Inherent stochasticity, noise and limits of detection in continuous and time-gated fluorescence systems

Nicholas H. Vitale; Arjang Hassibi; Hyongsok Tom Soh; Boris Murmann; Thomas H. Lee

doi:10.1371/journal.pone.0313949

Abstract

We present a model for the noise and inherent stochasticity of fluorescence signals in both continuous wave (CW) and time-gated (TG) conditions. When the fluorophores are subjected to an arbitrary excitation photon flux, we apply the model and compute the evolution of the probability mass function (pmf) for each quantum state comprising a fluorophore’s electronic structure, and hence the dynamics of the resulting emission photon flux. Both the ensemble and stochastic models presented in this work have been verified using Monte Carlo molecular dynamic simulations that utilize the Gillespie algorithm. The implications of the model on the design of biomolecular fluorescence detection systems are explored in three relevant numerical examples. For a given system, the quantum-limited signal-to-noise ratio (QSNR) and limits of detection are computed to demonstrate how key design tradeoffs are quantified. We find that as systems scale down to micro- and nano- dimensions, the interplay between the fluorophore’s photophysical qualities and use of CW or TG has ramifications on optimal design strategies when considering optical component selection, measurement speed, and system energy requirements. While CW systems remain a gold standard, TG systems can be leveraged to overcome cost and system complexity hurdles when paired with the appropriate fluorophore.

Citation: Vitale NH, Hassibi A, Soh HT, Murmann B, Lee TH (2024) Inherent stochasticity, noise and limits of detection in continuous and time-gated fluorescence systems. PLoS ONE 19(12): e0313949. https://doi.org/10.1371/journal.pone.0313949

Editor: Yuan-Fong Chou Chau, Universiti Brunei Darussalam, BRUNEI DARUSSALAM

Received: April 21, 2024; Accepted: November 2, 2024; Published: December 23, 2024

Copyright: © 2024 Vitale et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: We have made our code available on a public repo which can be found here: https://github.com/vitale940/plosone_code.git All relevant data are within the manuscript and its Supporting Information files.

Funding: NSF GRFP National Science Foundation Graduate Research Fellowship Program. (Grant No. DGE-1656518),https://www.nsfgrfp.org/, and Stanford-Samsung Research Initiative.

Competing interests: The authors have declared that no competing interests exist.

I. Introduction

Fluorescence spectroscopy is a fundamental analysis technique used to identify, quantify, and continually monitor molecular interactions [1] as well as biological micro- and nano-constructs [2]. In the past two decades, the widespread adoption of fluorescence-based techniques in biotechnology has enabled many detection platforms that are ubiquitous today. Notable examples are high-density microarrays [3], quantitative polymerase chain reaction (qPCR) systems [4], and whole genome DNA sequencers [5]. Furthermore, such platforms have found utility beyond pure life science research and in molecular diagnostics to, for instance, detect unique genetic sequences of viruses and bacteria [6] or oncogenic mutations within human cells [7].

Historically, the initial adoption of fluorescence spectroscopy was stifled by two major impediments. The first is related to the practicality of the fluorescence reporters (fluorophores), specifically their restricted commercial availability [8], chemical and photochemical instability [9], and the limited spectra (colors) that they could offer [10]. The second is related to the inherent complexity, bulkiness, and overall cost of the optical instrumentation necessary for fluorescence analysis (i.e., excitation source, filters and lens systems, and photodetectors) [11, 12]. Throughout the years, advancements in photochemistry and molecular engineering have gradually addressed the former. Today, there is an impressive variety of commercially available fluorophores with optimized photophysical qualities that can be readily incorporated in almost any application [13, 14]. Yet, the complexity and cost of the instrumentation is still considered a hinderance and many modern fluorescence spectroscopy systems use relatively expensive laboratory-grade equipment thus limiting the feasibility for mass deployment.

In recent years, the advent of solid-state optoelectronic devices, specifically light emitting diodes (LEDs) [15], laser diodes (LDs) [16], silicon detectors [17], and CMOS scientific cameras [18], has created a unique opportunity to decisively address the instrumentation challenges of fluorescence spectroscopy. While solid-state optoelectronic devices can replace certain bulky optical components (e.g., arc lamps, gas lasers, and cooled photomultiplier tubes) at a fraction of the cost [19], their implementation calls for rigorous designs with system architectures that are compatible with solid-state device manufacturing and development. Such design approaches become increasingly challenging when the detection system is scaled down to realize micro- and nano-sensors. To do this, one must be able to not only model the behavioral dynamics of the fluorescence signals, but also understand the inherent uncertainty and noise to identify optimal designs within the architecture design space.

In this work, we construct a mathematical model to describe the ensemble average and stochastic dynamics of fluorescence through the use of the Markov Chain Model (MCM) and Monte Carlo molecular dynamic simulation techniques. By adopting this systematic approach, we can couple the photophysical qualities of a given fluorophore with the specification of the optical sensors and involved fluorescence instrumentation to identify the limitations as well as achievable detection performances. Specifically, we apply our model to generate closed-form expressions that quantify the quantum-limited signal-to-noise (QSNR) and signal-to-noise (SNR) for both continuous-wave (CW) and time-gated (TG) fluorescence methods. Through numerical examples, we furthermore demonstrate how this framework can be used to identify and establish quantitative tradeoffs in a variety of relevant applications in modern biotechnology.

This work is organized as follows. In Section II, an overview of the quantum mechanical phenomena governing fluorescence is provided. Subsequently in Section III, we introduce a stochastic modeling framework to compute the ensemble average (expected) and stochasticity of fluorescence systems in both CW and TG conditions. These models thoroughly account for a fluorophore’s intra-molecular processes and can be employed in all detection regimes, ranging from systems comprised of many fluorophores in which the quantum noise is essentially “averaged out” to systems with a small number of fluorophores in which “the noise is the signal”. To construct these models, we first describe the system’s expected behavior by establishing a set of ordinary differential equations (ODE) which account for the core quantum mechanical processes illustrated in the Jablonski energy diagram. By analytically solving the ODEs, we derive closed-form approximations to quantify particular characteristics of the system’s expected behavior when subjected to an arbitrary photonic excitation source. These metrics include emission photon flux, quantum state occupancy distribution, system response time-constants, and number of collected photons.

In Section IV, we quantify the randomness of the system and account for the probabilistic nature of fluorescence by building a homogenous continuous-time MCM based on the lifetimes associated with each quantum state. The analytical solutions of the MCM reveal the temporal evolution of the probability mass functions (pmf) that are associated with each quantum state. Subsequently, we compute the uncertainty and noise of the total collected photons for CW and TG measurements. Moreover, we prove that the emission photon flux, unlike other classical photonic sources, is not a Poisson random process. Finally, all the predicted ensemble and stochastic behavior are computationally verified using classical molecular dynamic simulation techniques that rely on the Gillespie algorithm.

Finally, in Section V, we utilize our model in three practical examples to analyze different fluorescence-based molecular detection systems that are subject to unique design constraints such as scaling, imperfect (non-ideal) optics, and stochastic molecular interactions. The underlying objective in each example is to identify the fundamental limits of detection, compute signal-to-noise ratio (SNR) and rationalize design tradeoffs to facilitate the optimal use of design resources and successful extraction of the targeted molecular information.

II. Quantum processes of fluorescence

The unique fluorescence phenomena observed in specific molecular structures, generally referred to as a fluorophore, is a product of three quantum mechanical processes which are typically depicted using the Jablonski energy diagram (Fig 1) [20]. In the ground state, n_g, the electrons constituting the electronic structure of the fluorophore predominately reside in spin-paired orientation at the lowest energy level (S₀). However, when a photon with energy hυ_x couples to and is absorbed by the fluorophore, the excess energy promotes an electron into the excited state, n_e, which is comprised of higher energy levels (S₁,S₂ or S₃). This process is referred to as excitation. In the second process, the excited electron descends into lower energy levels and eventually relaxes back to the S₀ state through two mechanistically identical non-radiative relaxation processes, vibrational relaxation (VR) and internal conversion (IC). Both processes are endothermic and exothermic (i.e., absorb and produce heat). Alternatively, in the third process, an excited electron can emit a photon with energy of hυ_e where hυ_x > hυ_e rather than undergoing one of previous non-radiative processes when relaxing from S₁ to S₀. This process is called radiative relaxation, and the combination of all three processes are collectively known as fluorescence. It is important to note that some fluorophores exhibit the presence of triplet states [21] which contribute to the phenomena of phosphorescence, a long-lived version of fluorescence.

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Fig 1. Abbreviated Jablonski energy diagram depicting intra- and extra-molecular quantum mechanical processes involved in fluorescence.

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The states mentioned above account for intra-molecular quantum processes. However, in practice, there are extra-molecular interactions provoked by the surrounding physical environment [14]. Such interactions, like fluorescence resonance energy transfer (FRET) [22] and collision quenching [23], can create alternative relaxation paths in addition to IC and VR. Alternatively, other interactions may permanently alter the electronic structure of the fluorophore and its photochemical characteristics. A well-known example of this is photobleaching which prohibits photonic emission from the affected fluorophore altogether [24]. Because of the destructive nature, we account for photobleaching by adding a possible path in the Jablonski diagram (Fig 1) from n_e to a permanent bleached state, n_b. It is imperative to note that accounting for all possible extra-molecular interactions is a daunting task as there are many possibilities which are all environmentally dependent. Yet, it is common to lump them into non-radiating relaxation or deactivating processes provided that their impact can be quantified.

III. Expected behavior of fluorophores

A. System of ordinary differential equations

The ensemble average behavior of the fluorescence signal generated by a system comprised of N fluorophores can be described by a set of first-order ordinary differential equations (ODEs) [25]. Each ODE describes the change in the number of fluorophores in each specified quantum state (e.g., n_i denotes the number of fluorophores in state i) according to the influx and outflux between all other connected states. The general form for each ODE is (1) where m is the total number of states in the system, and r_j→i,r_i→j are the flux rates from state j to state i and visa versa. While we use flux rates in (1), we will also refer to their equivalent lifetime (half-life), τ, where τ_i = 1/r_i.

Fig 2 depicts different flux paths representing the processes illustrated in Fig 1. The corresponding system of ODEs describing the rate of change in the number of fluorophores occupying the ground (n_g), excited (n_e), and bleached (n_b) states are listed in Table 1, where n_g+n_e+n_b = N at all t. As evident, the excitation rate, r_x, radiative and non-radiative relaxation rates, r_r and r_nr, and the bleaching rate, r_b, quantify the rate at which a fluorophore enters and leaves each quantum state. Subsequently, the total number of photons, n_ph, emitted over a duration of T can be computed from the emission photon flux, F_ph, which is a function of the number of fluorophores in the excited state.

(2)

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Fig 2. Representative flux path state diagram describing the core quantum states illustrated in the Jablonski energy diagram.

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Table 1. Expected behavior system of ODEs.

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It is important to recognize that r_r,r_nr, and r_b represent the intra- and extra-molecular relaxation processes and are therefore fluorophore and environment dependent. For most commercially available fluorophores, their values are measured and reported in specific conditions (e.g., in biological buffers and solvents at specific temperatures). In Table 2, select fluorophores are listed with commonly reported figure-of-merits (FoM). One common FoM is the fluorescence lifetime, τ_l, which describes the average time the fluorophore spends in the excited state before relaxing to ground. Naturally, it is given by the reciprocal of the sum of r_r and r_nr. While r_r cannot be measured directly, it can be indirectly assessed from the fluorophore’s quantum efficiency, Φ, which specifies the probability of emitting a photon.

(3)

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Table 2. Example fluorophores.

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The rate r_x is somewhat different than other flux rates as it is not a sole function of fluorophore, and it is minimally dependent on the environment. Instead, it is proportional to the incident excitation flux, F_x, and the molar extinction coefficient, ϵ(λ_x), of the fluorophore at wavelength λ_x. A simplified formula for r_x is (4) where γ_x is a unit conversion factor and k_x is the proportionality constant that represents the coupling of the electric field of F_x with the dipole within the fluorophore [26]. The ϵ(λ_x) that is reported in literature describes the ensemble average of fluorophores with uniformly distributed dipole angles [27] in the solution, and therefore we can safely assume k_x≈1.

B. Simulation vs. Closed-form formulations

Numerically simulating the ODEs of Table 1 can be informative as we can find the dynamic response of the system to any arbitrary F_x. More importantly, we can compute F_ph and n_ph for both continuous-wave (CW) or time-gated (TG) measurements. Both CW and TG provide different advantages depending on the application at hand. For example, CW is widely used within biosensor applications whereas TG is more prevalent in fluorescence imaging applications. In CW systems, n_ph is collected by measuring F_ph for T_int seconds under a constant F_x. On the other hand, in TG systems, n_ph is collected by measuring F_ph starting at ΔT_D seconds after F_x is turned off, for a duration of T_int seconds. Because n_ph is typically small following one TG measurement, it is common to accumulate n_ph over K independent measurements in TG systems.

Fig 3 shows a transient simulation of [Ru(bpy)₃]Br₂ fluorophores (N = 100,000) that are subjected to a finite-time excitation pulse (see Fig 3A). As mentioned previously, fluorophores can experience bleaching which results in irreversible damage. To highlight the impact of this phenomenon, we simulate both CW and TG measurements under two scenarios. In one case, the bleaching is negligible (r_b∼ 0), while in the other, bleaching is significant (r_b = 6.66×10⁴ s^-1). For each bleaching rate, the observed F_ph (Fig 3B), quantum state occupancy levels (Fig 3C and 3E), and total number of photons captured (Fig 3F and 3G) are plotted.

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Fig 3. Simulation of the expected behavior for [Ru(bpy)₃]Br₂ fluorophores (N = 100,000) subjected to a finite-time excitation pulse in two different bleaching conditions.

For CW, T_int = 20μs, whereas for TG, T_int = 4.75μs and ΔT_D = 250ns.

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As evident, the fluorescence system is inherently non-linear with dynamics that are function of F_x. While we can set the rise time of F_x to be extremely fast, the response of the fluorophore in terms of F_ph is slower and dominated by its lifetime with a finite delay before the emission flux reaches steady state. If we consider bleaching to be negligible, then the system finds a steady-state solution resulting in a constant F_ph. However, with bleaching there is no steady state F_ph and there is a permanent decline in ground and excited states. In both cases, the n_ph captured under CW linearly increases over the duration of T_int as F_ph is approximately constant during the observation period. On the other hand, the n_ph captured under TG exhibits an exponential increase over the duration of T_int as F_ph follows the exponential relaxation of the excited fluorophores back to the ground state.

In addition to numerically solving the ODEs, we can analytically solve the ODEs in Table 1 under the assumption of r_b∼0. This is a practical approximation in many situations where the excitation power and/or exposure time is low, which is common in many detection platforms. We provide numerical benchmarks for the fluorophores listed Table 2 and specific closed-form solutions including n_ph in Table 3. It is important to note that the formulation for n_ph under CW is in its approximate form which applies for systems where T_int is much greater than the rise-time time constant of F_ph.

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Table 3. Closed-form formulations of ensemble average behavior.

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IV. Stochastic behavior of fluorophores

A. Deriving Probability Mass Function (PMF)

To quantify the stochasticity of F_ph, one needs to consider the lifetimes associated with each quantum state of a fluorophore. Since it is accurate to claim that intra-molecular transitions and most inter-molecular collision events are fundamentally memoryless processes [28], we model the sequence of transitions between quantum states with a homogenous continuous-time Markov Chain Model (MCM) [29]. Each state in the MCM is discrete and one can compute the probability of occupying a state at a time, t. Specifically, the probability of occupancy in state i is denoted by Pr{S(t) = i}, where for a system comprised of m states. Fig 4 depicts the MCM of the example fluorophore system described in Fig 2, noting m = 3.

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Fig 4. Corresponding Markov Chain Model representing the probability of quantum state transition over an infinitesimal time step.

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To ultimately compute the probability of being in a quantum state at any t, we first define the transition probability from state i to j,p_i→j, in an infinitesimal period of time, δt, by (5) which has the following relationship with the flux rate r_i→j (6)

By expanding (6) for all states, we create the transition probability matrix, P_m×m, of the fluorophore system as (7) where λ_i is the total outflux rate for state i, described by . Now, if vector S_m×1 describes the probability mass function (pmf) of states, i.e., its entry j is Pr{S(t) = j}, then the temporal evolution of S_m×1 for every δt is (8) in which I is an identity matrix, whereas Q, conventionally referred to as the generator matrix, is (9)

Subsequently, (8) can be rearranged to (10)

By taking the limit of δt→0, (10) converges to the differential equation that describes the dynamics of S (referred to as the forward Chapman-Kolmogorov Eq [29]) (11)

There are three significances for (11) that are important within the context of this work. The first is that if the probability mass function (pmf) for the states at t = 0 is S(0), then S(t) at any t≥0 can be computed by (12)

The second is that, if that for any N number of fluorophores with pmf of S_N(t), we can have S_N(t) = NS(t). This relationship assumes that fluorophores are statistically independent which is valid for most practical applications in which the concentration of fluorophores is small, i.e., fluorophore-fluorophore interactions are rare. Third, all the parameters required to assemble Q can be derived by the macroscopic flux path rates depicted in Fig 2. Specifically, the representative Q matrix of system in Fig 2, is (13)

It is equally critical to note that the PMF given by S(t) simultaneously allows one to compute the uncertainty and noise of F_ph and subsequently n_ph. However, the validity of S(t) given by (12) is based on Q remaining constant to obey the assumption of homogeneity in the MCM [30]. Specifically, any system parameter that forces Q to be a function of time, like a change in F_x, means that the solution to S(t) may not be given exactly by (12). Consequently, the resulting uncertainty and noise of the system must be analyzed separately for CW and TG measurements as F_x inherently changes with time.

B. Emission fluctuations in CW

Let’s first consider a system in which we want measure F_ph subject to a constant F_x, assuming steady state conditions, and negligible bleaching. Here, our main objective is to compute the inherent fluctuations of F_ph to estimate the variation (and uncertainty) in n_ph for a given integration time, T_int.

Classically, photons emitted by ideal light sources are known to follow a Poisson random process with emission inter-arrival times that are characterized by an exponential probability density function (pdf) [31]. Nevertheless, this assumption does not apply to a fluorescence system that follows the MCM given in Section IV. A. Therefore, to formulate n_ph, we first need to formulate the pdf of the photon emission inter-arrival time generated by a single fluorophore.

According to the MCM, the total time it takes for a fluorophore to be excited and subsequently return to its ground state, regardless of photon emission, is the sum of two sequential independent events (i.e., random variables), specifically excitation and relaxation. We represent this total time by the random variable t_θ such that (14) where t_x and t_l are exponentially distributed random variables and represent the sojourn times spent in the ground and excited states, respectively. Since t_x and t_l are statistically independent, we can define the pdf of t_θ as a hypoexponential distribution [32], φ_θ(t), given by (15)

It is important to recognize that while t_θ represents the inter-arrival time between excitation-relaxation events, it does not account for photon emission. When an excited fluorophore relaxes to its ground state, it has a probability p of emitting a photon equal to the quantum efficiency, Φ, which was dervied earlier in (3) as the ratio between r_r and r_l. Now by using (3) and (15), we can formulate φ_ph(t), the pdf of the inter-arrival time between photon emissions by (16) where ψ_m is (17)

It can be shown that φ_ph(t) can be simplified into another hypoexponential pdf in the form of (18) where λ₁,λ₂ = 1/2(r_x+r_l±γ) and .

By applying (18), we can calculate the mean and variance of F_ph and subsequently the variations of the observed n_ph. If we assume that T_int is much larger than the mean inter-arrival time, 〈φ_ph(t)〉, we can apply the elementary renewal theorem [33] to calculate 〈F_ph〉, the average emission photon flux (19) and , the variance of the emission photon flux (20)

Consequently, 〈n_ph〉, the average of n_ph, becomes (21) and , the variance of n_ph (22)

The formulations of (21) and (22) demonstrate that the ratio between the excitation and relaxation rate, as well as the fluorophore quantum efficiency, strongly influences the observed statistics. For example, if one rate were to dominate relative to the other, φ_ph(t) would converge to an exponential distribution and result in statistics that follow a Poisson process, i.e., . Yet in most practical systems, fluorophores are excited sufficiently enough that a distinguishable F_ph is ellicited. As a result, the excitation and relaxation rates tend to compete, and the observed statistics generally adhere to . More importantly, this statistical relationship demonstrates that the fluorescence phenomenon follows a sub-Poisson random process.

It is also logical to evaluate the power spectral density (PSD) of F_ph to better understand the spectral characteristics of the fluctuations. To do this, we first model F_ph by x(t), a random pulse train with t_n denoting the arrival time of its n^th pulse. The Fourier transform, X(jω), of this pulse train becomes (23) where G(jω) is the Fourier transform of the pulse shape. For the stochastic signal x(t), the corresponding single-sided PSD is (24)

To take the expectation of the double summation, we introduce the characteristic function, ϕ(jω), which is the Fourier transform of φ_ph(t) (25)

By substituting (25) into (24), it can be shown that the closed-form single-sided PSD of x(t) is given by (26) where v is the mean pulse rate of x(t). If we assume that the pulse shape of x(t) follows a Dirac-delta function, we can explicitly write (26) in terms of F_ph as (26)

Again, assuming statistical independence of each fluorophore, the corresponding PSD of a system comprised of N fluorophores is given by NS(ω).

Fig 5 depicts the PSD spectrum for an arbitrary fluorophore. The shape of the resulting PSD (ignoring the DC component), exhibits two distinct regimes which are dependent on the ratio between r_x and r_l as mentioned previously. At low frequencies, ω→0, the spectrum follows a sub-Poisson process with a spectral density, i.e., magnitude, equal to . However, as the frequency increases, the spectral density begins to rise and eventually converges to 2〈F_ph〉 which is characteristic of a Poisson process. This transition occurs around 10 times the frequency location obtained from the numerator of (26) which is given by f = (λ₁+λ₂)/2π. Since T_int maps to the frequency f = 1/2T_int, the high frequency behavior of the PSD demonstrates that for a small enough T_int the probability of seeing multiple photons, and thus any correlation between inter-arrival times, diminishes and results in random process that appears Poisson. However, practical applications operate with a T_int that resides in the sub-Poisson regime.

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Fig 5. Plot of the predicted power spectral density of fluorescence for an arbitrary fluorophore.

The spectrum is divided into its sub-Poisson and Poisson regions by the transition frequency located at f_c.

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C. Emission fluctuations in TG

Now, let’s consider a TG system with negligible bleaching where we aim to measure F_ph approximately ΔT_D seconds after F_x is shutoff over the duration T_int. Our objective is to predict the variance of the total n_ph resulting from the accumulation of K independent TG measurements. Here we make the critical assumption that the on and off duration of F_x is such that in each phase, the distribution of fluorophores in the ground and excited states reach their steady state values before a measurement is made.

To formulate the variance of n_ph, we first start by computing the probability that a photon is emitted in the time window [ΔT_D,T_int] given that there is exactly one excited fluorophore immediately after F_x turns off. Subsequently, we can apply the MCM where Q is modified with r_x,r_b = 0 and S(0) is initialized with one fluorophore in the excited state. It can be shown that the probability of photon emission, p, is the integral of the excited state’s pmf given by S(t) over the time [ΔT_D,T_int+ΔT_D] (27)

Now, while (27) represents the probability that a photon is emitted given there is one excited fluorophore, we must account for the fact that every time a measurement is made it is not guaranteed an excited fluorophore is present. As a result, the variance of n_ph is a product of both the uncertainty in n_e and the probability of photon emission.

Assuming that the system is in steady state before F_x shuts off, we can leverage the MCM to formulate 〈n_e〉, the probability of there being an excited fluorophore (28) and , the variance of n_e (29)

By utilizing (27)–(29), it can be shown through the Burgess Variance Theorem [34] that the mean and variance of n_ph collected over K measurements is given by (30) (31) where, again, the statistics of n_ph for a system comprised of N fluorophores are given by N〈n_ph〉 and .

D. Molecular dynamic simulation of fluorescence systems

To verify the formulations predicting the ensemble average and stochastic behavior of a fluorescence system, we conduct Monte Carlo molecular dynamic simulations by employing the Gillespie Algorithm (GA). The GA is widely used to simulate exact stochastic trajectories of chemical and biochemical reactions and their participating reactants across a large range of concentrations. In a system where there are i different reaction types, the GA determines what reaction, c_i, occurred at time t and then generates the time to the next reaction, t+τ, by sampling the probability distribution formed by each reaction’s propensity function, a_i(x) [35]. Depending on the type of reaction that occurred, the GA tracks the state of each reactant through a corresponding state update vector, u_i.

In Table 4, we associate each reaction, propensity function, and corresponding state update vector with the individual macroscopic flux paths and quantum states of the fluorescence system depicted in Fig 2. Based on the simulation parameters of Table 4, Figs 6 and 7 depict the result of 10,000 GA simulation trials of [Ru(bpy)₃]Br₂ fluorophores (N = 100) subjected to the same excitation photon flux intensity illustrated in Fig 3 for both CW and TG measurements in the scenario where bleaching is not considered.

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Fig 6.

Results of the quantum state occupancy in the exited (A) and ground (B) states and evolution of its pmf over time (C, D) simulated by the Gillespie algorithm (subset shown is 5 trials) compared to that predicted by the Markov model for [Ru(bpy)₃]Br₂ fluorophores (N = 100) subjected to the same excitation photon flux as in Fig 3.

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Fig 7.

Results of the emission photon flux (A) and cumulative emitted photons (B, C) simulated by the Gillespie algorithm (subset shown is 10 trials) compared to that predicted by the Markov model for [Ru(bpy)₃]Br₂ fluorophores (N = 100) subjected to the same excitation photon flux as in Fig 3. For CW, T_int = 18μs, whereas for TG, T_int = 3.75 μs and ΔT_D = 250ns.

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Table 4. Gillespie algorithm simulation parameters.

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In Fig 6, we depict the stochastic trajectories of the ground (Fig 6A) and excited (Fig 6B) states generated by the GA. The average of the GA trials and the trajectory predicted by the MCM are overlaid to demonstrate the validity of the MCM. Initially, in the first microsecond, the fluctuation of n_e is relatively small due to the fluorophore’s lifetime being on the order of this observation period. For fluorophores with small τ_l, the observed fluctuation would increase more rapidly as their response time to an excitation signal is enhanced. After a sufficient number of time-constants have elapsed (~5τ_l), the system reaches its steady state. At that point, the fluctuation reaches its maximal value as the average number fluorophores residing in n_e at any given time is approximately constant. In steady state, the fluctuation of n_e scales directly with the intensity of F_x as pointed out in Section IV.B. Once the excitation pulse is turned off, the fluctuation of n_e begins to slowly taper as the number of fluorophores decreases at an exponential rate defined by τ_l.

In Fig 6C and 6D, histograms of the number of fluorophores in the excited and ground states are plotted at five selected time points during the rising and falling edges of the excitation pulse as well as steady state. These histograms represent the simulated pmf of the quantum states. We overlay the pmf predicted by the MCM and it can be observed that all the simulated pmfs follows the distribution predicted by the MCM at each time point. This further validates that the MCM correctly predicts the temporal evolution of the pmf for a fluorescence system and demonstrates that the GA is a suitable tool for simulating fluorophore populations.

In addition to tracking the quantum state occupancies, F_ph and the subsequent statistics of n_ph for CW and TG can be extracted from the photon emission timing information generated by the GA simulation. In Fig 7A, the simulated emission photon flux agrees well with the trajectory predicted by the MCM. On the other hand, Fig 7B and 7C depict the simulated trajectories of n_ph under CW and TG conditions, respectively. As expected, CW generates more n_ph on average due to the steady F_ph and longer T_int. Alternatively, TG generates less n_ph due to the measurement window being placed during the exponential decay of F_ph. It should be noted that the fluctuation observed in F_ph may further increase if subjected to a “noisy” F_x, which is not considered in this simulation.

Finally, we further confirm the validity of the proposed MCM and GA method by constructing the PSD via the simulated photon emission times. Fig 8, shows the PSD generated by the GA and that derived in Section IV.B. As predicted, the simulated PSD exhibits the transition from sub-Poisson to Poisson behavior around the center frequency determined by the values of F_x,τ_l, and Φ. More importantly, the observation of the predicted PSD via stochastic numerical simulation validates the hand-derivation of the PSD presented in (26). The agreement between the simulated results of Figs 6–8, and the predicted expected and stochastic behavior computationally validates the proposed modeling framework for fluorescence systems.

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Fig 8. Results of power spectral density simulated by the Gillespie algorithm compared to that predicted by the Markov model for [Ru(bpy)₃]Br₂ fluorophores (N = 100) subjected to the same excitation photon flux as in Fig 3.

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V. Application to fluorescence system design

A. Scaling impacts on SNR and measurement speed

In many fluorescence systems, such as DNA microarray assays [36] and next-generation sequencing (NGS), hundreds of thousands to millions of fluorescently labeled spots are imaged on a surface. These systems, which commonly employ CW, encounter a hard tradeoff between the speed at which surfaces can be imaged and signal-to-noise ratio (SNR) that can be achieved. Here, the SNR is defined as the ratio of signal intensity to measured background noise intensity. Many systems aim to achieve minimum SNRs of > 6dB while the maximum SNR is set by either the well-capacity of the imager (typically > 40dB), or maximum image acquisition time [37, 38].

In recent years, the average size of a spot has isomorphically scaled to achieve higher sample density [39]. However, to image increasingly smaller spots, the accompanying imaging hardware must compensate to maintain a specified target SNR. This is often manifested by adjusting the imaging optics to ensure that each spot sample maps to a certain M×M region of pixels on the imager (see Fig 9). Consequently, this optical adjustment constrains the field of view and can result in an overall longer time to image a surface of a given size.

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Fig 9. Pictorial representation of isomorphically scaling down spots within a fixed field of view and corresponding imaging configuration.

https://doi.org/10.1371/journal.pone.0313949.g009

In order to study the SNR tradeoffs, we first analyze the quantum-limited SNR (QSNR) which is defined as the SNR in the absence of any extrinsic transduction or background noise, normalized to the total array image acquisition time, T_C [40]. The QNSR establishes the upper bound on the achievable SNR under theoretically perfect operating conditions. Fig 10 plots the QSNR for this CW system for FITC and [Ru(bpy)₃]Br₂ fluorophores. The corresponding simulation conditions are outlined in Table 4. Additionally, we provide an analytical expression for the QSNR in Table 5. The achievable QNSR within one second of acquisition time is computed for different isomorphic spot scaling ratios, α. It can be observed that the QSNR decreases with decreasing α. As one attempts to image more spots on a given surface, the time that the imager can spend per spot decreases with T_cα/2. Unsurprisingly, less photons are captured per spot thus negatively impacting the QSNR. To overcome this limitation, one must either increase the number of fluorophores within each spot, N, or T_C.

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Fig 10. The quantum-limited SNR, normalized a total image acquisition time of 1s, is plotted as a function of spot scaling versus the number of fluorophores contained with the spot area for [Ru(bpy)₃]Br₂ and FITC fluorophores.

https://doi.org/10.1371/journal.pone.0313949.g010

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Table 5. Closed-form formulations for signal-to-noise ratios.

https://doi.org/10.1371/journal.pone.0313949.t005

It is desirable to operate at the QSNR limit to achieve high dynamic range (DR) which directly trades off with the imaging acquisition time as observed in Fig 10. In practical system implementations the achievable SNR and DR can be significantly constrained due to extrinsic noise and signal transduction inefficiencies [37, 40]. For example, one primary noise source emanates from the imager’s transduction noise which is a combination of the electronic read noise and dark current leakage. Another prominent noise source arises from the presence of the excitation source which introduces background noise as imagers cannot distinguish excitation-emission wavelength differences with high selectivity. To combat background interference, precious imaging optics with high optical densities (OD) are commonly employed to reject indecent photons outside of the targeted fluorophore emission wavelength band [41, 42]. These optical components are often bulky, expensive, and complex and thus it is crucial that the right filter OD is selected for the application at hand. Finally, as the complexity of the optical components (e.g., coupling guides, focusing lenses) increase, the amount of photons coupled to the imaging pixels decreases substantially and can often be < 25% at state-of-the-art.

To demonstrate the impact of optical components, specifically filter OD, and extrinsic noise sources on the SNR (see formulation in Table 5), Fig 11 depicts the CW SNR for a fixed population of [Ru(bpy)₃]Br₂ fluorophores subject to different filter OD magnitudes. The simulation specifications are listed in Table 7, and the same imaging performance parameters listed in Table 6 are used. The SNR curve exhibits two distinct regions. Initially, for low filter OD values, the SNR decays by 10 dB per decade with decreasing filter OD as the influence of the background noise competes with the measured n_ph. As the filter OD is increased to higher values (> OD 6), the SNR begins to flatten and indicates that the background noise contribution is negligible, i.e., background-free. If the imaging hardware contributes negligible noise, adequate signal averaging is employed, and a large enough integration time is achieved, CW systems can produce measured SNRs that approach the QSNR limit.

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Fig 11. The impact of extrinsic noise sources on the measured SNR is plotted for both CW and TG against varying optical filter magnitudes.

For TG, four different scenarios are plotted where the ratio between ΔT_D and τ_s is increased to observe enhanced background interference rejection.

https://doi.org/10.1371/journal.pone.0313949.g011

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Table 6. Imaging system performance specifications for example applications.

https://doi.org/10.1371/journal.pone.0313949.t006

On the other hand, it is common to employ TG over CW as it brings different advantages to the design space when considering proper selection of filter OD. Fig 11 also depicts the SNR for a TG system with different ΔT_D (grey lines) subject to the same filter ODs used in the CW system. It should be noted that the number of TG measurements, K, in these examples is constrained by T_C. At first glance, the magnitude of the background free SNR of a TG system is almost always lower than that of the CW system primarily because TG results in lower n_ph. Additionally, because of the smaller n_ph, one may have to accumulate more images than expected to acquire a discernable signal level otherwise the limit of detection could be compromised. Moreover, the complexity of a TG system can surpass that of CW as the lifetime of the fluorophore decreases since it requires high-bandwidth processing equipment. Despite these setbacks, a major advantage TG offers is that it extends the background free region over substantially lower filter OD (< OD 4) values thus increasing its DR in comparison to CW. This can be advantageous when attempting to physically scale down the size of an imaging system or operating with low-well capacity imagers.

B. Excitation power requirements with non-ideal optics

While large-scale imaging systems may benefit from high precision optics, expensive scientific cameras, and relatively unlimited power, there are many instances where this is not always the case. For instance, the deployment of point-of-care (POC), implantable, or single-use systems tend to have stringent limits on cost, space, and available power supplies. Consequently, these systems encounter distinct tradeoffs between the achievable minimum number of detectable fluorophores, referred to as the minimum detection limit (MDL) [43, 44], and the required input excitation power when employing CW or TG. One practical example of a resource constrained system is POC real-time quantitative polymerase chain reaction (RT-qPCR) which is commonly employed for detection of viruses and/or bacteria [45]. Fig 12 depicts how the fluorescence is generated using a fluorogenic probe [46] (TaqMan probe).

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Fig 12.

(A) Pictorial representation of RT-qPCR process with the fluorogenic TaqMan probe where fluorescence is generated upon successful extension of primers and cleavage of probe complex. (B) Example of typical RT-qPCR fluorescence measurements as a function of the amplification cycle number and concentration of fluorophores present.

https://doi.org/10.1371/journal.pone.0313949.g012

Fig 13 presents the SNR, normalized to the measurement time, T_c, versus average input excitation power for both CW and TG where we define the MDL to be when the SNR = 0. Again, the number of TG measurements, K, in these examples is constrained by T_C. The simulation is performed for a fixed population of Eu³⁺L₂ fluorophores with the same general simulation specifications outlined in Table 7, however the decay time time-constant of the excitation source is increased to 500 ns to represent a leaky source as one might encounter with an LED or other non-laser source. The SNR for the CW system is plotted for filter OD 6–8 whereas the SNR for the TG system is plotted with a fixed OD 4 but with varying ΔT_D. Over the range of excitation power, CW more readily achieves the specified MDL level with lower excitation power granted that it has a sufficiently high filter OD (> OD 8). In reality, many optical filters can suffer degradation to their OD due to material imperfections or environmental influences like high optical scattering. Therefore, it is advisable to consider CW performance under less-than-ideal OD conditions (< OD 6) which ultimately results in having to burn more excitation power to meet the MDL. However, a TG system, even with an OD 4, can meet with the specified MDL level with significantly less excitation power than that of a CW system with an OD 6 filter. In fact, with an appropriate ΔT_D and T_int, the TG system can outperform a CW system equipped with an OD 8 filter at moderate to high excitation power levels.

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Fig 13. Predicted SNR normalized to the acquisition time of 1s versus input excitation power for CW and TG systems.

In the CW case, three scenarios are considered where the utilized optical filter has an OD > 6. Conversely, in the TG case, the utilized optical filter is fixed at OD 4 while the ratio between ΔT_D and τ_s is swept.

https://doi.org/10.1371/journal.pone.0313949.g013

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Table 7. Simulation specifications for figure.

https://doi.org/10.1371/journal.pone.0313949.t007

C. Stochastic modulation through molecular interactions

In many applications, such as DNA hybridization assays [47], the observed fluorescence is modulated by external molecular interactions such as binding. Therefore, the spectral characteristics of the fluctuations in F_ph will be impacted. To study how the PSD derived in Section IV.B changes as a function of molecular interaction kinetics, let’s consider a simple system of a single-strand DNA (ssDNA) tagged with a fluorophore-quencher pair. As depicted in Fig 14, a ssDNA fragment will constantly change conformation between its unquenched (random coil) and quenched (hairpin) states. To express the influence of hairpin formation on the PSD of F_ph, we first need to formulate the PSD describing the fluctuation of ssDNA fragments in their unquenched state.

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Fig 14. Depiction of the two possible quenched (hairpin) and unquenched (random coil) conformational states ssDNA can switch between at equilibrium.

https://doi.org/10.1371/journal.pone.0313949.g014

Let’s consider a system at equilibrium where the concentration of ssDNA strands, [X₀], is equal to the sum of the concentration of unquenched strands, [X_S1], and quenched states, [X_S2]. The rate of change in [X_S1] can be expressed by the rates at which an strands forms or loses it hairpin structure denoted as k_on and k_off, respectively (32) (33)

Therefore, the equilibrium concentration of unquenched strands, , can be expressed in terms of [X₀] (34)

With (33) and (34), it can be shown that the impulse response, h(t), of [X_S1] when subjected to a small change in [X_S1] is given by (35)

Consequently, we can apply Carson’s Theorem [48] to formulate the PSD of [X_S1] as (36)

Now, because quenching due to hairpin formation can be treated as a linear process, the resulting PSD of the modulated F_ph is given by the multiplication of (36) with the PSD of F_ph dervied in (26). Given there are N ssDNA strands in the solution the modulated PSD is given (37)

Fig 15 shows a simulation of (37) for varying concentrations of ssDNA with different temperatures. The simulation specifications and reaction kinetics are outlined in Table 8. The resulting PSD profile follows a Lorentzian profile with a spectral density proportional to the equilibrium value of unquenched ssDNA fragments and a 3dB corner frequency determined by (k_on+k_off)/2π. Moreover, the presence of binding imposes a bandwidth limit to the spectral characteristics of the PSD. This bandwidth influences the observed noise and appropriate integration time and thus should be considered when predicting the system performance metrics mentioned in the prior two examples.

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Fig 15. Predicted power spectral density of fluorescence under the influence of molecular binding kinetics for system with 1 nM and 100 nM of ssDNA against temperature.

https://doi.org/10.1371/journal.pone.0313949.g015

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Table 8. Simulation parameters of hairpin formation kinetics.

https://doi.org/10.1371/journal.pone.0313949.t008

It is imperative to recognize that the resulting PSD in (37) relies on the assumption of linearity of the quenching process. In most molecular beacon systems, such as the depicted ssDNA hairpin example, the dominant quenching process can be explained by static quenching which is exactly described by the complex rate formation in (32) [50]. However, in some systems, the quenching can be a combination of both static and dynamic quenching processes. We assume dynamic to be negligible here in this example; yet the presence of dynamic can introduce inverse quenching dependence as temperature and concentration are increased [14]. Thus, the overall quenching relationship might deviate from the linear case and lead to a more complex PSD than that presented in (37).

VI. Conclusion

Fluorescence systems can be broken down into their ensemble average and stochastic components which are critical in determining the performance of optical biosensor systems. We present a stochastic modeling framework using ODEs and homogenous continuous-time Markov Chains, characterized by measurable photophysical properties of a fluorophore, and applied it to CW and TG measurements. We then distilled closed-form formulations to compute various expected performance metrics without the need for direct numerical simulation. Additionally, we employed the Markov chain to predict the fluctuations in emission photon flux under CW and TG measurements. Notably, we proved that fluctuations in the emission photon flux do not follow a Poisson random process and derived their spectral characteristics. Finally, we computationally verified the validity of the proposed modeling framework through Monte Carlo molecular dynamic simulations of a fluorescence system with the use of the Gillespie Algorithm.

The models presented throughout this work were used to identify system performance metrics such as quantum and extrinsically limited signal-to-noise ratios as well as detection limits for a given system. Furthermore, we elucidated core design tradeoffs involved in three different biosensing applications where we explored the influence of system scaling, imperfect optics, power constraints, and molecular interactions. In general, we observed that CW systems are superior in their limits of detection (i.e., high SNR), imaging speed, excitation power requirements, and ability to approach quantum-limited performance given adequate imaging hardware performance. However, these benefits all come at the price of a high OD filter which may not be feasible either due to cost/space constraints or from unavoidable complications imposed by material imperfections or environmental conditions. On the other hand, TG systems can substantially relax the filter OD requirement while still maintaining comparable SNR, acquisition speed, and required excitation power. In some cases, TG even outperforms CW systems outfitted with high OD optics indicating that TG systems should be carefully considered even if one has access to the correct optics. A critical consideration relates to the minimum number of TG measurements required to digitize a discernable reading which may not be possible depending on the fluorophore’s relaxation lifetime and the total time allotted to make the measurements. While it is crucial to select the proper measurement modality for the application at hand, it is even more important to judiciously select a fluorophore with photochemical qualities that complements the specified system metrics such as total acquisition time, power consumption, and or SNR.

While the system performance metrics did not consider molecular interactions, we demonstrated that these interactions do impact the observed fluctuation in the emission flux. Notably, we show that molecular binding band-limits the spectral characteristics of the emission flux which should be equally considered when predicting system performance or setting rational performance metrics.

Together, these insights allow one to better understand how system performance metrics can be prioritized and set within a variety of different biosensing applications. More importantly, the proposed modeling framework fuses the phenomena of fluorescence with the conventional design space by quantitatively identifying how certain photophysical characteristics impact the observed signal behavior. As one collects more information of the system’s behavior in a variety of environments, the model can progressively improve its predictive power and result in a deeper understanding of the fundamental limitations governing fluorescence detection.

Acknowledgments

The authors would like to thank Yasser Gidi and Vladimir Kesler for their technical discussions.

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Figures

Abstract

I. Introduction

II. Quantum processes of fluorescence

III. Expected behavior of fluorophores

A. System of ordinary differential equations

B. Simulation vs. Closed-form formulations

IV. Stochastic behavior of fluorophores

A. Deriving Probability Mass Function (PMF)

B. Emission fluctuations in CW

C. Emission fluctuations in TG

D. Molecular dynamic simulation of fluorescence systems

V. Application to fluorescence system design

A. Scaling impacts on SNR and measurement speed

B. Excitation power requirements with non-ideal optics

C. Stochastic modulation through molecular interactions

VI. Conclusion

Acknowledgments

References