Spatiotemporal model for FRET networks with multiple donors and acceptors: multicomponent exponential decay derived from the master equation

Masaki Nakagawa; Yuki Miyata; Naoya Tate; Takahiro Nishimura; Suguru Shimomura; Sho Shirasaka; Jun Tanida; Hideyuki Suzuki

doi:10.1364/JOSAB.410658

1. INTRODUCTION

Förster resonance energy transfer (FRET) has served as local signaling in nanoscale without molecular diffusion processes. For example, local signaling by FRET between fluorescent molecules on DNA substrates is used as photonic logic gates [1–4]. Kinetics of single-step FRET is well understood by Förster theory even in special environments such as membranes and solutions (see, e.g., Chapters 13–15 in [5]), and that of multistep FRET (a cascade of FRETs) can also be understood by Förster theory in principle. Multistep FRET is experimentally shown in heterogeneous fluorescent dyes [6] and even in homogeneous fluorescent dyes [7] on linear DNA scaffolds. Furthermore, multistep FRET occurs in hetero- and homogeneous quantum dots (QDs) [8,9].

In principle, multistep FRET can occur over multiple locations and times if spatially distributed fluorescent molecules are excited simultaneously. In such “FRET networks,” fluorescent molecules often play roles of both donors and acceptors according to temporally changing situations. Thus, FRET networks are very interesting from the viewpoint of their rich spatiotemporal dynamics and have great potential for applications including information processing and computing. In fact, the spatiotemporal dynamics of FRET between fluorophores on a DNA substrate is used as principles for designing components of intelligent systems such as unclonable physical keys [10] and photonic logic gates [1–4]. Furthermore, some networks of energy transfer, probably due to FRET, between spatially distributed QDs can generate diverse spatiotemporal signals, which can be useful for information processing [11]. The key to designing their applications is to understand the spatiotemporal behavior of FRET networks. Here, we develop a spatiotemporal model for FRET networks and uncover its temporal characteristic behavior, i.e., the network-induced multicomponent exponential decay of the fluorescence intensity, even for FRET networks of fluorophores with an identical single exponential decay.

The prototype of our model for FRET networks is a continuous-time Markov chain (CTMC) model developed for multistep FRET with a single donor and multiple acceptors [12,13] (hereafter called the single-donor model). The single-donor model assumes the situation where the system has only one excited molecule and hence cannot consider the “level occupancy effect,” which means that already excited molecules are effectively forbidden from energy absorption. On the other hand, FRET networks involve multiple excited and non-excited molecules, and hence the consideration of the level occupancy effect is essential. The level occupancy effect may significantly affect the behavior of FRET networks. In the following section, we propose a novel CTMC model of FRET networks that takes into account the level occupancy effect. Although similar models considering the level occupancy effect already exist, e.g., [14,15], their approaches are different from ours in the following points. (i) Their main aim is to present a Monte Carlo simulation algorithm using their models. On the contrary, our main aim is to produce the theoretical results using our model and thus understand the spatiotemporal behavior fundamentally. (ii) They introduce the level occupancy effect as a complete exclusion of already excited molecules from their roles as acceptors, whereas our model incorporates such roles by considering the Auger recombination. (iii) Their models handle mainly decay processes, whereas our model additionally covers the light-induced excitation process. We note that our model can be applied to any fluorescent molecules, but here, we focus mainly on QD-based FRET networks because QDs are expected to be notable fundamental elements to realize compact and energy-efficient information-processing systems [16–19].

2. METHOD

A. Formulation of the Model

Here, we propose a spatiotemporal model for FRET networks based on the CTMC model [12,13]. We consider mainly FRET networks consisting of QDs, but not excluding other types of fluorescent molecules. First, we assign either a ground state “0” or an excited state “1” to each QD, which is denoted as ${i_n} = 0$ (or ${S_n}$) or ${i_n} = 1$ (or $S_n^*$), respectively, for the $n$th QD. Now, consider the probabilities ${P_{{i_1} \cdots {i_N}}}(t)$ for the system states $({i_1},{i_2}, \cdots ,{i_N}) \in {\{0,1\} ^N}$ at time $t$. Naturally, $\sum\nolimits_{({{i}_{1}},\cdots ,{{i}_{N}})\in {{\{0,1\}}^{N}}}{{P}_{{{i}_{1}}\cdots {{i}_{N}}}}(t)=1\;(\forall \, t\in \mathbb{R})$ holds. As the whole system state is considered at each time, we can take into account the level occupancy effect in FRET between QDs, as shown later. Next, we define three state transitions for each QD as follows:

(1a)$$S_n^*\mathop{\to}\limits^{k_n^{\rm F}} {S_n}\;(+ h{\nu _{\rm f}}),$$(1b)$$S_n^*\mathop \to \limits^{k_n^{\rm N}} {S_n},$$(1c)$$S_n^* + {S_m}\mathop {\longrightarrow}\limits^{k_{\textit{nm}}^{{\rm FRET}}} {S_n} + S_m^*,$$

where $k_n^{\rm F},k_n^{\rm N}$ denote the rate constants of radiative (fluorescence) decay and nonradiative decay for the $n$th QD, respectively, and $k_{\textit{nm}}^{{\rm FRET}}$ denotes the rate constant of FRET from the $n$th QD to the $m$th QD. $h{\nu _{\rm f}}$ denotes a fluorescence photon. The rate constants are given using the fundamental physical constants as follows: $k_n^{\rm F} = {Q_n}/{\tau _n}$, $k_n^{\rm N} = ({1 - {Q_n}})/{\tau _n}$, $k_{\textit{nm}}^{{\rm FRET}} = ({3/2})({\kappa _{\textit{nm}}^2/{\tau _n}}){({{R_{\textit{nm}}}/{r_{\textit{nm}}}})^6}$, where ${Q_n},{\tau _n}$ denote the quantum yield and fluorescence lifetime for the $n$th QD, respectively. $\kappa _{\textit{nm}}^2$ is the orientation factor between the $n$th and $m$th QDs. ${R_{\textit{nm}}},{r_{\textit{nm}}}$ denote the Förster and physical distances from the $n$th QD to the $m$th QD, respectively. Note that the CTMC model defined by Eq. (1) is essentially equivalent to the single-donor model [12,13]. Finally, we define additional two-state transitions for each QD as follows:

(2a)$$S_n^* + S_m^*\mathop {\longrightarrow} \limits^{k_{\textit{nm}}^{{\rm FRET}}} {S_n} + S_m^*,$$(2b)$${S_n}\;(+ h{\nu _{\rm e}})\mathop {\longrightarrow} \limits^{k_n^{\rm E}(t)} S_n^*,$$

where $k_n^{\rm E}(t)$ denotes the rate constant of the excitation process with irradiation of time-dependent excitation light for the $n$th QD. $h{\nu _{\rm e}}$ denotes an excitation photon. The rate constant of the excitation process is given as $k_n^{\rm E}(t) = {\sigma _n}{I_{{\rm ex},n}}(t)$, where ${\sigma _n}$ denotes the collision cross section for the $n$th QD, and ${I_{{\rm ex},n}}(t)$ denotes the irradiation photon density for the $n$th QD. Equation (2a) describes an energy transfer by FRET and the subsequent Auger recombination. The Auger recombination is a nonradiative decay process from a higher energy exciton $S_m^{**}$ to the first level exciton $S_m^*$, which can be modeled as several interactions, e.g., [20]. For simplicity, we assume that this decay process is quite rapid, i.e., $S_m^{**}\mathop \to \limits^\infty S_m^*$. Finally, Eq. (2b) is the state transition due to the light-induced excitation process.

In this paper, for simplicity of theoretical analyses, we assume that the orientation factors $\kappa _{\textit{nm}}^2$ and physical distances ${r_{\textit{nm}}}$ are constant in time, i.e., the fluorophores have low anisotropies and minimal lateral motions during their excited-state lifetimes. We note that our model can also be applied to such situations for (i) any values of the orientation factor other than 2/3 commonly assumed or even dynamic ones, and (ii) diffusion of fluorophores during their excited states. However, when a much faster rotation or diffusion is compared to their excited-state lifetimes, one may need to use simpler models (see, e.g., Chapter 4 in [21], or [22] for the dynamic averaging regime, and [23] for the rapid-diffusion limit).

B. Master Equation

Considering the inflow and outflow of probability, the master equation for the CTMC model defined by Eqs. (1) and (2) is given as follows:

(3)$$\begin{split}&\frac{{\rm d}}{{{\rm d}t}}{P_{{i_1} \cdots {i_N}}}(t)\\[-2pt] &= - \sum\limits_{n = 1}^N {i_n}(k_n^{\rm F} + k_n^{\rm N}){P_{{i_1} \cdots {i_N}}}(t) \\[-2pt]&\quad {+}\sum\limits_{n = 1}^N {\bar i_n}(k_n^{\rm F} + k_n^{\rm N}){\cal S}_n^ + {P_{{i_1} \cdots {i_N}}}(t) - \sum\limits_{n,m = 1}^N {i_n}k_{\textit{nm}}^{{\rm FRET}}{P_{{i_1} \cdots {i_N}}}(t) \\[-2pt] &\quad{+}\sum\limits_{n,m = 1}^N {\bar i_n}{i_m}k_{\textit{nm}}^{{\rm FRET}}{\cal S}_n^ + {P_{{i_1} \cdots {i_N}}}(t) + \sum\limits_{n,m = 1}^N {\bar i_n}{i_m}k_{\textit{nm}}^{{\rm FRET}}{\cal S}_n^ + {\cal S}_m^ - {P_{{i_1} \cdots {i_N}}}(t) \\[-2pt]&\quad {-}\sum\limits_{n = 1}^N {\bar i_n}k_n^{\rm E}(t){P_{{i_1} \cdots {i_N}}}(t) + \sum\limits_{n = 1}^N {i_n}k_n^{\rm E}(t){\cal S}_n^ - {P_{{i_1} \cdots {i_N}}}(t),\end{split}$$

where ${\bar i_n}$ denotes the inverted binary for the $n$th QD’s state ${i_n}$, i.e., ${\bar i_n} = 1 - {i_n}$, and ${\cal S}_n^ \pm$ denotes the shift operator for states $({i_1}, \cdots ,{i_N})$, i.e., ${\cal S}_n^ \pm {P_{{i_1} \cdots {i_N}}}(t) = {P_{{i_1} \cdots {i_n} \pm 1 \cdots {i_N}}}(t)$. Note that if the state $({i_1}, \cdots ,{i_n} \pm 1, \cdots ,{i_N})$ is not proper, i.e., not in ${\{0,1\} ^N}$, ${P_{{i_1} \cdots {i_n} \pm 1 \cdots {i_N}}}(t) = 0$. In addition, we set $k_{\textit{nn}}^{{\rm FRET}} = 0$. The time-dependent fluorescence intensity $I(t)$ is expressed as

(4)$$I(t) = \sum\limits_{({i_1}, \cdots ,{i_N}) \in {{\{0,1\}}^N}} \left[{\sum\limits_{n = 1}^N {i_n}k_n^{\rm F}{P_{{i_1} \cdots {i_N}}}(t)} \right].$$

Note that Eq. (3) of our model includes spatial information through the rate constants $k_{\textit{nm}}^{{\rm FRET}}$, i.e., the network structure of QDs. In the following sections, we focus on the temporal behavior of the FRET network and analyze our model as well as the single-donor model.

C. Relation to the Single-Donor Model

Wang et al. [12,13] introduced a mathematical model for multistep FRET that assumes networks with at most one excited molecule. Namely, only the state transition rules described in Eq. (1) are considered. Therefore, considering the inflow and outflow of probability, the master equation for the single-donor model [12,13] is summarized as follows:

(5)$$\begin{split}\frac{{\rm d}}{{{\rm d}t}}{P_n}(t) &= - (k_n^{\rm F} + k_n^{\rm N}){P_n}(t) - \sum\limits_{m = 1}^N k_{\textit{nm}}^{{\rm FRET}}{P_n}(t)\\&\quad + \sum\limits_{m = 1}^N k_{\textit{mn}}^{{\rm FRET}}{P_m}(t),\end{split}$$

where ${P_n}(t)$ denotes the probability that the $n$th QD is in an excited state. Now, we will show that Eq. (3) of our model with at most one excited molecule and without excitation light, i.e., ${i_1} + \cdots + {i_N} \le 1$ and $k_n^{\rm E} = 0$, coincides with Eq. (5) of the single-donor model. To show this, assume ${i_1} + \cdots + {i_N} \le 1$ and put $P_{0 \cdots 1 \cdots 0}^{\hspace{0.82em}n}(t) = {P_n}(t)$. Then one can easily transform each term in the right-hand side of Eq. (3) as follows: ${ (\text{the first term})} = - (k_n^{\rm F} + k_n^{\rm N}){P_n}(t)$, ${(\text{the second term})} = 0$, ${(\text{the third term})} = - \sum\nolimits_{m = 1}^N k_{\textit{nm}}^{{\rm FRET}}{P_n}(t)$, ${(\text{the fourth term})} = 0$, ${ (\text{the fifth term})} = \sum\nolimits_{m = 1}^N k_{\textit{mn}}^{{\rm FRET}}{\cal S}_m^ + {\cal S}_n^ - {P_n}(t) = \sum\nolimits_{m = 1}^N k_{\textit{mn}}^{{\rm FRET}}$${P_m}(t)$, ${(\text{the sixth term})} = 0$, ${(\text{the seventh term})} = 0$.

3. RESULTS

A. Multicomponent Exponential Decay

Here, we present the fundamental temporal property of the decay process of the fluorescence intensity derived from our model with ${k^{\rm E}}(t) = 0$. That is, we show that nontrivial network-induced properties of the fluorescence intensity decay occur when multiple donors are considered. Now consider the simplest situation where the network consists of only one type of QDs, i.e., $k_n^{\rm F} = {k^{\rm F}}$, $k_n^{\rm N} = {k^{\rm N}}$, ${k^{\rm F}} + {k^{\rm N}} = 1/\tau$, $k_{\textit{nm}}^{{\rm FRET}} = k_{\textit{mn}}^{{\rm FRET}}$. First, we will show that the single-donor model (5) implies the single-exponential decay. Because the fluorescence intensity in the single-donor model (5) becomes $I(t) = {k^{\rm F}}\sum\nolimits_{n = 1}^N {P_n}(t)$, the derivative of $I(t)$ is

(6)$$\begin{split}\frac{{\rm d}}{{{\rm d}t}}I(t) &= - \frac{1}{\tau}I(t) - \sum\limits_{n,m = 1}^N k_{\textit{nm}}^{{\rm FRET}}{k^{\rm F}}{P_n}(t) \\&\quad+ \sum\limits_{n,m = 1}^N k_{\textit{mn}}^{{\rm FRET}}{k^{\rm F}}{P_m}(t).\end{split}$$

The sum of the second and third terms in the right-hand side of Eq. (6) is zero because of the symmetricity of $k_{\textit{nm}}^{{\rm FRET}}$. Therefore, the single-donor model (5) implies the single-exponential decay, i.e., $I(t) = I(0)\exp (- t/\tau)$.

Next, we will show that Eq. (3) of our model implies the multicomponent exponential decay, i.e., $I(t) = \sum\nolimits_j {\alpha _j}\exp (- t/{\tau _j})$. To show this, we define the $l$-excited states as $\{({i_1}, \cdots ,{i_N}) \in {\{0,1\} ^N}:{i_1} + \cdots + {i_N} = l\}$ and the time-dependent fluorescence intensity from the $l$-excited states as follows:

(7)$${I_l}(t) = \sum\limits_{{\scriptstyle({i_1}, \cdots ,{i_N}) \in {\{0,1\} ^N}\atop\scriptstyle({i_1} + \cdots + {i_N} = l)}} \left[{\sum\limits_{n = 1}^N {i_n}k_n^{\rm F}{P_{{i_1} \cdots {i_N}}}(t)} \right].$$

Obviously, $I(t) = \sum\nolimits_{l = 1}^N {I_l}(t)$ holds from Eqs. (4) and (7). The following expression for the all-excited-state probability ${P_{1 \cdots 1}}(t)$ can be easily derived from Eq. (3) in the case where the network consists of only one type of QDs:

(8)$$\frac{{\rm d}}{{{\rm d}t}}{P_{1 \cdots 1}}(t) = - \frac{N}{\tau}{P_{1 \cdots 1}}(t) - \left({\sum\limits_{n,m = 1}^N k_{\textit{nm}}^{{\rm FRET}}} \right){P_{1 \cdots 1}}(t).$$

Therefore, ${P_{1 \cdots 1}}(t)$ shows the single-exponential decay ${P_{1 \cdots 1}}(t) = {P_{1 \cdots 1}}(0)\exp ({- t/\tau _*^{(N)}})$, where the decay time is

(9)$$\tau _*^{(N)} = {\left({\frac{N}{\tau} + \sum\limits_{n,m = 1}^N k_{\textit{nm}}^{{\rm FRET}}} \right)^{- 1}}.$$

Because ${I_N}(t) = N{k^{\rm F}}{P_{1 \cdots 1}}(t)$, the resulting fluorescence intensity ${I_N}(t)$ also shows the single-exponential decay ${I_N}(t) = {I_N}(0)\exp ({- t/\tau _*^{(N)}})$. Similarly, the following expression for the $(N - 1)$-excited-state probability $P_{1 \cdots 0 \cdots 1}^{\hspace{0.78em}n}(t)$ can be derived from Eq. (3) by a straightforward calculation:

(10)$$\begin{split}\frac{{\rm d}}{{{\rm d}t}}P_{1 \cdots 0 \cdots 1}^{\hspace{0.78em}n}(t) &= \sum\limits_{m = 1}^N {A_{\textit{nm}}}P_{1 \cdots 0 \cdots 1}^{\hspace{0.73em}m}(t) \\&\quad+ \left({\frac{N}{\tau} + \sum\limits_{m = 1}^N k_{\textit{nm}}^{{\rm FRET}}} \right){P_{1 \cdots 1}}(t),\end{split}$$

where the elements of the matrix $A$ are

(11)$${A_{\textit{nm}}} = - \left({\frac{{N - 1}}{\tau} + \sum\limits_{{n^\prime ,m^\prime = 1\atop (n^\prime \ne n)}}^N k_{n^\prime m^\prime}^{{\rm FRET}}} \right){\delta _{\textit{nm}}} + k_{\textit{nm}}^{{\rm FRET}}(1 - {\delta _{\textit{nm}}}).$$

${\delta _{\textit{nm}}}$ denotes the Kronecker delta. The matrix $A$ is real symmetric, and hence has real eigenvalues and is diagonalizable. Let ${\lambda _1}, \ldots ,{\lambda _N}$ be the real eigenvalues (with multiplicity) of the matrix $A$. One can show that the matrix $A$ is negative definite, i.e., ${{\boldsymbol x}^T}A{\boldsymbol x} \lt 0$ for all non-zero ${\boldsymbol x} \in \mathbb{R}^N$, by a straightforward calculation and rearrangement of the terms:

(12)$$\begin{split}\sum\limits_{n,m = 1}^N\! {x_n}{A_{\textit{nm}}}{x_m} &= - \frac{{N - 1}}{\tau}\sum\limits_{n = 1}^N x_n^2 - \sum\limits_{n = 1}^N\! \left({\sum\limits_{{ n^\prime \lt m^\prime \atop (n^\prime ,m^\prime \ne n)}}^N 2k_{n^\prime m^\prime}^{{\rm FRET}}} \!\right)\!x_n^2 \\ &\quad-\sum\limits_{n \lt m} k_{\textit{nm}}^{{\rm FRET}}{({{x_n} - {x_m}} )^2}\quad ( \lt 0).\end{split}$$

Therefore, all of the eigenvalues ${\lambda _j}$ are strictly negative. Thus, one can see that the solution to Eq. (10), $P_{1 \cdots 0 \cdots 1}^{\hspace{0.78em}n}(t)$, is a linear sum of $\exp ({- t/\tau _*^{(N - 1,1)}}), \cdots ,\exp ({- t/\tau _*^{(N - 1,N)}})$ and $\exp ({- t/\tau _*^{(N)}})$, where the decay times are $\tau _*^{(N - 1,j)} = |{\lambda _j}{|^{- 1}}$ labeled in ascending order. Because ${I_{N - 1}}(t) = (N - 1){k^{\rm F}}\sum\nolimits_{n = 1}^N P_{1 \cdots 0 \cdots 1}^{\hspace{0.78em}n}(t)$, the resulting fluorescence intensity ${I_{N - 1}}(t)$ shows the multicomponent exponential decay including these exponential decay components. Finally, we will show that the fluorescence intensity ${I_1}(t)$ includes the exponential decay component $\exp (- t/\tau)$. The following expression for the 1-excited-state probability $P_{0 \cdots 1 \cdots 0}^{\hspace{0.82em}n}(t)$ can be derived from Eq. (3) by a straightforward calculation:

(13)$$\begin{split}\frac{{\rm d}}{{{\rm d}t}}P_{0 \cdots 1 \cdots 0}^{\hspace{0.82em}n}(t) &= - \frac{1}{\tau}P_{0 \cdots 1 \cdots 0}^{\hspace{0.82em}n}(t) \\ &\quad{+}\sum\limits_{{m = 1\atop(m \ne n)}}^N \left({\frac{1}{\tau} + k_{\textit{mn}}^{{\rm FRET}}} \right)P_{0 \cdots 1 \cdots 1 \cdots 0}^{\hspace{0.82em}n\hspace{0.49em}m}(t).\end{split}$$

Therefore, the 1-excited-state probability $P_{0 \cdots 1 \cdots 0}^{\hspace{0.82em}n}(t)$ includes the exponential decay $\exp (- t/\tau)$, as $P_{0 \cdots 1 \cdots 1 \cdots 0}^{\hspace{0.82em}n\hspace{0.49em}m}(t)$ goes asymptotically to zero faster than $P_{0 \cdots 1 \cdots 0}^{\hspace{0.82em}n}(t)$. Because ${I_1}(t) = {k^{\rm F}}\sum\nolimits_{n = 1}^N P_{0 \cdots 1 \cdots 0}^{\hspace{0.82em}n}(t)$, the resulting fluorescence intensity ${I_1}(t)$ also includes the exponential decay component $\exp (- t/\tau)$. Note that in the argument for Eq. (13), we used a physical insight, and hence it is not a rigorous proof. In summary, Eq. (3) of our model implies that the fluorescence intensity $I(t)$ shows the multicomponent exponential decay including at least $N + 2$ exponential decay components, i.e., $\exp ({- t/\tau _*^{(N)}})$, $\exp ({- t/\tau _*^{(N - 1,1)}}), \cdots ,\exp ({- t/\tau _*^{(N - 1,N)}})$, and $\exp ({- t/\tau})$ if ${P_{1 \cdots 1}}(0) \ne 0$ and $P_{1 \cdots 0 \cdots 1}^{\hspace{0.78em}n}(0) \ne 0\;(n = 1, \cdots ,N)$. We expect that each fluorescence intensity ${I_l}(t)$ from the $l$-excited states has potentially up to $\left({\begin{array}{*{20}{c}}N\\l\end{array}}\right)$ exponential decay components, and hence the resulting fluorescence intensity $I(t)$ has potentially up to $\sum\nolimits_{l = 1}^N \left({\begin{array}{*{20}{c}}N\\l\end{array}}\right){= 2^N} - 1$ exponential decay components. Note that the observable number of exponential decay components can be smaller if fewer QDs are initially excited. Furthermore, from specific calculations for $N = 2,3,4$, we can infer that $\tau _*^{(N)} \lt \tau _*^{(N - 1,1)} \le \cdots \le \tau _*^{(N - 1,N)} \lt \tau$, i.e., the network-induced decay times are smaller than the natural decay time.

In Fig. 1(a), we show the simulation results of the fluorescence intensity $I(t)/I(0)$ obtained from our model. In the simulation, we adopted a computationally efficient stochastic simulation algorithm, the time-dependent rejection-based stochastic simulation algorithm (tRSSA) [24], to handle the time-dependent transition rate (2b). Note that in this paper, we do not show the simulation under the presence of time-dependent excitation light, but we confirmed that our model can reproduce the frequency-domain fluorometry that uses sinusoidally modulated excitation light (see Chapter 5 in [5] for the fluorometry details), which will be shown in a future paper. One may also consider deterministic simulation algorithms such as the Euler method to solve the system of equations Eq. (3), but we did not select such deterministic simulation algorithms for the following reasons. (i) The system of equations Eq. (3) is too large (${2^N}$) to solve using deterministic simulation algorithms. (ii) We consider that a simulation that can generate sample paths of Eq. (3) is better for comparison with actual experimental results. The simulation was conducted in the following settings: QDs were located on a $50 \times 50$ lattice, and the lattice spacing was $\sqrt c \times {R_0}$ (${R_0}$: the Förster distance between QDs), where $c = 5$ to 1/5, as shown in the legend. The parameters of QDs were set to $Q = 0.40$ (quantum yield), $\tau = 19.5\;{\rm ns}$ (natural decay time), and ${R_0} = 6.18\; {\rm nm}$ (Förster distance), as assuming QD585. Furthermore, ${\kappa ^2}$ (orientation factors) were set to 2/3 assuming that our QD-experimental system is in the dynamic averaging regime for the three-dimensional spatial and orientational case (see Chapter 4 in [21], or [22] for effective kappa-squared values). In Fig. 1(a), “strong excitation” and “weak excitation” mean that initially excited QDs comprise 90% and 10% of the entire amount, respectively. For simplicity, we assumed that QDs are points without volume in the simulation. We performed ${10^4}$ independent simulation runs and averaged the results. We accumulated photons in a time interval of 0.1 ns at each time point to evaluate the fluorescence intensity. We confirmed that the simulation results show multicomponent exponential decays. To be more specific, the decays were fast in earlier times, slow in later times, and finally, with the natural decay time $\tau$, as stated above as the inference from the case of $N = 2,3,4$. Moreover, the result shows faster decays as the density of excited QDs increases or the excitation becomes stronger, as expected due to the level occupancy effect. The effect promotes the emission of the transferred and saturated energy between excited QDs through a nonradiative process such as heat dissipation, as seen from Eq. (2a). As a result, the radiative energy dissipation becomes faster as the density of excited QDs increases or the excitation becomes stronger (see also Section 4.1 in [11] for a more intuitive explanation).

Fig. 1. (a) Simulation results of the fluorescence intensity obtained using tRSSA [24]. The dots represent intensity data accumulated every 1 ns, and the solid lines represent the intensity data for every 0.1 ns. (b) Experimental results of the fluorescence intensity acquired using actual QDs. The dots and solid lines represent experimental data of the photon counts accumulated every 1 ns, and the dotted lines are fitted curves with two-component exponential functions. The common features found in (a) and (b) are as follows: (i) multicomponent exponential decay, (ii) fast decay in earlier times and slow decay in later times, and (iii) slower decay with higher dilution or lower excitation.

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B. Experimental Confirmation of the Multicomponent Exponential Decay

In Fig. 1(b), we show the experimental results of the fluorescence intensity obtained from spatially distributed single-type CdSe/ZnS QDs (NN-labs, CZ580, fluorescence wavelength: 580 nm, mass molar: 1.0 nmol/mg). We used a microspectroscope (HRS-300MS-NI-KS, TELEDYNE PRINCETON INSTRUMENTS) with a pulse laser (NR-K-PIL-515NM, Advanced Laser Diode Systems), microscope (NR-K-BX53F, OLYMPUS), and single-pixel photon counter (R-K-SPD-050-CTE, MICRO PHOTON DEVICES). The figure shows the case of three densities of QD solutions (solvent: Toluene), i.e., two-, three-, and fivefold dilutions relative to the reference concentration in which the mass concentration is 5 mg/mL. In the figure, “strong excitation” and “weak excitation” mean that the intensities of pulsed laser beams are relatively strong (intensity: $67.1 \times {10^3}\;{\rm W}/{{\rm cm}^2}$ pulse width: 77 ps) and relatively weak (intensity: $9.6 \times {10^3}\,\,{\rm W}/{{\rm cm}^2}$, pulse width: 140 ps). The solid curves in the figure are curves fitted by the two-component exponential functions. (See [11] for the details of this experiment.) One can see that in highly dense QD solutions, the fluorescence intensity decays faster, which is also shown by the simulation of our model and probably due to the level occupancy effect, as stated in the previous section.

Table 1. $\chi _R^2$ and $P$ Values for Each Number of Exponential Decay Components ($n$)^a

View Table

We checked whether the experimental results were single- or multicomponent exponential decays. The data in Fig. 1(b) were fitted with exponential functions whose numbers of decay components are from $n = 1$ to 6 by the nonlinear least-square Marquardt–Levenberg algorithm implemented in Gnuplot. (See Chapter 4 in [5] for details of this type of data analysis.) In Table 1, we show the resulting $\chi _R^2$, i.e., the reduced sum of squared errors ${\chi ^2}/({N_{{\rm data}}} - p)$ where $p$ is the number of parameters, and $P$ values for the chi-squared test. Note that when $\chi _R^2$ is close to one, the fitness is better. By evaluating the $P$ values against a significance level of 5% ($P = 0.05$), we can conclude that the experimental results do not show single-component exponential decays, which supports our model.

4. CONCLUSION

We developed a spatiotemporal model for FRET networks and uncovered its temporal characteristic behavior. We theoretically showed that our model can generate the network-induced multicomponent exponential decay of the fluorescence intensity even in the case of only one type of QDs. The experimental results using QDs confirmed the theoretical predictions. Our model can be applied to any FRET networks consisting of any fluorescent molecules and can greatly contribute to the development of possible applications, for example, temporal information processing such as reservoir computing. In future research, we will test our model in detail in the case of time-dependent excitation light and examine temporal information-processing applications using FRET networks.

Funding

JST, Core Research for Evolutional Science and Technology (CREST) (Grant Number JPMJCR18K2), Japan.

Disclosures

The authors declare no conflicts of interest.

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$n$	1	2	3	4	5	6
2-Fold Dilution, Strong Excitation ( $N_{d a t a} = 2733$ )
$χ_{R}^{2}$	1.550	1.023	1.014	1.015	1.016	1.016
$P$	$< 10^{- 4}$	0.193	0.299	0.290	0.282	0.272
3-Fold Dilution, Strong Excitation ( $N_{d a t a} = 2733$ )
$χ_{R}^{2}$	1.137	0.965	0.964	0.965	0.966	0.968
$P$	$< 10^{- 4 *}$	${0.097}^{*}$	${0.092}^{*}$	${0.096}^{*}$	${0.106}^{*}$	${0.115}^{*}$
5-Fold Dilution, Strong Excitation ( $N_{d a t a} = 2340$ )
$χ_{R}^{2}$	1.228	1.012	1.012	1.013	1.014	1.015
$P$	$< 10^{- 4}$	0.341	0.338	0.326	0.318	0.305
2-Fold Dilution, Weak Excitation ( $N_{d a t a} = 2733$ )
$χ_{R}^{2}$	1.287	1.022	1.020	1.020	1.019	1.022
$P$	$< 10^{- 4}$	0.209	0.232	0.226	0.245	0.210
3-Fold Dilution, Weak Excitation ( $N_{d a t a} = 2733$ )
$χ_{R}^{2}$	1.177	1.049	1.050	1.046	1.047	1.048
$P$	$< 10^{- 4}$	0.035	0.034	0.046	0.043	0.041
5-Fold Dilution, Weak Excitation ( $N_{d a t a} = 2340$ )
$χ_{R}^{2}$	1.206	1.154	1.154	1.156	1.157	1.158
$P$	$< 10^{- 4}$	$< 10^{- 4}$	$< 10^{- 4}$	$< 10^{- 4}$	$< 10^{- 4}$	$< 10^{- 4}$

Spatiotemporal model for FRET networks with multiple donors and acceptors: multicomponent exponential decay derived from the master equation

Abstract