Dissect interaction kinetics through single-molecule interaction simulation

Abstract

The ability to extract kinetic interaction parameters from single-molecule fluorescence resonance energy transfer trajectories without the need for solving complex single-molecule differential equations has the potential to address some of the critical biophysical questions. Here, we provide a single-molecule interaction simulation (SMIS) tool to give the expected dwell-time distributions and relative populations of each FRET states based on the assigned kinetic model and to dissect kinetic interaction parameters from single-molecule FRET trajectories. The method provides the expected dwell-time distributions, averaged transition rates, and relative populations of each FRET states based on the assigned kinetic model. Comparing extensive simulated data with experimental data enables the quantification of the kinetic rate and equilibrium constants. We have also demonstrated that SMIS is useful to quantify the interaction kinetic rate constants that were originally unobtainable through the traditional single-molecule analytical solution approach.

Graphical Abstract

Single-molecule fluorescence resonance energy transfer (or smFRET) is a powerful biophysical technique to dissect stochastic interactions of protein-substrate, ^[1–3] protein-protein, ^[3–5] and protein-DNA complexes,^[6–10] as well as the folding behaviors of DNA^[11,12], RNA^[13], and protein^[14–17]. smFRET typically quantifies the number of FRET states and the waiting times of specific state transitioning to other states. These pieces of information provide microscopic insight into the interaction kinetics. Many methodologies have been created to reliably distilled out the FRET state identifications and transitions^[18–22]. These approaches enable the experimental determination of the probability density function of waiting times PDF(τ), which is typically obtained from the histogram of microscopic dwell times. It represents the overall lifetime of species within a FRET state. Potentially, analyzing the PDF(τ) enables the quantification of rate constants^[23], which reveals the mechanism of the interaction process.

Extracting out the interaction rate constants from the PDF(τ) is typically achieved via assigning a kinetical model and fitting the PDF(τ) by the analytical probability density function f(τ). For example, Jain’s laboratory studied the kinetics of self-assembled monolayer formation on individual nanoparticles and extracted the formation rate constants of self-assembled thiol monolayer^[24]. Chen’s laboratory discovered how transcription factors regulate transcription process in vitro and in cells^[25–27]. They also provided a detailed single-molecule kinetic theory for heterogeneous catalysis^[28–30]. Scherer’s laboratory revealed the kinetics and mechanism of the physical passing of particles in an optical ring trap with an adjustable driving force^[31]. Landes’ laboratory reported the multistep desorption kinetics of α-lactalbumin from Nylon, which provided insight into the mechanisms driving protein-polymer interactions^[32].

However, the determination of the kinetical model and the derivation of analytical solutions for kinetic parameters may not always be straightforward. In general, the process begins with the selection of the kinetic model based on the features of observed FRET states. This kinetic model generates single-molecule rate equations. With sufficient boundary conditions, one can solve the rate equations to obtain the analytical probability density function f(τ) for transitions between different FRET states, as well as the analytical solutions of relative populations of each FRET state. Unfortunately, having sufficient boundary conditions could be difficult due to too many species co-existing in the same FRET or the kinetic model containing repeated differential equations. The difficulties getting the f(τ) and relative populations of each FRET state hinder the mechanistic study of interaction processes. One possible alternative way to address this issue is to compare the experimental PDF(τ) with the corresponding simulated results of various kinetic models with wide-range rate constants. This approach does not require any boundary conditions and thus can broadly apply to dissect stochastic interaction kinetics. This idea has been demonstrated at the ensemble level. Software such as Berkeley Madonna has been developed for modeling and visualization of chemical reactions^[33–36]. However, a systematic way to simulate these kinetic parameters using single-molecule interaction FRET events is still lacking.

Here, we provide a single-molecule interaction simulation (SMIS) tool to provide the expected kinetic characteristics of each FRET state based on the assigned kinetic model, and to dissect interaction kinetics from single-molecule FRET trajectories. In the EXPERIMENTAL SECTION, using the two-state kinetic model, we derived the analytical probability density function and the relative equilibrated populations of each FRET state. We then introduced SMIS and provided a step-by-step simulation of the single-molecule interaction FRET events, the probability density of dwell times, average transition rates, and the relative population of each FRET state. In the RESULTS and DISCUSSIONS, the general solutions from the two-state kinetic model were then converted into the analytical solutions of the well-known Michaelis-Menten enzyme kinetic model to validate the SMIS. Finally, we further demonstrated a successful application of SMIS to quantified kinetic rate constants that were originally unobtainable through the traditional single-molecule analytical solution approach.

EXPERIMENTAL SECTION

The FRET data directly associates the interacting species in the proposed kinetic model with the observed FRET states. Using a kinetic model containing three interacting species in two FRET states (Figure 1a) as an example, we demonstrated the derivation of dwell-time distributions, average transition rate, and relative populations of each interacting species through the analytical approach or the simulation approach.

Figure 1. Kinetic scheme, exemplary single-molecule trajectory, and kinetic and thermodynamic results of a two-state FRET model.

(a) Kinetic model describes three interacting species (E, ES*, and ES) associated with high (I_H) and low (I_L) FRET states, (b) Single-molecule turnover trajectory shows stochastic transitions between I_H (blue curve) and I_L (red curve) states, (c) Rate constants and physical situations used to obtain f(τ), ⟨τ⟩⁻¹, and P([S]) in d-f (d-f) Analytical solutions of the (d) probability density function f_HL(τ) and f_LH(τ), (e) average transition rates ⟨τ_HL⟩⁻¹ and ⟨τ_HL⟩⁻¹ and (f) relative populations P_HL and P_LH under 5 conditions.

Derive analytical solutions of kinetic properties of the kinetic model

Figure 1a shows the kinetic model describing an enzyme existing as one of the three interacting species (E, ES*, and ES) associated with the high (I_H) and low (I_L) FRET states. The substrate (S) bound to the enzyme (E) to form the interacting complex ES through an intermediate ES* with the forward (k₁, k₂, and k₃) and reversed (k₋₁, k₋₂, and k₋₃) rate constants annotated. Assuming in the single-molecule FRET experiment, we observed the highly fluorescent enzyme E interacts substrates, forms a fluorescent intermediate ES*, and eventually generate a weakly-fluorescent product ES. Under constant laser illumination, one can detect these steps at the single-molecule level in real-time. Figure 1b shows the typical single-molecule trajectory that reflects these processes through the stochastic on-off burst like signals. Each FRET-efficiency increase marks the presence of E or ES*; each decrease marks a formation of ES. The τ_HL, the dwell time on the I_H state before transitioning to the I_L state, reports the microscopic dwell times for completing steps that involve k₁° (i.e., k₁[S]), k₋₁, k₂, and k₃° (i.e., k₃[S]). τ_LH, the dwell time on the I_L state before transitioning to the I_H state, reports the microscopic dwell times for completing steps that involve k₋₂ and k₋₃. The distributions of these two stochastic observables (τ_HL and τ_LH) provide valuable kinetic information and molecular insights to the interaction mechanism.

To obtain the kinetic rate constants analytically, we derived the probability density function of dwell times f(τ) by solving the single-molecule rate equations under proper initial conditions. For example, to derive the probability density function of τ_HL, f_HL(τ), we wrote out the single-molecule kinetic equations based on the processes (Figure 1a middle) that lead to the transition from the I_H to I_L state. Entering the I_L state can occur either through the E to ES (pathway involves k₃°) or ES* to ES (pathway involves k₂) and relevant to P_E(t)₁, P_ES*(t)₁, P_ES(t)₁, P_E(t)₂, P_ES*(t)₂, and P_ES(t)₂. The overall probability function of time is a linear combination of both pathway P_i(t) = C₁P_i(t)₁ + C₂P_i(t)₂ (i ∈ [E, ES*, ES]), which C₁ and C₂ are the probability coefficients for two different initial conditions (Supporting Information S1.1). We evaluated the probability density function f_HL(τ) from the P_i(t). f_HL(τ) dictates the probability density of finding a dwell time τ_HL for the transition from the I_H to the I_L state. The probability for finding a particular dwell time τ_HL is f_HL(τ)Δτ, which equals the probability of switching from I_H to I_L between t = τ and τ + Δτ. Since the transitions from the I_H to the I_L state only occur via E to ES or ES* to ES pathways, f_HL(τ)Δ τ can be estimated from the overall ΔP_ES(τ). In the limit of infinitesimal Δτ, the analytical expression of overall probability density function f_HL(τ) is $f_{\mathrm{HL}}(\tau )=C_1\frac{d{P}_{\mathrm{ES}}{(\tau )}_1}{d\tau }+C_2\frac{d{P}_{\mathrm{ES}}{(\tau )}_2}{d\tau }$ (Supporting Information S1.2). Eq 1 shows the final analytical expression of f_HL(τ).

$$f_{\mathrm{HL}}(\tau )=D_1{\mathrm{e}}^{(\mathrm{B}+\mathrm{A})\tau }+D_2{\mathrm{e}}^{(\mathrm{B}-\mathrm{A})\tau }$$ (1)

Where $A=\frac{\sqrt{{(k_{-1}+{k_3}^{°}+k_2+{k_1}^{°})}^2-4\left({k_3}^{°}{k}_{-1}+k_2({k_1}^{°}+{k_3}^{°})\right)}}{2}$, $B=\frac{-({k_3}^{°}+k_2+{k_1}^{°}+k_{-1})}{2}$, $D_1=\frac{(k_2(A+B+{k_1}^{°}+{k_3}^{°})+{k_3}^{°}{k}_{-1})k_{-2}+({k_1}^{°}{k}_2+{k_3}^{°}(A+B+k_2+k_{-1}))k_{-3}}{2\mathrm{A}(k_{-2}+k_{-3})}$, and $D_2=\frac{(k_2(A+B+k_2+k_{-1})-{k_3}^{°}{k}_{-1})k_{-2}+(-{k_1}^{°}{k}_2+{k_3}^{°}(A+B+{k_1}^{°}+{k_3}^{°}))k_{-3}}{2\mathrm{A}(k_{-2}+k_{-3})}$.

Finally, with the f_HL(τ), we evaluated the average transition time ⟨τ_HL⟩ by $\langle {\tau }_{\mathrm{HL}}\rangle ={\int }_0^{\infty }\tau f_{\mathrm{HL}}(\tau )d\tau$, whose reciprocal value reports the average transition rate (Eq 2).

$${\langle {\tau }_{\mathrm{HL}}\rangle }^{-1}=\frac{(k_{-2}+k_{-3})(k_1{k}_2+k_3(k_2+k_{-1}))[\mathrm{S}]}{k_{-3}(k_2+k_{-1}+k_1[\mathrm{S}])+k_{-2}(k_{-1}+(k_1+k_3)[\mathrm{S}])}$$ (2)

Similarly, we evaluated the P_LH(t), f_LH(τ), and ⟨τ_LH⟩⁻¹ for the transitions from the I_L to I_H state (Supporting Information S2). Eq 3 and Eq 4 summarize the final analytical expression of f_LH(τ) and ⟨τ_LH⟩⁻¹, respectively.

$$f_{\mathrm{LH}}(\tau )=(k_{-2}+k_{-3})e^{-(k_{-2}+k_{-3})\tau }$$ (3)

$${\langle {\tau }_{\mathrm{LH}}\rangle }^{-1}={({\int }_0^{\infty }\tau f_{\mathrm{LH}}(\tau )d\tau )}^{-1}=k_{-2}+k_{-3}$$ (4)

With these analytical solutions, we can predict the f_HL(τ) and f_LH(τ) and test the effect of substrate concentration, [S], on the ⟨τ_HL⟩⁻¹ and ⟨τ_LH⟩⁻¹. Figure 1d–f show these predictions for the below conditions (Figure 1c): (1) all rate constants are similar; (2) ES* existing as a transient complex; (3) ES* existing a stable complex; (4) ES existing as a transient complex; and (5) ES existing as a stable complex. Since most of the rate constants range from 10⁻² to 10², we used rate constants of 1 s⁻¹ to predict the kinetic properties. We then varied the rate constants to simulate ES* and ES existing as the transient or stable complexes. For the stable complex condition (i.e., formation rate constant is larger than dissociation rate constant), we set the formation rate constants to ^~ 10² s⁻¹ and the dissociation rate constants to ^~ 10⁻² s⁻¹. On the other hand, formation rate constants are ^~ 10⁻² s⁻¹ and the dissociation rate constants to ^~ 10² s⁻¹ for the transient complex condition. Figure 1d and 1e top panels show the f_HL(τ) and ⟨τ_HL⟩⁻¹ for these conditions. The f_HL(τ) predicts the multiple exponential binding behaviors and the ⟨τ_HL⟩⁻¹ predicts the hyperbolic dependence of [S] on the ES formation rate. On the other hand, Figure 1d and 1e bottom panels show the f_LH(τ) and ⟨τ_LH⟩⁻¹. Since the transitions from the I_L to I_H state represent the dissociation of ES, the log-log plot of f_LH(τ) predicts that the dissociation of ES follows the unimolecular dissociation mechanism.

Thermodynamic parameters such as the relative subpopulation of each species were also derived. P_E([S]), P_ES([S]), and P_ES*([S]) are the relative subpopulation of E, ES*, and ES, respectively. Using the two-state kinetic model (Figure 1a), the single-molecule rate equation of each species were drafted based on the generation and consumption of the species (Supporting Information S1.3). When reaching equilibrium, the relative population of each species remains constant. This fact means that $\frac{d{P}_{\mathrm{i}}(t)}{\mathit{dt}}=0$ (i ∈ [E, ES, ES*]). Furthermore, the sum of the relative subpopulation of all species is always equal to 1. By setting up experiments on different [S], together with these boundary conditions, we obtained the populations of E, ES*, and ES. One can expect the system reaches equilibrium by considering the time-dependent populations at t = ∞. We summarized P_E([S]), P_ES*([S]), and P_ES([S]) in Eq S36–Eq S38. Since E and ES* contribute to the I_H while ES to the I_L, we also obtained the P_HL([S]) and P_LH([S]) as shown in Eq 5 and Eq 6, respectively. Figure 1f summarized the [S] dependent P_HL([S]) and P_LH([S]).

$$P_{\mathrm{HL}}([S])=\frac{k_2{k}_{-3}+k_3{k}_{-2}[S]+(k_{-2}+k_{-3})(k_{-1}+k_1[S])}{k_2(k_{-3}+(k_1+k_3)[S])+k_3{k}_{-2}[S]+k_1(k_{-2}+k_{-3})[S]+k_{-1}(k_{-2}+k_{-3}+k_3[S])}$$ (5)

$$P_{\mathrm{LH}}([S])=\frac{(k_1{k}_2+k_3(k_2+k_{-1}))[S]}{k_2(k_{-3}+(k_1+k_3)[S])+k_3{k}_{-2}[S]+k_1(k_{-2}+k_{-3})[S]+k_{-1}(k_{-2}+k_{-3}+k_3[S])}$$ (6)

Simulate kinetic properties of the kinetic model using single-molecule interaction simulation (SMIS)

To obtain the kinetic properties through simulation, we created the single-molecule interaction simulation (SIMS) tool. We simulated single-molecule interaction trajectories in MATLAB by following the procedures below (Figure 2):

Figure 2.

Workflow of SMIS

Define the kinetic model and transition probability from the rate constants (Step 1).

We first define the kinetic model by assigning the number of interacting species in each FRET state and assign rate constants for transitions between interacting species. For example, E, ES*, and ES specify the species in the model (Figure 2a). The k_I,J represents the rate constant for interconversion from the state I to J (J ≠ I; and I, J ∈ [E, ES*, ES]). In other words, k_E,ES* is the k₁°, k_ES*,ES is the k₂, k_ES,E is the k₋₃, and vice versa. The transition from species I to a J follows the relative probability k_I,J/Σ_I k_I,J.

Build a sequence of dwell time for each species (Step 2).

With the kinetic model defined, we can define the sequence of species (e.g., E→ES*→E→ES*→ES→E→ES*→E→ES*→ES) based on the transition probability. The dwell times of each species follows Σ_I k_I,J exp (−Σ_I k_I,Jt), where the Σ_I k_I,J is a rate-constant sum of all competing pathways leaving from species I to J (J ≠ I). We randomly sample one dwell time from the dwell-time distribution of Σ_I k_I,J exp (−Σ_I k_I,Jt) and assign it to each species and generated the sequence of dwell time (e.g., τ→τ_ES*→τ_E→τ_ES*→τ_ES→τ_E→τ_ES*→τ_E→τ_ES*→τ_ES).

Associate species with FRET states and generate a single-molecule FRET trajectory (Step 3).

The dwell time for each species in the sequence is associated with FRET states. For example, by assigning E and ES* in the high; ES in the low FRET state, and combining dwell times belonging the same FRET state, we can convert the τ_E→τ_ES*→τ_E→τ_ES*→τ_ES→τ_E→τ_ES*→τ_E→τ_ES*→τ_ES sequence into single-molecule FRET trajectory of τ_HL→τ_LH→τ_HL→τ_LH.

Generate the probability density function of dwell time PDF(τ), average transition rates and, relative populations.

With single-molecule trajectories, we can extract microscopic dwell times (i.e., τ_HL and τ_LH). Normalizing the histograms of the dwell times by the overall area generates the probability density function of dwell times. The average FRET-state transition time can be calculated from $\langle {\tau }_{\mathrm{i}}\rangle =\frac{\sum {\tau }_{\mathrm{i}}}{N_{\mathrm{i}}}$ (i ∈ [HL, LH], N_i,: number of dwell-times), whose reciprocal value reports the average transition rate between FRET states. Using single-molecule FRET trajectories, we can generate the relative population of each species P_i (i ∈ [E, ES*, and ES] ). P_i, is calculated by dividing the sum of microscopic dwell times (i.e. Στ_E, Στ_ES* or Σ τ_ES) with the length of the trajectory (i.e., $P_{\mathrm{i}}=\frac{\sum {\tau }_{\mathrm{i}}}{\sum \tau }$). Similarly, we can also generate P_HL and P_LH (i.e., $P_{\mathrm{HL}}=\frac{\sum {\tau }_{\mathrm{HL}}}{\sum \tau }$, $P_{\mathrm{LH}}=\frac{\sum {\tau }_{\mathrm{LH}}}{\sum \tau }$).

RESULTS AND DISCUSSIONS

Validation of SMIS using Michaelis-Menten enzyme kinetic model

We validated the SMIS by comparing our simulation with 200,000 dwell times to the well-known Michaelis-Menten enzyme kinetic model with rate constants varies from 0.01 to 100 s⁻¹. Michaelis-Menten model is a simplified condition of our two-state model where k₋₂ and k₃ equal to zero. By replacing k₋₂ and k₃ in Eq 1 to Eq 6, we obtained the f_HL(τ), f_LH(τ), P_HL, P_LH, ⟨τ_HL⟩⁻¹, and ⟨τ_LH⟩⁻¹ of the Michalis-Manton model in Eq 7 to Eq 12, respectively.

$$f_{\mathrm{HL}}(\tau )=\frac{{k_1}^{°}{k}_2}{2A}({\mathrm{e}}^{(\mathrm{B}+\mathrm{A})\tau }-{\mathrm{e}}^{(\mathrm{B}-\mathrm{A})\tau })$$ (7)

Where $\mathrm{A}=\frac{\sqrt{{(k_{-1}+k_2+{k_1}^{°})}^2-4k_2{k_1}^{°}}}{2}$, $\mathrm{B}=\frac{-(k_2+{k_1}^{°}+k_{-1})}{2}$

$$f_{\mathrm{LH}}(\tau )=k_{-3}{e}^{-k_{-3}\tau }$$ (8)

$$P_{\mathrm{HL}}([S])=\frac{k_2{k}_{-3}+k_{-1}{k}_{-3}+k_1{k}_{-3}[S]}{k_2(k_{-3}+k_1[S])+k_1{k}_{-3}[S]+k_{-1}{k}_{-3}}$$ (9)

$$P_{\mathrm{LH}}([S])=\frac{k_1{k}_2[S]}{k_2(k_{-3}+k_1[S])+k_1{k}_{-3}[S]+k_{-1}{k}_{-3}}$$ (10)

$${\langle {\tau }_{\mathrm{HL}}\rangle }^{-1}=\frac{k_1{k}_2[S]}{k_2+k_{-1}+k_1[S]}$$ (11)

$${\langle {\tau }_{\mathrm{LH}}\rangle }^{-1}={({\int }_0^{\infty }\tau f_{\mathrm{LH}}(\tau )d\tau )}^{-1}=k_{-3}$$ (12)

Our simulation accurately recovers the characteristics predicted by the Michaelis-Menten equations. We first compared the analytical and simulated results using a model where E, ES*, and ES are equally stable (i.e., condition 1 in Figure 1c). The other conditions mimic the ES* and ES existing as transient and stable complexes are summarized in Supporting Information S2. Figure 3b–d shows the comparison of dwell-time distributions (Figure 3b), average transition rates (Figure 3c), and relative populations (Figure 3d) of each species. In all conditions, the simulations nicely overlap with the predictions, validating that SMIS successfully generate single-molecule trajectories for the target kinetic model.

Figure 3. Comparisons between simulated and analytical results for the Michaelis-Menten enzyme kinetic model.

(a) Rate constants for the SMIS simulation and analytical solution of the Michaelis-Menten model. (b-d) Comparisons between simulated (blue bar or circle) and analytical (red curve) probability density function f_HL(τ) and f_LH(τ) (b), average transition rate ⟨τ_HL⟩⁻¹ and ⟨τ_LH⟩⁻¹ (c), and relative populations P_HL and P_LH (d) under condition 1. (e) Percent error (% ERR) of extracted rate constants as a function of the number of dwell times.

To determine how many dwell times are needed to robustly recovery the rate constants, we compared how the extracted rate constants deviate from the input. Here, we used the transition from I_L to I_H state as the testing model. Using k₋₃ with the input value of 5 s⁻¹, we simulated the distribution of τ_LH with the number of transition varies from 30 to 10,000. Figure 3e shows the percent error (% ERR) of extracted rate constants comparing to the input value of 5 s⁻¹ as a function of the number of dwell times. The % ERR decreases and gets below 10% once the number of dwell time is larger than 300.

Application of SMIS to kinetic model without analytical solutions

f(τ) plays a crucial role in quantifying kinetic rate constants in the typical single-molecule FRET approach. However, the derivation of f(τ) requires a system of interest to fulfill many criteria. General procedures to derive f(τ) involve (1) formulating the kinetic model from experimental results; (2) dissecting kinetic model to specific transitions between FRET states; and (3) solving the differential equations with proper initial conditions (see example in Supporting Information S1). However, solving f(τ) could be difficult or impossible when there are repeated differential equations, which results in a deficit of useful equations for the variables (i.e., number of useful equations is less than the number of variables). Difficulty can also be originated from undefined initial conditions, such as multiple species co-existing in the same FRET state with relative population undefined. Figure 4a shows one example whose analytical solutions are unobtainable. The kinetic model describes an enzyme existing as one of the four interacting species (E, ES*, ES**, and ES) where E and ES* associates with the FRET high (I_H) state, and ES** and ES with the low (I_L) state. The substrate S binds to the enzyme E to form the interacting complex ES through different configurational intermediates ES* and ES** with the forward (k₁, k₂, k₃, and k₄) and reversed (k₋₁, k₋₂, k₋₃, and k₋₄) rate constants annotated.

Figure 4. Application of SMIS to kinetic model without analytical solutions.

(a) Kinetic model describes four interacting species (E, ES*, ES**, and ES) associated with high (I_H) and low (I_L) FRET states, (b) Summary of input and extracted rate constants for SMIS. (c) Experimental data (blue bars) used to compare with the SMIS simulations (red curve). Pannels from top to bottom show the PDF_HL(τ) under [S]= 2, 10, and 50 μM, PDF_LH(τ), and [S] dependent population (P_HL, and P_LH). (d) Example of % RSD calculation. % RSD is estimated by the ratio of the residue area (Orange) to the total area under experimental data (Blue shaded area). (e) Histogram of simulations based on their $\overline{\%\mathit{RSD}}$ from the first-round screening. Red bar hight light the top 10% data with the smallest error. (f) Histogram of rate constants from the top 10% simulations in e. (g) Bisection method to extract rate constant k₂ through simulations. In each simulation cycle, red bars highlight the most probable k₂. Green area indicates the boundary used for the next round of screening. (h) Progression of identifying correct rate constants. Using the bisection method in g, we quantified each rate constant after seven-round screening. The yellow area highlights the screening range of k in each screening. Bluelines indicate the input rate constants for the experimental data. Redlines show the most probable rate constant extracted from SMIS.

Even though the analytical solutions are not available, in principle, one may still extract the rate constants and species population through simulations with a range of rate constants. To test how effective SMIS can extract rate constants, we applied SMIS to a kinetic model without analytical solutions. We randomly selected a set of rate constants as specified in Figure 4b. With the number of transition set to 500,000 (equivalent to the number of dwell time of 217,000), we generated the PDF_HL(τ), PDF_LH(τ), P_HL, and P_LH under three substrate concentrations ([S] = 2, 10, and 50 μM) as shown in Figure 4c. This set of data serves as the experimental data, on which we applied SMIS to extract the rate constants.

To extract out the rate constants, we performed an extensive simulation using SMIS and search for most probable rate constants by minimizing the average of percent residue. We adapted SMIS to simulate the PDF_HL(τ), PDF_LH(τ), P_HL, and P_LH under [S] = 2, 10, and 50 μM with all eight rate constants vary from 0.2, 0.6, 2, 6, and 20 s⁻¹ in the first search. Since there are eight rate constants for each simulation, this step creates 32768 (8⁵) simulations. Deviation of each simulation from the experimental data was individually estimated through the averaged % RSD, $\overline{\%\mathit{RSD}}$. For each simulation, deviations of simulated PDF_HL(τ), PDF_LH(τ), P_HL, and P_LH from the experimental data were first calculated by the ratio of residue to the total area under the curve (Figure 4d). Averaging of all deviations gave the averaged % RSD, $\overline{\%\mathit{RSD}}$, which serves as the goodness of simulation for each input rate constant set. Since the simulation has a ^~10% error, we consider the simulations with $\overline{\%\mathit{RSD}}<10\%$ are equally accurate. We thus sorted and picked the simulations with $\overline{\%\mathit{RSD}}<10\%$ (Figure 4e). With the selected simulations, we generated the histogram for each rate constant to identify the most probable rate constants (Figure 4f).

To search for the most probable rate constants, we applied the repeated bisection method (Figure 4g) to each rate-constant histogram. In the simulation step, we used SMIS to generate simulations at five different rate constants, used $\overline{\%\mathit{RSD}}$ to select the top 10% simulations, and generated the histogram of each rate constant (Figure 4f). Using the histogram, we bisected the most probable range for each rate constant when the sum of the possibility of simulated values (starting from high to low) is more than 50%. Take k₂ as an example; we picked the second and the first bin (red bars) to define a new range for the most probable k₂. This approach provides the boundaries of rate constants for the subsequent screening. In the subsequent screening, the rate-constant space between boundaries was further divided into five zones to repeat the searching process (Figure 4g). Figure 4h shows the searching results for each rate constant after seven searches. With this approach, we extracted most rate constants (k₁, k₂, k₃, k₋₁, k₋₂, and k₋₃) which were originally unobtainable (Figure 4b). Unfortunately, SMIS still can not recover k₄ and k₋₄. This is most likely due to the lack of useful data since only PDF_LH(τ) contains the k₄ and k₋₄ information. In contrast, from the [S] dependent PDF_HL(τ), PDF_LH(τ), P_HL, and P_LH, we identified the k₁, k₂, k₃, k₋₁, k₋₂, and k₋₃ with only seven-round simulations (extracted values were summarized in Figure 4b).

CONCLUSION

The single-molecule interaction simulation (SMIS) opens new possibilities for objective characterization of interaction kinetics based on the kinetic model of interest, regardless of whether the analytic solutions are available or not. With the two-state model, we derived, as well as used SMIS to simulate, the probability density function (i.e., f_HL(τ), f_LH(τ)), average transition rates (⟨τ_HL⟩⁻¹ and ⟨τ_LH⟩⁻¹), and the relative populations of high and low states (i.e., P_HL and P_LH). These derived analytical solutions justified the feasibility of SMIS to recovery important kinetic distributions using the Michaelis-Menten enzyme kinetic model. To test how effective SMIS can extract rate constants, we applied SMIS to a kinetic model without analytical solutions. We extracted most rate constants that were originally unobtainable. These results indicate the SMIS is useful in providing important characteristics of kinetic parameters for an assigned kinetic model. Comparison between the experimentally determined distributions of kinetic parameters and the simulations crossing a wide range of rate constants can robustly quantify the rate constants. Our findings here contribute to the quantitative analysis of smFRET data, which is an essential step toward understanding biophysical problems using the smFRET approach.