63, 25 (2012).
  * Keidel  _et al._ [2018] R. Keidel, A. Ghavami, D. M. Lugo, G. Lotze, O. Virtanen, P. Beumers, J. S. Pedersen, A. Bardow, R. G. Winkler, and W. Richtering, Time-resolved structural evolution during the collapse of responsive hydrogels: The microgel-to-particle transition, Sci. Adv. 4, eaao7086 (2018).
  * Stuart  _et al._ [2010] M. A. C. Stuart, W. T. S. Huck, J. Genzer, M. Müller, C. Ober, M. Stamm, G. B. Sukhorukov, I. Szleifer, V. V. Tsukruk, M. Urban, F. Winnik, S. Zauscher, I. Luzinov, and S. Minko, Emerging applications of stimuli-responsive polymer materials, Nat. Mater. 9, 101 (2010).
  * Mittal and Best [2010] J. Mittal and R. B. Best, Dependence of protein folding stability and dynamics on the density and composition of macromolecular crowders, Biophys. J. 98, 315 (2010).
  * Denton and Schmidt [2002] A. R. Denton and M. Schmidt, Colloid-induced polymer compression, J. Phys. Cond. Matt. 14, 12051 (2002).
  * Lin  _et al._ [2020] Y.-C. Lin, B. Rotenberg, and J. Dzubiella, Structure and position-dependent properties of inhomogeneous suspensions of responsive colloids, Phys. Rev. E 102, 042602 (2020).
  * Ilie  _et al._ [2016] I. M. Ilie, W. K. den Otter, and W. J. Briels, A coarse grained protein model with internal degrees of freedom. application to $\alpha$-synuclein aggregation, J. Chem. Phys. 144, 085103 (2016).
  * Kapteijns  _et al._ [2019] G. Kapteijns, W. Ji, C. Brito, M. Wyart, and E. Lerner, Fast generation of ultrastable computer glasses by minimization of an augmented potential energy, Phys. Rev. E 99, 012106 (2019).
  * Berthier  _et al._ [2019] L. Berthier, E. Flenner, C. J. Fullerton, C. Scalliet, and M. Singh, Efficient swap algorithms for molecular dynamics simulations of equilibrium supercooled liquids, J. Stat. Mech. Theory Exp. 2019, 064004 (2019).
  * Ciarella  _et al._ [2021] S. Ciarella, M. Rey, J. Harrer, N. Holstein, M. Ickler, H. Löwen, N. Vogel, and L. M. C. Janssen, Soft particles at liquid interfaces: From molecular particle architecture to collective phase behavior, Langmuir 37, 5364 (2021).
  * Baul and Dzubiella [2021] U. Baul and J. Dzubiella, Structure and dynamics of responsive colloids with dynamical polydispersity, J. Phys. Condens. Matter 33, 174002 (2021).
  * Choi  _et al._ [2011] U. B. Choi, J. J. McCann, K. R. Weninger, and M. E. Bowen, Beyond the random coil: Stochastic conformational switching in intrinsically disordered proteins, Structure 19, 566 (2011).
  * Dhiman  _et al._ [2017] S. Dhiman, A. Jain, and S. J. George, Transient helicity: Fuel-driven temporal control over conformational switching in a supramolecular polymer, Angew. Chem., Int. Ed. 56, 1329 (2017).
  * Yoshida  _et al._ [1996] R. Yoshida, T. Takahashi, T. Yamaguchi, and H. Ichijo, Self-oscillating gel, J. Am. Chem. Soc. 118, 5134 (1996).
  * Heinen  _et al._ [2017] L. Heinen, T. Heuser, A. Steinschulte, and A. Walther, Antagonistic enzymes in a biocatalytic ph feedback system program autonomous dna hydrogel life cycles, Nano Lett. 17, 4989 (2017).
  * Che  _et al._ [2018] H. Che, S. Cao, and J. C. M. van Hest, Feedback-induced temporal control of breathing polymersomes to create self-adaptive nanoreactors, J. Am. Chem. Soc. 140, 5356 (2018).
  * Hänggi  _et al._ [1990] P. Hänggi, P. Talkner, and M. Borkovec, Reaction-rate theory: fifty years after kramers, Rev. Mod. Phys. 62, 251 (1990).
  * Hansen and McDonald [2013] J.-P. Hansen and I. R. McDonald,  _Theory of Simple Liquids_ , 4th ed. (Academic Press, Oxford, 2013).
  * Rovigatti  _et al._ [2019] L. Rovigatti, N. Gnan, A. Ninarello, and E. Zaccarelli, Connecting elasticity and effective interactions of neutral microgels: the validity of the hertzian model, Macromolecules 52, 4895 (2019).
  * Urich and Denton [2016] M. Urich and A. R. Denton, Swelling, structure, and phase stability of compressible microgels, Soft Matter 12, 9086 (2016).
  * Bergman  _et al._ [2018] M. J. Bergman, N. Gnan, M. Obiols-Rabasa, J.-M. Meijer, L. Rovigatti, E. Zaccarelli, and P. Schurtenberger, A new look at effective interactions between microgel particles, Nat. Commun. 9, 5039 (2018).
  * Mohanty  _et al._ [2014] P. S. Mohanty, D. Paloli, J. J. Crassous, E. Zaccarelli, and P. Schurtenberger, Effective interactions between soft-repulsive colloids: Experiments, theory, and simulations, J. Chem. Phys. 140, 094901 (2014).
  * Berendsen [2007] H. J. C. Berendsen,  _Simulating the Physical World_ (Cambridge University Press, Cambridge, 2007).
  * Allen and Tildesley [2017] M. P. Allen and D. J. Tildesley,  _Computer Simulations of Liquids_ , 2nd ed. (Oxford University Press, Oxford, 2017).
  * Yamamoto  _et al._ [2021] E. Yamamoto, T. Akimoto, A. Mitsutake, and R. Metzler, Universal relation between instantaneous diffusivity and radius of gyration of proteins in aqueous solution, Phys. Rev. Lett. 126, 128101 (2021).
  * [39] See the Supplemental Material at http://link.aps.org/ supplemental/xxx for details on the simulation parameters, perturbation theory, fit parameters for emergent bimodal distributions, calculation of mean first passage times, and verification of the Arrhenius behavior in the simulations.
  * Hinczewski  _et al._ [2010] M. Hinczewski, Y. von Hansen, J. Dzubiella, and R. R. Netz, How the diffusivity profile reduces the arbitrariness of protein folding free energies, J. Chem. Phys. 132, 245103 (2010).
  * Bechinger  _et al._ [2016] C. Bechinger, R. Di Leonardo, H. Löwen, C. Reichhardt, G. Volpe, and G. Volpe, Active particles in complex and crowded environments, Rev. Mod. Phys. 88, 045006 (2016).
  * Moncho-Jordá and Dzubiella [2020] A. Moncho-Jordá and J. Dzubiella, Controlling the microstructure and phase behavior of confined soft colloids by active interaction switching, Phys. Rev. Lett. 125, 078001 (2020).
  * Bley  _et al._ [2021] M. Bley, A. Moncho-Jordá, and J. Dzubiella, Active binary switching of soft colloids: stability and structural properties, Soft Matter 17, xxxx (2021).