11affiliationtext: School of Physics and Astronomy, University of Leeds,
Leeds, LS2 9JT22affiliationtext: School of Biological and Chemical Sciences,
Queen Mary University of London, Mile End, London E1 4NS**affiliationtext:
<EMAIL_ADDRESS>
# Unravelling the fluorescence kinetics of light-harvesting proteins with
simulated measurements

Callum Gray  Lekshmi Kailas  Peter G. Adams  Christopher D. P. Duffy

###### Abstract

The plant light-harvesting pigment-protein complex LHCII is the major antenna
sub-unit of PSII and is generally (though not universally) accepted to play a
role in photoprotective energy dissipation under high light conditions, a
process known Non-Photochemical Quenching (NPQ). The underlying mechanisms of
energy trapping and dissipation within LHCII are still debated. Various
proposed models differ considerably in their molecular and kinetic detail, but
are often based on different interpretations of very similar transient
absorption measurements of isolated complexes. Here we present a simulated
measurement of the fluorescence decay kinetics of quenched LHCII aggregates to
determine whether this relatively simple measurement can discriminate between
different potential NPQ mechanisms. We simulate not just the underlying
physics (excitation, energy migration, quenching and singlet-singlet
annihilation) but also the signal detection and typical experimental data
analysis. Comparing this to a selection of published fluorescence decay
kinetics we find that: (1) Different proposed quenching mechanisms produce
noticeably different fluorescence kinetics even at low (annihilation free)
excitation density, though the degree of difference is dependent on pulse
width. (2) Measured decay kinetics are consistent with most LHCII trimers
becoming relatively slow excitation quenchers. A small sub-population of very
fast quenchers produces kinetics which do not resemble any observed
measurement. (3) It is necessary to consider at least two distinct quenching
mechanisms in order to accurately reproduce experimental kinetics, supporting
the idea that NPQ is not a simple binary switch switch.

##  1 Introduction

The light-harvesting antenna system of Photosystem II (PSII) in higher plants
is a large, modular assembly of pigment-protein complexes, the major one being
the cyclic heterotrimer LHCII [1]. Under low intensity illumination the PSII
antenna operates with a quantum efficiency of 0.8 - 0.85 [2] though this drops
significantly at higher intensities [3]. This reflects the non-photochemical
quenching (NPQ) mechanism, a photoprotective response involving the (mostly)
reversible down regulation of PSII antenna efficiency following a sudden
increase in light levels [4, 5]. The purpose of this is to mitigate
_photoinhibition_ of PSII [6], the slowly reversible (and metabolically
costly) oxidative damage to the PSII reaction centres.

Why the exact mechanism of NPQ is still the matter of debate but a general
(though not universal) consensus on the basic scheme has emerged in the last
decade. The primary trigger is the formation of a steep $\Delta\text{pH}$
across the thylakoid membrane [7]. This interacts with the PSII antenna,
predominantly LHCII trimers but possibly also in minor (monomermic) antenna
complexes and even the PSII core [8], resulting in both subtle conformational
change in the individual LHCII complexes [9, 10, 11] and their mutual
clustering/aggregation [12, 13, 14, 15, 16, 17]. This somehow alters the
interactions between the pigment molecules within LHCII, creating molecular
states (known as _quenchers_) that trap and dissipate excitations where before
they promoted rapid excitation transfer to the reaction centres. Also involved
are the PSII subunit PsbS [18, 19] and the enzymatic de-epoxidation of the
LHCII-associated carotenoid violaxanthin to zeaxanthin (the xanthophyll cycle
[20]), though both are allosteric regulators of NPQ rather than pre-requisite
triggers [21].

The molecular details of the quencher states and the structural changes that
form them are still unclear, and for a comprehensive discussion of the various
proposed models the reader is directed to several reviews [22, 4, 21] and a
collected work [23]. Briefly, most models assume that the quenchers involve
the various carotenoid pigments bound by the PSII antenna, which are
attractive candidates due to their inherently short excited state lifetimes,
$\tau_{\text{car.}}\sim 10\text{ps}$ [24], compared to that of chlorophyll
(Chl), $\tau_{\text{Chl}}\sim 4ns$. Some suggest quenching via excitation
energy transfer (EET) from a low energy cluster of Chls to the carotenoid
lutein within LHCII [9, 25, 10, 26, 27, 27], while others propose chlorophyll-
carotenoid charge transfer (CT) [28, 29] or _excitonic_ [30, 31, 32] states.
Not all models involve carotenoids, instead proposing excitation quenching
Chl-Chl CT states [33, 34]. These models all differ in the kinetics, location
and density of quenchers, and experimentally determining which are involved in
_in vivo_ NPQ (at least definitively) is difficult.

Experimental measurements of quenching roughly fall into two categories:
ultra-fast time-resolved spectroscopy applied to isolated complexes, or lower-
resolution fluoresence techniques applied to both isolated LHCII and more
intact systems. The first includes transient absorption (TA) measurements of
LHCII aggregates/oligomers in aqueous suspension [9, 33, 35], trimers
immobilized in gel [36], and more recently two-dimensional (2D) electronic
spectroscopy measurement on LHCII in lipid nanodiscs [26, 27, 37]. The
measured kinetics have been fit to various kinetic models or qualitatively
compared to quantum mechanical simulations of a single LHCII trimer [25, 38,
39, 29, 40], but they have yet to reveal a definitive mechanism (or
mechanisms). Moreover, induction of quenching in these in vitro LHCII systems
generally requires low detergent conditions, meaning these quenching processes
maybe different to those that occur _in vivo_. The second category,
fluoresence techniques, includes Pulse Amplitude Modulation (PAM) fluorescence
measurements which were originally used to probe the kinetics of NPQ formation
and relaxation in intact plants and membranes [41] and various fluorescence
lifetime measurements of leaves and chloroplasts [42, 43, 15], LHCII
aggregates in solution [44, 38], in liposomes [45, 46, 17, 47] and on mica
surfaces [48].

While fluorescence lifetime measurements have a lower time resolution than
TA/2D, they do probe a critical part of the NPQ mechanism: the competition
between the quenchers and long-range energy transfer between neighbouring
LHCII trimers. Analysis of the fluorescence decay kinetics therefore involve
course grained models of energy transfer and quenching/trapping within large
LHCII assemblies [49, 50, 15]. In 2011, Valkunas _et al._ published such a
model, addressing whether fluoresence decay kinetics could discriminate
between different proposed quenching mechanisms [49]. They assumed that the
aggregate was composed of quenched and unquenched LHCII that could exchange
energy. They defined three characteristic timescales: the migration time,
$\tau_{mig}$, which characterized energy transfer from the antenna bulk to the
vicinity of the quencher, the _transfer-to-trap_ time, $\tau_{tt}$, for
transfer to the quenched site from its nearest neighbour, and the quenching
time, $\tau_{Q}$, the actual timescale of excitation dissipation. They then
considered two limiting quenching scenarios, $\tau_{tt}>>\tau_{Q}$, termed
_slow/fast_ (S/F), and $\tau_{tt}<<\tau_{Q}$ (_fast/slow_ or F/S). They showed
that at low-excitation density (much less than one excitation per LHCII) the
fluorescence decay kinetics of the two models were both essentially mono-
exponential. However, at higher excitation densities, where quenching must
compete with excitation annihilation, the S/F kinetics became bi-exponential,
which is consistent with experimental observation. This was judged to support
the idea of quenching via EET to a carotenoid over a mechanism such as Chl-Chl
CT states, since EET is relatively slow ($10-100$ ps) and dissipation by the
Car excited is relatively fast ($<10$ ps). .

Here we revisit the idea of identifying the quenching mechanism from the
fluorescence decay kinetics of LHCII aggregates, addressing several
limitations in previous models. Firstly, previous models tend to simulate the
_excitation_ decay kinetics of the aggregate following instantaneous
excitation. Here we simulate the actual _fluorescence_ decay kinetics as
measured by Time-Correlated Single Photon Counting (TCSPC), which is a wide-
spread and accessible experimental technique used to (indirectly) measure
excitation decay. In TCSPC, sample excitation is via a laser pulse of finite
($\sim 10-100$ ps) temporal width which will overlap with fast processes such
as annihilation. Measured traces are also noisy and analysed by fitting
lifetime components which may obscure any fine kinetic differences between
different quenching mechanisms. Secondly, thanks to various TA/2D measurements
and atomistic models of LHCII, we have more detailed hypotheses about the
mechanism of excitation trapping and dissipation. By combining more detailed
models of the quenching mechanism with a realistic simulation of a typical
measurement we extract a surprising amount of detail from simple fluorescence
measurements. In a meta-study of several published fluorescence experiments we
show that observations are consistent with most LHCII trimers switching to a
relatively weak quenching state and that at least two, kinetically distinct,
mechanisms are invovled.

Figure 1: a. Diagram of how we model a single LHCII trimer inset, shown side
and top view. On the left is a sketch of a typical excitonic model of the
complex. A ladder of delocalized exciton states are formed by mixing of the
Chl excited states. States that are predominantly Chl a/b are represented in
green/blue, with teal indicating mixed states. Following photo-excitation the
exciton (red dot) rapidly relaxes to the Chl a states. On the right is the
thermodynamic model where exciton states are replaced with a smaller number of
_thermodynamically equivalent_ effective states. The excitons can internally
sample each of these states. They can either relax to the ground state (via
fluorescence or internal conversion) or, if there is more then one exciton,
non-radiatively via singlet-singlet annihilation (explosion icon). b.
Simulated and experimental TCSPC traces for LHCII trimers isolated in dilute
solution in annihilation free conditions. Experimental data is from [48],
while simulated trace assumed $n_{ex}=0.05$. c. Simulated traces at increasing
excitation density.

##  2 Theoretical Framework

Our analysis of TCSPC kinetics of LHCII aggregates is based on a model with
two couplled sub-layers. The first layer is the kinetic Monte-Carlo simulation
of photon absorption, energy migration, exciton annihilation, non-radiative
decay, fluorescence and quenching within a lattice of coarse-grained LHCII
trimers. These are the processes that reflect the internal function of the
LHCII assembly but, apart from fluorescence, they are not observable. The
second layer deals with the observable, binning of fluorescence decays in the
same manner as in a TCSPC experiment and performing multiple re-convolution
fits of the resulting decay curves, effectively treating it as if it were
experimental data.

###  2.1 Thermodynamic model of an isolated LHCII trimer

The basic assumption of this work is that the internal excitation
equilibration within an LHCII timer are much faster than energy transfer
between trimers and deactivation processes like fluorescence, internal
conversion and annihilation. In detailed models of the former [51], the
coupling between Chl electronic transitions gives rise to a manifold of
delocalized exciton states. Following photo-excitation the exciton rapidly
($1-2$ ps) equilibrates across the lowest energy (predominantly Chl a) states
(see the left side of Fig. 1 a.). In our model we replace the $N_{\epsilon}$
exciton states with $\tilde{N}<N_{\epsilon}$ thermodynamically equivalent
states (see right side of Fig. 1 a.),

$\tilde{N}=\Biggl{\lceil}\sum_{i=1}^{N_{\epsilon}}\exp{\left(-\frac{\epsilon_{i}}{k_{B}T}\right)}\Biggr{\rceil}$
(1)

where $\epsilon_{i}$ is energy of the $i^{th}$ exciton state relative to the
lowest one, which are taken from [51]. At $T=300$ K $\tilde{N}\approx 15$ (5
states per monomer subunit). If $n(t)\leq\tilde{N}$ is the number of
excitations within the trimer then the equation of motion for an ensemble of
of isolated LHCII trimers is,

$\frac{d}{dt}n=\sigma
J\left(\tilde{N}-n\right)-\sigma_{\text{SE}}J(t)n-k_{\text{decay}}n-\gamma
n\left(n-1\right)$ (2)

where $J(t)$ is the photon flux from the excitation pulse, and $\sigma$ and
$\sigma_{\text{SE}}$ are the absorption and stimulated emission (SE) cross-
sections of the trimer respectively. $k_{decay}$ and $gamma$ are rate
constants for excitation decay (fluorescence and internal conversion) and
annihilation respectively. Generally, we assume $\sigma\sim 2\times
10^{-15}\text{cm}^{2}$, although this varies strongly with excitation
wavelength. In Fig. 1 b. we show the a TCSPC trace for LHCII in solution with
an excitation wavelength of $\lambda=485$ nm [48], where absorption by
carotenoids will be significant. Here, rather than estimate a particular value
for $\sigma$, we define,

$n_{ex}=\sigma\int_{-\infty}^{\infty}dtJ(t)$ (3)

as the average number of excitations per trimer one would have _if_ all of the
energy delivered by the pulse arrived simultaneously
($J(t)\rightarrow\delta(t)$). We can therefore define excitation conditions in
terms of $n_{ex}$ which is agnostic of any particular experimental conditions.
As a rough guide, if the excitation wavelength is $485$ nm and we assume
$\sigma\sim 1\times 10^{-15}\text{cm}^{2}$ (see [52]) then
$n_{\text{ex}}=0.1$, $1.0$ and $5.0$ would require laser fluences,
$\int_{-\infty}^{\infty}dtJ(t)\sim 0.04$, $0.4$, and $2.0$ mJ cm-2
respectively. These laser fluences are typical of those used in spectroscopy
experiments although annihilation is generally purposely avoided in
measurements by keeping to the lower end of this range.

$\sigma_{\text{SE}}$ is also difficult to define absolutely since complex
excitation equilibration within the pulse duration. We assume that
$\sigma_{\text{SE}}\sim\sigma$, meaning that SE becomes more likely than
absorption when the trimer is half-filled with excitations.

$k_{\text{decay}}=k_{IC}+k_{F}$ is composed of internal conversion and
fluorescence but rather than specify both we realise that $k_{IC}$ effectively
rescales $k_{F}$ and $k_{\text{decay}}$ is an effective fluorescence rate. In
Fig. 1 b. we fit a model with $n_{ex}=1$ and $k_{\text{decay}}^{-1}=3.6$ ns to
an experimental TCSPC trace for LHCII in solution, assuming a $200$ ps (FWHM)
Gaussian pulse (i.e. the Instrument Response Function, IRF). The fluorescence
kinetics are purely mono-exponential.

For $n_{ex}>1$ annihilation effects become apparent, though typical TCSPC
experiments will tune laser fluence to avoid this. The annihilation time
constant, $\gamma^{-1}$, in LHCII aggregates has been measured via TA, with
values $16-24$ ps depending on the model used to analyse the TA kinetics [53].
We take $16$ ps throughout this work. Fig. 1 c. show how annihilation
introduces a second faster component to the TCSPC kinetics of an isolated
LHCII trimer. Interestingly, it is a very small effect, even for very high
fluences ($n_{ex}=5$), which arises from two effects. Firstly, the
annihilation is very fast and energy delivery occurs over a finite (and quite
long) laser pulse. Secondly, at high fluences absorption competes with SE
which means the excitation density never actually reaches $n_{ex}$. Although
the second component is visually detectable, bi-exponential re-convolution
fits (see below) are not very stable and so are omitted.

The time-evolution of the system is simulated via a kinetic Monte Carlo
algorithm. We define an array of $200$ unconnected LHCII trimers, turn on the
simulated laser pulse, and perform Monte Carlo sweeps every $dt=1ps$. In a
given step each trimer is considered once on average. The occupancy of the
trimers and the various rate constants determine the probability of different
processes occurring and we consider each process on average once per time-
step, $dt$. All processes are assume to be Poissonian and so the probability
of process $m$ occurring in the interval $t\rightarrow(t+dt)$ is,

$p_{m}\left(dt\right)=\nu(dt)e^{-\nu_{m}\left(dt\right)dt}$ (4)

where $\nu_{m}\left(dt\right)$ is the calculated rate. We do not consider the
possibility of the same process occurring multiple times in the same trimer in
the same time-step and choose $dt=1$ ps to ensure the likelihood of that
occurring is $<0.1\%$.

We compute the TCSPC trace by binning the times at which ’emissive decays’,
that is decays via the $k_{\text{decay}}$ channel (though we separately bin
all other ’non-emissive’ decay processes for book-keeping), until either
$J(t)=0$ and $n=0$ for all trimers (i.e. nothing else can happen). A histogram
bin width of $\Delta t=50$ ps is used here, similar to typical experiments. We
then reseed the Monte Carlo algorithm, trigger another pulse, and repeat the
process until one of the emissive time bins exceeds a threshold count (here
chosen to be 10,000 in line with common TCSPC data collection). Due to the
definition of the probabilities in Eqn. (4), the traces automatically
reproduce the Poissonian noise of TCSPC measurements.

Lastly, the simulated traces are subject to a reconvolution fit in which an
exponential decay series,

$F(t)=H(t_{0})\sum_{m}=A_{m}exp\left(-\frac{t-t_{0}}{\tau_{m}}\right)$ (5)

is convolved with the IRF, which in our case is the laser pulse. $H(t_{0})$ is
simply the Heaviside function used to define the start time, $t_{0}$. Mono-,
bi-, and tri-exponential fits are generated and the best is selected on the
basis of the variances on the fit parameters and whether the addition of a
component visually improves the fit. We then calculate the amplitude-weighted,

$\tau_{\text{amp}}=\frac{\sum_{m}A_{m}\tau_{m}}{\sum_{m}A_{m}}$ (6)

and intensity-weighted,

$\tau_{\text{int}}=\frac{\sum_{m}A_{m}\tau_{m}^{2}}{\sum_{m}A_{m}\tau_{m}}$
(7)

lifetimes. Throughout we will consider $\tau_{\text{amp}}$ as this is
generally the quantity quoted in the literature. The reason for this is that
$\tau_{\text{amp}}$ is directly proportional to the fluorescence quantum yield
(and inversely proportional to the overall extent of quenching) whereas
$\tau_{\text{int}}$ is not [54].

###  2.2 Extension of the model to aggregated LHCII (without quenchers)

Figure 2: a. An AFM image of a pseudo-2D LHCII aggregate on a mica surface
[48] (inset), rendered into a discrete coarse-grained model of photon
absorption, energy transfer, annihilation, quenching and fluorescence. Each
LHCII trimer is treated as a thermodynamic system (see Fig. 1). b. The
different lattice geometries considered in our aggregate models. c. The
simulated TCSPC kinetics of the different lattice geometries (colour coded as
in b.) in the absence ($n_{e}x=0.5$) and presence ($n_{ex}=5$) of
annihilation. The kinetics for different geometries are hard to discern as
they are essentially identical. d. An example of a reconvolution fit for a
hexagonal lattice and $n_{ex}=2$, shown with the weighted residuals.

Fig. 2 a. shows a typical laminar aggregate of LHCII, deposited on mica, as
visualized by AFM [48]. By simply packing circles of the approximate diameter
of an LHCII timer ($~{}5$ nm) into the area we can construct a very
approximate map. It shows that aggregates are small (10-100 trimers) and
heterogeneous in terms of coordination number, $n_{c}$, of each trimer.
Instead of considering specific aggregates we consider ideal lattices of 100
trimers with $n_{c}=2$, $3$, $4$, and $6$ (see Fig. 2 b.). In the absence of
any quenchers the coupled equations of motion of the aggregate become,

$\begin{split}\frac{d}{dt}n_{i}&=\sigma
J\left(\tilde{N}-n_{i}\right)-\sigma_{\text{SE}}J(t)n_{i}-k_{\text{decay}}n_{i}-\gamma
n_{i}\left(n_{i}-1\right)\\\
&-\sum_{j=1}^{n_{c}}{\left(K_{n_{i},n_{j}}^{n_{i}-1,n_{j}+1}n_{i}-K_{n_{i},n_{j}}^{n_{i}+1,n_{j}-1}n_{j}\right)}\end{split}$
(8)

where $K_{n_{i},n_{j}}^{n_{i}-1,n_{j}+1}$ and
$K_{n_{i},n_{j}}^{n_{i}+1,n_{j}-1}$ are the rate constants for the transfer of
an excitation from trimer $i$ to neighbouring trimer $j$ and vice versa. They
depend on the initial occupancies of the two trimers due to an _entropic
repulsion_ effect in which the transfer of an excitation to an already crowded
trimer is thermodynamically penalized. We define,

$K_{n_{i},n_{j}}^{n_{i}-1,n_{j}+1}=\begin{cases}\tau_{hop}^{-1}\quad\text{for}\quad\Delta
S_{n_{i},n_{j}}^{n_{i}-1,n_{j}+1}\leq 0\\\ \\\
\tau_{hop}^{-1}exp\left(\frac{1}{k_{B}}\Delta
S_{n_{i},n_{j}}^{n_{i}-1,n_{j}+1}\right)\quad\text{for}\quad\Delta
S_{n_{i},n_{j}}^{n_{i}-1,n_{j}+1}>0\\\ \end{cases}$ (9)

where $\tau_{hop}$ is a phenomenological hopping time. Values of
$5<tau_{hop}<25$ ps have been variously assumed in course-grained models of
the PSII antenna [55] and we use $\tau_{hop}=25$ ps, since variations in this
range have almost no effect on the simulated TCSPC kinetics other than a tiny
re-scaling of the mean lifetime. $\Delta S_{n_{i},n_{j}}^{n_{i}-1,n_{j}+1}$ is
the entropy change of for the transfer of the excitation,

$\Delta
S_{n_{i},n_{j}}^{n_{i}-1,n_{j}+1}=k_{B}\text{ln}\left[\frac{n_{i}\left(\tilde{N}-n_{j}\right)}{\left(n_{j}+1\right)\left(\tilde{N}-\left(n_{i}-1\right)\right)}\right]$
(10)

where this reduces to $K_{1,0}^{0,1}=K_{1,0}^{0,1}\equiv\tau_{hop}^{-1}$ in
the single excitation limit. A derivation of Eqn. (10) is included in the
Supplementary Material.

Fig. 2 c. shows the (lack of) dependence on the TCSPC kinetics on aggregate
geometry for low ($n_{ex}=0.5$) and very high ($n_{ex}=5$) fluences. We see
that the effect of annihilation is enhanced with respect to the isolated
trimers, since the excitations can now hop around the aggregate and encounter
each other. Fig. 2 d. shows a typical re-convolution fit of these kinetics
(shown is $n_{ex}$, the onset of annihilation) including the weighted
residuals. From $0<n_{ex}<2$ the trace is mono-exponential with
$\tau_{\text{amp}}=3.6$ ns (by construction). For $n_{ex}\geq 2$ a second
component, $\tau_{2}=40-140$ ps, is needed and $\tau_{\text{amp}}=1-2$ ns.
This is shown in Fig. S1 of the Supplementary material. An excellent quality
of fit and low residuals as shown in this example were achieved for all
simulations.

###  2.3 Extension of the model to include quenching mechanisms

Figure 3: a. A schematic of a the generalized thermodynamic model of a
quenched trimer (next to an unquenched trimer to the left). The LHCII trimer
consists of a reduced number of non-quencher states known as the ’antenna
pool’. They can exchange energy with a ’pre-quencher’ (or states), which
irreversibly transfers energy to the actual ’quencher’. The quencher state
dissiaptes energy on timescale of $\Gamma=10$ ps. b. The simulated TCSPC
traces for the various quenching models outlined in Table (1) at
$n_{ex}=0.05$. We have adjusted the quencher density, $\rho_{q}$, in each case
to achieve $\tau_{amp}\sim 500$ ps (fits are not shown for clarity). For the
slow (SL, SD) and moderate (ML, MD) models a pulse-width of 200 ps (wide IRF)
and bin width of $\Delta t=50$ ps was used. For FL and FD well-behaved fits
were only obtained for a $50$ ps pulse, as discussed in main text and shown in
Fig. S2 of the Supplementary Material. c. The same as in b. but at high
excitation density, $n_{ex}=5$

Experimentally, aggregation of LHCII is always accompanied by excitation
quenching ($\tau_{\text{amp}}<2$ ns) and it is assumed that this is due to a
sub-population of trimers that are in some quenched state. Rather than address
particular published quenching models we adopt a generalized kinetic scheme
which is sketched in Fig. 3 a.. We assume that there is a quencher with a very
short lifetime, $\Gamma$, which we assume is $\sim 10$ ps throughout. This is
roughly the excitation lifetime of the various carotenoids proposed as
quenchers [9, 10] or the decay time of the Chl-carotenoid CT states in other
models [40] and measurements on LHCII aggregates imply that the quenching
state is very short-lived [50]. Next, we have a pre-quencher which is a sub-
set, $\tilde{N}^{pq}$, of the antenna states that the quencher can directly
access. Transfer of energy from the pre-quencher to the quencher is assumed to
be irreversible, since the quencher functions as an excitation trap. Lastly,
we have the antenna pool, the rest of the antenna states,
$\tilde{N}^{a}-\tilde{N}-\tilde{N}^{pq}$, left unchanged by the formation of
the quencher but are coupled to it via the pre-quencher. Energy transfer
between the quencher and pre-quencher state is assumed to be bi-directional,
depending largely on the balance of $\tilde{N}^{a}$ and $\tilde{N}^{pq}$. The
equations of motion are,

$\begin{split}\frac{d}{dt}n_{i}^{a}&=\sigma
J\left(\tilde{N}^{a}-n_{i}^{a}\right)-\sigma_{\text{SE}}J(t)n_{i}^{a}-k_{\text{decay}}n_{i}^{a}-\gamma
n_{i}^{A}\left(n_{i}^{a}-1\right)\\\ &-\sum_{j\neq
i}{\left(K_{n_{i}^{a},n_{j}^{a}}^{n_{i}^{a}-1,n_{j}^{a}+1}n_{i}^{a}-K_{n_{i}^{a},n_{j}^{a}}^{n_{i}^{a}+1,n_{j}^{a}-1}n_{j}^{a}\right)}-k_{a,pq}n_{i}^{a}+k_{pq,a}n_{i}^{pq}\end{split}$
(11)

$\begin{split}\frac{d}{dt}n_{i}^{pq}&=\sigma
J\left(\tilde{N}^{pq}-n_{i}^{pq}\right)-\sigma_{\text{SE}}J(t)n_{i}^{pq}-k_{\text{decay}}n_{i}^{pq}-\gamma
n_{i}^{pq}\left(n_{i}^{pq}-1\right)\\\
&-\left(k_{pq,a}+k_{pq,q}\right)n_{i}^{pq}+k_{a,pq}n_{i}^{a}\end{split}$ (12)

$\frac{d}{dt}n_{i}^{q}=k_{pq,q}n_{i}^{pq}-\frac{1}{\Gamma}n_{i}^{q}$ (13)

where $n_{i}^{a}(t)\leq\tilde{N}_{i}^{a}$, $n_{i}^{pq}(t)\leq\tilde{N}^{pq}$,
and $n_{i}(t)\leq 1$ are excitation occupations of the antenna pool, pre-
quencher and quencher respectively, $k_{a,pq}$ is rate constant for energy
transfer from the antenna pool to the pre-quencher ($k_{pq,a}$ is the
reverse), and $k_{pq,q}$ is the transfer rate from the pre-quencher to the
quencher.

There is enough flexibility in the model to qualitatively represent most
proposed quenching mechanisms, though here we are focusing on those mediated
by a carotenid-like (short-lived, optically-forbidden) state. We then consider
fast, moderate, and slow quenchers (as compared to $\tau_{\text{hop}}$) as
$k_{pq,q}^{-1}=1$, $25$, and $100$ ps respectively. In a fast quencher an
excitation is more likely to be quenched than it is to hop to a neighbouring
LHCII trimer and vice versa in a slow quencher. The $<1\text{ps}$ energy
transfer from Chl a to lutein observed in the 2D electronic spectra of LHCII
in native membranes [27], or the Chl-carotenoid excitonic quenching mechanism
based on two-photon excitation spectroscopy [31], would be examples of fast
quenchers. By contrast, quenching by incoherent Chl-lutein EET ($\sim 10-50$
ps) [9, 56] or by the formation of Chl-lutein CT states [57, 29], would be a
moderate to slow mechanisms. If we assume the antenna pool and pre-quencher
are iso-energetic then $k_{a,pq}/k_{pq,a}$ is dictated the relative sizes of
the two domains (essentially entropy)

$\frac{k_{a,pq}}{k_{pq,a}}=exp\left(\frac{1}{k_{B}}\Delta
S_{n^{a},n^{pq}}^{n^{a}-1,n^{pq}+1}\right)$ (14)

If we assume that the base rate of energy relaxation amongst the Chls in
trimer is $\tilde{k}^{-1}\sim 1$ ps then we can define two limiting cases. An
entropically-limited or local quencher,

$k_{a,pq}=\frac{1}{\tilde{N}-1}\tilde{k}$ (15) $k_{pq,a}=\tilde{k}$ (16)

is one in which the quencher is connected only to a single Chl state which
results in an entropic ’bottleneck’ between the antenna Chls and the quencher.
Models that propose quenching by the Chl a612-Lutein620 heterodimer [9, 39,
27] are examples of local quenchers. We also have delocalised quenchers in
which the antenna-pool and the pre-quencher are comparable in size,

$k_{a,pq}=k_{pq,a}=\tilde{k}$ (17)

Mechanisms in which the carotenoid is coupled to several Chls [40] are
delocalised. Table 1 summarises the different parameter schemes.

To implement quenching in our TCSPC simulation we assume that quenching in our
aggregate arises from a random fraction, $\rho_{q}=N_{q}/N_{LHCII}$ of the
quenched trimers. This random distribution is reinitialized before every
excitation pulse ensuring we average over a large number of possible
configurations.

Quenching Model |  $k_{a,pq}^{-1}$ (ps) |  $k_{pq,a}^{-1}$ (ps) | $k_{pq,q}^{-1}(\text{ps})$
---|---|---|---
Slow Localised (SL) | 5 | 1 | 100
Slow Delocalised (SD) | 1 | 1 | 100
Medium Localised (ML) | 5 | 1 | 25
Medium Delocalised (MD) | 1 | 1 | 25
Fast Localised (FL) | 5 | 1 | 1
Fast Delocalised (FD) | 1 | 1 | 1
Table 1: Relevant parameters for the various quenching models used in our
LHCII aggregate simulations. In all cases $\Gamma=10$ ps.

##  3 Results

###  3.1 Simulated TCSPC kinetics of different quenching models

Figure 4: a. A plot of $\tau_{\text{amp}}$ as a function of $\rho_{q}$ for an
LHCII aggregate with FD quenchers and an assumed pulse-width (IRF) of
$\text{FWHM}=200$ ps (red,left vertical axis). For all data points the decay
kinetics are purely mono-exponential. Also shown (blue, right vertical axis)
is the total fraction of all excitation that are eventually quenched,
$\Sigma_{q}$. We see that though $\Sigma_{q}$ increases monotonically with
$\rho_{q}$, the ’detected’ lifetime is discontinuous with respect to
$\rho_{q}$ and never drops below $\tau_{\text{amp}}\sim 2.2$ ns. b. The
simulated traces for $0.8\leq\rho_{q}\leq 1.0$. The re-convolution fits were
very poor and so it is difficult to assign a $\tau_{\text{amp}}$ in this
range. Note the sharp transition from mono-exponential, to sharply multi-
exponential, to the pure IRF. c. $\Sigma_{q}$ for the various quenching
models. The blue plot corresponds to annihilation-free conditions
($n_{ex}=0.05$) with $\rho_{q}$ chosen to yield $\tau_{\text{amp}}\sim 500$ ps
(see Table 2). The green plot corresponds to the same values of $\rho_{q}$ but
in conditions of significant excitation annihilation ($n_{ex}=5$, see Table
3).

Initially we considered quenching in annihilation-free conditions, choosing
$n_{\text{ex}}=0.05$ for an aggregate of $200$ LHCII trimers, meaning that
each pulse delivers an average of $10$ excitations. We then took each
quenching model listed in Table (1) and adjusted the fraction of quenched
LHCII, $\rho_{q}$, until we (approximately) obtained a representative quenched
lifetime of $\tau_{\text{amp}}\sim 500$ ps (although $200-1200$ ps is
generally considered ’quenched’ [44]). Getting an exact match between models
was quite difficult as $\tau_{\text{amp}}$ is as sensitive to fit quality as
it is to the underlying excitation dynamics. Regardless, it is only
qualitative differences in the kinetics that we are interested in as these are
probably all that are observable experimentally. The traces are shown in Fig.
3 b. (on a logarithmic scale) and the quencher densities and re-convolution
fit parameters are listed in Table 2.

For SL and SD a quencher concentration of $\rho_{q}=0.9$ and $0.85$
respectively are required to reduce the lifetime to $\tau_{\text{amp}}\sim
380-530$ ps. Since the kinetics are marginally bi-exponential (particularly
SL) we listed both mono- and bi-exponential fits Table 2, with the traces and
fits plotted in Figs. S3 and S4 of the Supplementary Material. The bi-
exponential fits are visually closer and yield $\tau_{\text{amp}}\sim 530$ and
$521$ ps for SL and SD respectively. However, the slow component,
$\tau_{1}\sim 4.3-5.7$ ns, is significantly slower than the free decay of an
unquenched LHCII trimer ($k_{\text{decay}}^{-1}=3.6$ ns), probably indicating
that neither mono- nor bi-exponential fits are perfect. The fast component is
$\tau_{2}\sim 264-383$ ps, indicating that the ’slow’ mechanisms can
significantly quench an isolated LHCII trimer (as previously noted [39]).

For ML and MD a quencher density of $\rho_{q}=0.80$ yields
$\tau_{\text{amp}}\sim 568$ and $584$ ps respectively, and kinetics that are
obviously bi-exponential (Fig. 3 b.). The slow component, $\tau_{1}=3.7-3.9$
ns is essentially just free decay of unquenched LHCII while the fast
component, $\tau_{2}=167-264$ ps, is comparable to the timescale of the IRF.
Interestingly, despite the ML and MD quenchers being much faster than SL and
SD ($k_{pq,q}^{-1}=25$ ps compared to $100$ ps) a high density of quenchers is
still needed to quench the whole aggregate. This is due to the fact that,
since $\tau_{\text{hop}}=k_{pq,q}=25$ ps, an excitation still has a fairly
high probability of evading a quencher by hopping to a neighbouring LHCII
trimer.

The behaviour of FL and FD is complicated by the fact that the kinetics of the
quencher are significantly faster than the $200$ ps pulse-width (IRF)
initially used in our model, with the decay kinetics having a seemingly
counter-intuitive dependence on $\rho_{q}$. In Fig. 4 a. (red line) we show
the dependence of $\tau_{\text{amp}}$ on $\rho_{q}$ in the range
$0.01\leq\rho_{q}\leq 0.8$, for the FD mechanism (the behaviour of FL is
identical). The blue line shows the ratio of total excitations quenched,
$\Sigma_{q}$, to the total excitations that decay emissively,
$\Sigma_{\gamma}$. For very low quencher density ($\rho_{q}\sim 0.01$) the
aggregate is essentially unquenched and $\tau_{\text{amp}}\approx
k_{\text{decay}}^{-1}$. For $0.02\leq\rho_{q}\leq 0.2$, $\tau_{\text{amp}}$
drops to and plateaus at $\sim 2.2$ ns, indicating a small degree of overall
quenching. At $\rho_{q}\sim 0.2$, $\tau_{\text{amp}}$ discontinuously jumps to
$3.6ns$ and remains at this level up to $\rho_{q}\sim 0.8$. Despite the fact
that the amount of excitation quenching increases significantly (blue line,
Fig. 4 a.), it is not ’detected’ in the emission histogram. Finally, in the
region $0.8\leq\rho_{q}\leq 1.0$ the kinetics become sharply multi-exponential
and the reconvolution fits become very unreliable. The interpretation of this
is actually straightforward: At low quencher densities there is simply very
little quenching. Although FL and FD are labelled ’fast’ they are not
irreversible traps and therefore excitations have a reasonable chance of
evading the quencher. The level of quenching increases sharply at some
critical point around $\rho_{q}\sim 0.2$ though it is effectively masked by
the wide IRF. Excitations are delivered to the aggregate over $200$ ps but are
typically quenched within the first detection bin ($t<50$ ps). All that is
detected is the small fraction of excitations that evade the traps and undergo
free decay. Essentially, the kinetics are bi-exponential, with the short
component completely masked by the IRF. At $\rho_{q}=1.0$, all LHCII trimers
are equivalent and deeply quenched and the TCSPC signal is simply the IRF. In
the range $0.8\leq\rho_{q}<1.0$ there is a sharp transition between these two
regimes.

The only way to clearly resolve the FL and FD mechanisms is to reduce the
pulse width to $\text{FWHM}=50$ ps and the bin width to $\Delta t=10$ ps
(reducing the bin width alone does not significantly alter the kinetics as
shown in Fig. S2 of the Supplementary Material). This higher resolution is
still experimentally feasible but requires high performance instrumentation.
At a quencher density of $\rho_{q}\sim 0.8$ the FL and FD mechanisms result in
sharply bi-exponential TCSPC kinetics with $\tau_{\text{amp}}\sim 652$ and
$538$ ps respectively. The slow component is, as with ML and MD, $\tau_{1}\sim
k_{\text{decay}}^{-1}=3.6$ ns, while the fast component is $\tau_{2}\sim 34$
and $25$ ps. The fits are shown in Fig. S6 of the supplementary material and
it is apparent that longer lifetime of the FL model ($\tau_{\text{amp}}\sim
652$ ns) is the result of poor fit quality.

Lastly, to confirm that we were truly comparing comparable quenching regimes
across the different models, we compared $\Sigma_{q}$ at
$\tau_{\text{amp}}\sim{500}$ ps (see Fig. 4 c.). In all cases there is
significant quenching, with $0.84\leq\Sigma_{q}\leq 0.89$ for all models. This
shows that all mechanism have the some functional consequences, quenching
80-90% of the energy absorbed by the aggregate, the experimental signatures
are very different.

$n_{ex}=0.05$
---
Model | $\rho_{q}$ |  $\tau_{\text{amp}}$ (ps) | $A_{1}$ |  $\tau_{1}$ (ps) | $A_{2}$ |  $\tau_{2}$ (ps)
SL${}_{\text{mono}}$ | 0.9 | $434\pm 5$ | $1.683\pm 0.006$ | $434\pm 5$ | — | —
SL${}_{\text{bi}}$ | 0.90 | $530\pm 35$ | $0.049\pm 0.002$ | $5750\pm 1132$ | $1.729\pm 0.005$ | $383\pm 5$
SD${}_{\text{mono}}$ | 0.85 | $386\pm 11$ | $1.765\pm 0.016$ | $386\pm 11$ | — | —
SD${}_{\text{bi}}$ | 0.85 | $521\pm 16$ | $0.137\pm 0.000$ | $4298\pm 219$ | $2.008\pm 0.007$ | $264\pm 3$
ML | 0.80 | $568\pm 18$ | $0.284\pm 0.004$ | $3932\pm 135$ | $2.382\pm 0.024$ | $167\pm 4$
MD | 0.80 | $584\pm 11$ | $0.403\pm 0.003$ | $3774\pm 60$ | $2.806\pm 0.032$ | $126\pm 3$
FL | 0.80 | $652\pm 4$ | $0.436\pm 0.001$ | $3645\pm 14$ | $2.113\pm 0.012$ | $34\pm 4$
FD | 0.80 | $538\pm 3$ | $0.470\pm 0.001$ | $3618\pm 10$ | $2.824\pm 0.018$ | $25\pm 3$
Table 2: Fit parameters for each quenching model, assuming $n_{ex}=0.05$ and
tuning $\rho_{q}$ to achieve an amplitude-weighted lifetime,
$\tau_{\text{amp.}}$ of roughly $500$ ps. We show SL and SD parameters for
both mono- and bi-exponential fits and note that $\tau_{\text{amp}}\sim 650$
ps for the FL model is due to a poor fit rather than any significant
differences in the trace itself.

###  3.2 Different quenching models at high excitation density

Using the same excitation densities, $\rho_{q}$, as in Table 2, the excitation
density was increased to $n_{ex}=5$ to introduce annihilation (traces shown in
Fig. 3 c.). Visually, the SL and SD models do not appear to have changed, the
ML and MD traces seem to decay a little faster and the FL and FD traces are
noticeably altered. This is confirmed by the reconvolution fits, for which the
parameters are listed in Table (3) and the plots shown in Figs. S7 - S10 in
the Supplementary Material. The SL and SD mechanisms remain essentially
unchanged by the presence of annihilation, though there is some slight
difference for SD. The reason for this is the large difference in timescales
between SL/SD quenching and annihilation, coupled with the fairly wide
excitation pulse. Annihilation is fast and with relatively slow excitation
delivery, is largely over by the end of the pulse. SD/SL quenching occurs
later and so to two processes do not compete in a way that is detectable in
the signal. ML/MD and particularly FL/FD do compete with annihilation and so
their kinetics are visibly dependent on excitation density. This can also be
seen in Fig. 4 c. which shows the decrease in $\Sigma_{q}$ due to
annihilation, which is most significant for the SL and SD quenchers.

$n_{ex}=5.0$
---
Model | $\rho_{q}$ |  $\tau_{\text{amp}}$ (ps) | $A_{1}$ |  $\tau_{1}$ (ps) | $A_{2}$ |  $\tau_{2}$ (ps)
SL${}_{\text{mono}}$ | 0.9 | $433\pm 5$ | $1.664\pm 0.008$ | $433\pm 5$ | — | —
SL${}_{\text{bi}}$ | 0.90 | $524\pm 36$ | $0.047\pm 0.003$ | $5723\pm 1186$ | $1.714\pm 0.009$ | $381\pm 5$
SD${}_{\text{mono}}$ | 0.85 | $377\pm 8$ | $1.759\pm 0.008$ | $377\pm 8$ | — | —
SD${}_{\text{bi}}$ | 0.85 | $516\pm 36$ | $1.900\pm 0.012$ | $5186\pm 702$ | $0.089\pm 0.004$ | $297\pm 6$
ML | 0.80 | $449\pm 31$ | $0.143\pm 0.004$ | $4699\pm 474$ | $2.410\pm 0.022$ | $197\pm 6$
MD | 0.80 | $428\pm 28$ | $0.169\pm 0.005$ | $4481\pm 402$ | $2.674\pm 0.028$ | $172\pm 6$
FL | 0.80 | $192\pm 9$ | $0.086\pm 0.002$ | $4275\pm 232$ | $2.407\pm 0.012$ | $46\pm 1$
FD | 0.80 | $193\pm 8$ | $0.095\pm 0.002$ | $4196\pm 211$ | $2.520\pm 0.014$ | $43\pm 1$
Table 3: Fit parameters for the same models (and model parameters) but for
$n_{ex}=5.0$.

###  3.3 Measured fluorescence kinetics of LHCII aggregates

TCSPC measurements on quenched LHCII aggregates have been reported in a
variety of experimental conditions. Here we compare (annihilation-free) traces
from LHCII crystals [44], LHCII trimers in gel [58], in proteoliposomes [45],
aggregates in solution [43], and laminar aggregates on mica [48]. The
published traces were digitised using the WebPlotDigitizer service [59] and
are plotted (normalized) in Fig. 5 a. Visually, they seem to resemble our SD
and SL model traces, lacking the sharply and obviously bi-exponential kinetics
of the faster quenching models. Still, the kinetics are clearly heterogeneous
and they are generally reported with a bi-exponential fit consisting of a
’slow’ ($\sim 2$ ns) and a ’fast’ ($\sim 200-600$ ps) component. The fast
component matches the $\tau_{2}\sim 300-400$ ps component of our SL and SD
models, which reflects migration to and then trapping by a very large
distribution of slow quenchers. In all of our models the slow component,
$\tau_{1}\sim 3-5$ ns, essentially reflects the free decay of excitations in
unquenched trimers.

Using our simulations we attempted to fit some of these experimental traces.
We focused on LHCII in proteolipsomes [45] and LHCII aggregates on a mica
surface [48] since these are quasi-2D arrays that should resemble our model.
LHCII crystals and aggregates in solution, despite being highly non-native
structures, produce very similar TCSPC kinetics to LHCII in proteolipsomes and
on mica respectively (but with a narrower IRF). LHCII in gel is equally non-
native, with gel-immobilization assumed to induce quenching in the absence of
aggregation. The fits are plotted in Fig. 5 b. Both experimental traces could
be fit to an SL quenching model, assuming $\rho_{q}=0.82$ and $0.97$
respectively. The dotted lines show the best fits we could obtain with the
constraint that $k_{\text{decay}}^{-1}=3.6$ ns (taken from fitting LHCII in
detergent in Fig. 1 b.) and while the short time kinetics are fit very well
the longer times are not. The closest visual fits (solid lines in Fig. 5 b.)
are obtained by setting $k_{\text{decay}}^{-1}=2.2$ ns implying that, in
addition to a large sub-population of slow, localized quenchers, there is some
global reduction of Chl excited state lifetime. A schematic diagram
illustrating this mixed quenching scenario is shown in Fig. 5 c.

Figure 5: a. A selection of TCSPC measurements on quenched LHCII in a variety
of experimental conditions. These include LHCII crystal [44], LHCII
immobilized in gel [58], and LHCII aggregates in proteoliposomes [45],
solution [43], and on a mica surface [48]. Data was obtained by the
WebPlotDigitizer service [59]. b. The measured TCSPC trace for LHCII
aggregates in proteoliposomes [45] (green dots) and on mica [48] (purple dots)
shown with various simulated traces for an aggregate with SL quenchers. The
only differences between the two traces appears to be the value of $\rho_{q}$.
The dotted lines are the best fits obtainable with the constraint
$k_{\text{amp}}^{-1}=3.6$ ns. The fits are significantly improved by relaxing
this constraint and setting $k_{\text{amp}}^{-1}=3.6$ ns (solid lines). c. A
schematic of different aggregate scenarios. On the left there is the
hypothetical aggregate in the absence of quenching. Each site is identical
with a free excitation decay time of $k_{\text{decay}}^{-1}\sim 4$ ns. In the
middle is a conventional picture of quenching in which a sub-population of
trimers adopt a quenching configuration. On the right is the scenario implied
by experimental measurements in which, in addition to the quenchers, there is
global decrease in the free Chl lifetime, $k_{decay}^{-1}$

##  4 Discussion

The essential aim of this work was to determine whether one could identify the
nature of the NPQ quencher from a relatively ’simple’ TCSPC measurement on an
LHCII aggregate. It had been previously suggested that this may be the case if
one considered excitation densities high enough to induce singlet-singlet
annihilation [49]. We sought to evolve this work in two ways. Firstly, we
aimed to model, as closely as possible, the actual experiment, including
finite instrument responses, limited temporal resolution and reconvolution
fitting of the resulting signal, all factors that could completely mask any
fine kinetic detail of the quenching mechanism. Secondly, various works
combining quantum chemical modelling of isolated LHCII trimers with ultra-fast
optical spectroscopies have provided us with a selection of plausible (if not
definitive) quenching mechanisms, meaning we can focus our model on a narrower
parameters space than previously considered.

###  4.1 The TCSPC kinetics of different quenching mechanisms

In previous work it was argued that two limiting models of quenching, fast
trapping and slow quenching (F/S) and vice-versa (S/F), have identical
fluorescence kinetics in the absence of annihilation but different
dependencies on increasing excitation density [49]. While it is difficult to
compare directly due to model difference, our results are rather different.
Firstly, we see stark differences in the simulated signals between different
quenching scenarios, even at very low excitation density. Our SL and SD
models, which are closest to the previous S/F model, result in fluorescence
kinetics that are almost mono-exponential, but for a small-amplitude, long-
time tail. Conversely, the faster quenching mechanisms are sharply bi-
exponential. For faster quenching models the difference in the lifetimes of
quenched and unquenched LHCII trimers is larger and fewer LHCII trimers need
to be quenched to obtain the same overall lifetime, $\tau_{\text{amp}}$, for
the aggregate.

When we introduced annihilation the TCSPC kinetics of the SL and SD models
barely change, while those of the faster models are associated with a
significant drop in $\tau_{\text{amp}}$. This is contrary to previous
theoretical study in which slow-trapping (S/F) quenching models were suggested
to be far more sensitive to excitation density than fast-trapping (F/S) models
[49]. The rational was that fast-trapping quenchers trap excitons before they
have a chance to annihilate while slow-trapping quenchers don’t. The reason we
see the opposite in our more realistic simulations is as follows:

  * •

In a TCSPC measurement excitations are delivered over a time period defined by
the pulse width, typically $\sim 50-200$ ps. Since annihilation occurs on the
order of ($\gamma^{-1}\sim 20$ ps) the majority of annihilation occurs inside
the pulse. Add to this the contribution of stimulated emission and it is quite
difficult to accumulate a large residual exciton population outside of the
pulse (remember $n_{ex}$ is crude estimate of excitation density). The
majority of the annihilation is effectively masked by the IRF, unless more
expensive/advanced lasers with ultra-narrow temporal width are used.

  * •

The remaining excitons can still diffuse about the aggregate but annihilation
is hindered by their mutual entropic repulsion, which means they have a
tendency to avoid occupying the same timer. This is why annihilation kinetics
have little dependence on aggregate topology (Fig. 2 c.).

  * •

For SL and SD almost every trimer is a quencher site meaning quenching is by
far the most likely dissipation pathway regardless, up to a point, of the
residual excitation density.

  * •

For ML and MD a slightly smaller population of quenchers is required to give
the same overall level of quenching, meaning the likelihood of two residual
excitons meeting each other and annihilating increases slightly. There is
therefore a small dependence of $\tau_{amp}$ on excitation density.

  * •

The very large decrease in lifetime for the FL and FD models at high
excitation density is primarily the result of the narrower laser pulse ($\sim
50$ ps) needed to resolve the quenching. Excitations are delivered over a
shorter timescale, meaning that a higher transient population is achieved and
more excitons are present after the pulse. This is also the reason
annihilation is so obvious in TA measurements which use sub-picosecond
excitation pulses.

This implies that annihilation may not be very useful in discriminating
between different quenching mechanisms in real measurements. Different
mechanisms should have different fluorescence kinetics even at low excitation
densities and the annihilation kinetics are more sensitive to the excitation
pulse width than to the quenching mechanism.

###  4.2 The identity of the quencher in NPQ

Surprisingly, TCSPC measurement, when analysed with our simulated
measurements, may yield significant information about the mechanism
responsible for quenching in various LHCII assemblies (Fig. 5 a. and b.). For
LHCII aggregates on mica surfaces [48] and in proteolipsomes [45], the
measured fluorescence kinetics are consistent with a very large sub-population
($\rho_{q}\sim 80-90\%$ or trimers) of ’slow quenchers’. That is some
quenching configuration in which energy is delivered slowly
($k_{pq,q}^{-1}\sim 100$ ps) but irreversibly to a trap state with a short
lifetime ($\Gamma\sim 10$ ps). Interestingly, the only difference betweenthe
two appears to be the quencher density, $\rho_{q}$. This seems in line with
very recent time-resolved fluorescence measurements on LHCII in model
membranes, which showed that $\rho_{q}$ depends on the lipid-LHCII ratio [47].
However, whether this slow NPQ trap state is coupled only to particular Chl
(SL) or accessible from many (SD) is probably not resolvable with TCSPC, since
the fluorescence kinetics for each scenario are _qualitatively_ identical.

Of course we did not explore the entire parameter space of our model. For
example, we did not consider a state that a quencher with a fast trapping time
($k_{pq,q}^{-1}\sim 1$ ps) and slow quenching time ($\Gamma>100ps$), mainly
because this is not suggested by the majority of plausible models. Most models
assume the quencher is a carotenoid [27, 40, 39, 9] or some short-lived Chl-
carotenoid excitonic [31] or CT state [28, 29]. Models aside, it was observed
in early TA measurements of LHCII aggregates [9] that the quencher did not
accumulate a detectable population, implying it was both short-lived and
slowly-populated (hence the initial hypothesis of a carotenoid as a quencher).

Our models do imply that a fast-trapping ($k_{pq,q}^{-1}<10$ ps), fast-
quenching ($\Gamma\sim 10$ ps) mechanism would yield very sharply bi-
exponential kinetics, something that doesn’t appear to be reported despite a
wide range of experimental conditions. Of course, one could argue that this
behaviour could be masked by a combination of a wide excitation pulse and low
detection resolution, but this seems unlikely given the number of independent
measurements in vastly different conditions. However, two recent measurements
have suggested some form of ultra-fast transfer to a carotenoid or carotenoid-
like state is involved in quenching. TA measurements on quenched LHCII
immobilized in gel and in low detergent conditions showed the instantaneous
(within resolution) appearance of a carotenoid-like signal upon excitation of
Chl a or Chl b [36]. Similarly, 2D measurements of LHCII in lipid nano-disks
showed $<500$ fs Chl-carotenoid EET [27]. While one could argue that the gel
immobilization represents a highly non-native set of conditions, it is harder
to argue this for lipid nano-disks.

The most interesting revelation is that at least two quenching mechanisms are
needed to reproduce the observed fluorescence kinetics. The reduction of
_both_ lifetime components has always been a feature of the measured
fluorescence kinetics of quenching but it was never clear how this related to
the actual internal processes of exciton migration, trapping and decay. The
’second’ quenching pathway appears to be rather slow ($\sim 2$ ns) and
uniformly distributed within the aggregate. We modelled this as an ad hoc
reduction in $k_{\text{decay}}^{-1}$, but we are not necessarily proposing
that some process radically alters the fundamental excited state lifetime of
Chl. We are not excluding this possibility either, given that it is well known
that non-planar distortions to tetrapyroles can significantly shorten their
fluorescence lifetimes (though generally by enhancing triplet formation) [60,
61]. Either way, this shortening of the LHCII lifetime from $\sim 4$ ns to
$\sim 2.2$ ns is also observed in the thylakoid membrane in the absence of
quenching. It was initially thought to be due to radical pair recombination in
the PSII reaction centre [62, 63]. However, the same lifetime reduction was
observed in plants depleted of PSII due to being grown on the PSII repair
inhibitor lincomycin, implying that $\sim 2.2$ ns lifetime comes from some
decay process in the antenna [64]. Our previous energy relaxation simulation
of LHCII suggested while simple EET to the carotenoid lutein could not explain
the level of quenching seen in NPQ, it may explain the ’background quenching’
responsible for the $\sim 2.2$ ns lifetime.

###  4.3 Why focus on Time-Correlated Single Photon Counting?

High resolution non-linear spectroscopies (TA, 2D, transient grating, etc.)
are the state-of-the art technique for unravelling the ultra-fast and
branching relaxation processes in isolated photosynthetic complexes, their
utility vastly enhanced by an extremely well-developed theoretical framework
[65, 66]. Still, they have yet to unambiguously reveal key functional aspects
such as the underlying mechanism of NPQ. This is possibly due to a combination
of a lack of fine structural information on the quenched and unquenched
states, and the fact that these measurements generally have to be performed in
highly non-native conditions.

To understand the actual functional characteristics of NPQ one must consider
its place within a larger network of complexes, the clusters of LHCII that
form in the membrane and even the thylakoid grana as a whole. These length-
and time-scales are better probed by techniques such as Fluorescence Lifetime
Imaging (FLIM) which, when correlated with visualization techniques such as
AFM, offer insight into the relationship between membrane organization and
function [48]. FLIM is based on TCSPC and direct data analysis is generally
limited to reconvolution fitting of the fluorescence decay kinetics. Models of
the underlying processes have in the past been highly phenomenological or
overlooked the technical limitations of the method. We have shown here that
while factors such as pulse width and detection resolution can have a profound
impact on the kinetics and the subsequent fits, this is not necessarily a
problem, so long as they are carefully factored into any model as they have
been here.

In conclusion, we here present a first look at new method for analysing the
_measured_ fluorescence kinetics of large networks of photosynthetic
complexes. The approach is general and easily extendable to include non-
trivial structures and additional decay pathways such as photochemical
quenching in the reaction centre.

##  5 Acknowledgements

All authors acknowledge the support of BBSRC (joint grant BB/T000023/1).

##  6 Software details

We use a mix of Fortran with OpenMPI for the TCSPC simulation and Python for
the fitting and analysis. The pRNG used in the Fortran code is xoshiro1024
(see here) and the code itself is available at https://github.com/QMUL-
DuffyLab/aggregates.

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