Limiting Energy Dissipation Induces Glassy Kinetics in Single-Cell High-Precision Responses

Abstract

Single cells often generate precise responses by involving dissipative out-of-thermodynamic-equilibrium processes in signaling networks. The available free energy to fuel these processes could become limited depending on the metabolic state of an individual cell. How does limiting dissipation affect the kinetics of high-precision responses in single cells? I address this question in the context of a kinetic proofreading scheme used in a simple model of early-time T cell signaling. Using exact analytical calculations and numerical simulations, I show that limiting dissipation qualitatively changes the kinetics in single cells marked by emergence of slow kinetics, large cell-to-cell variations of copy numbers, temporally correlated stochastic events (dynamic facilitation), and ergodicity breaking. Thus, constraints in energy dissipation, in addition to negatively affecting ligand discrimination in T cells, can create a fundamental difficulty in determining single-cell kinetics from cell-population results.

Introduction

Living systems are capable of generating surprisingly precise responses in noisy environments (1, 2, 3, 4, 5). For example, T cells, a major orchestrator of adaptive immunity in jawed vertebrates, can identify antigen-presenting cells displaying few pathogenic ligands (1–10 molecules) in a background of tens of thousands of self-ligands (2, 6). High-precision responses such as the above are often produced by (free) energy-dissipating nonequilibrium thermodynamic processes (2, 3, 5, 7, 8). Therefore, a continuous supply of energy is required to support execution of these processes.

However, the availability of energy (e.g., the ATP pool) for generating high-precision cell responses can depend on a number of factors, such as nutrient availability in the local environment (9), activation of specific signaling pathways regulating ATP generation (10), or prioritization of other cell functions over high-precision response (11, 12). For example, activation of the kinase AMPK in T cells during early-time signaling events leads to ATP generation that provides energy to execute the signaling processes (10). In another case, similar to the Warburg effect in tumor cells (11), effector T cells employ a less efficient ATP-producing glucose metabolism to prioritize cell proliferation (12).

The mechanistic details regarding how variation of energy supply affects energy-consuming responses in single cells are not well understood. The relation between energy dissipation and speed and error in high-precision responses such as kinetic proofreading (KPR) (7, 8, 13) or chemotaxis (4) has been investigated in mathematical models for cell populations. These studies suffer from two major drawbacks. 1) It is unclear to what extent the results obtained at the level of cell populations will generalize to single cells where the signaling kinetics (1), as well as the dissipated energy (14), is affected by intrinsic and extrinsic noise fluctuations (1). 2) Some of these studies are carried out at the steady states of the kinetics. Since energy restriction could slow down the kinetics, the steady states could occur on timescales that are unrealistically large for a biological function of interest. Thus, the steady-state results are unlikely to hold in such situations. I address the above issues here in the context of a KPR model of ligand discrimination in single immune cells (T cells) and demonstrate that the single-cell responses in the dissipation-limited case can be fundamentally different from their counterparts with unrestricted dissipation. The KPR model in the presence of intrinsic and extrinsic noise fluctuations pertaining to single cells has been analyzed previously (15), but the role of limiting dissipation in affecting the signaling kinetics has not been investigated before.

This concept of KPR, originally proposed by Hopfield (3) and Ninio (5), was applied by McKeithan (16) to explain the remarkable ability of immune cells (such as T and B cells) to sensitively discriminate between closely related antigens. These cells are able to distinguish between similar antigens whose half-lives differ by only a few seconds (2, 17). A key biochemical step in McKeithan’s scheme is that upon ligand (antigen) unbinding from the receptor, any activated state of the receptor is reset to the neutral state (Fig. 1 A). Although this step increases the sensitivity of the response, it also breaks the detailed balance condition (1, 18, 19, 20), creating a need for constant (free) energy supply that is dissipated by the system to sustain a nonvanishing probability current in the biochemical network. Biophysical models (2, 15, 21) for antigen discrimination in T cells that provide better agreement with experimental data than the original model proposed by McKeithan use KPR as the core concept and require breakdown of the detailed balance condition and continuous energy dissipation. Therefore, the results obtained here regarding the role of energy dissipation in regulating the kinetics will have implications for these models.

Figure 1.

(A) Schematic diagram displaying the biochemical reactions in the minimal model. The KPR step (TM^∗→T + M) is shown in green. The transition T + M→T^∗M (dotted line), occurring at a much smaller rate than the rest of the reactions, is assumed to keep the entropy calculations finite. (B) The exact solution (solid line) with two boundary conditions at E = 3 and E = 1 for N_T0 = N_M0 = 1, k_on = 1/e, k_off = 1/e², k_p = 1, k_d = 1/e, k₁ = 1/e², and k_off^′ = 1/e (rate for the KPR step) is compared with the developed continuous-time MC scheme for p₂(t) (circles) and p₃(t) (squares). (C) Schematic diagrams showing the arrested and mobile states in the case with a TCR (T) interacting with a single pMHC (M) molecule. The arrows indicate the states where the system arrives at the dissipation limit. To see this figure in color, go online.

Here, I investigated the role of limiting energy dissipation in a minimal model involving a KPR scheme for ligand discrimination in single T cells using semianalytical calculations and continuous-time Monte Carlo (MC) simulations. Dissipation in single cells is quantified by calculating the rate of entropy production in single microscopic trajectories (22). I specifically investigated cases where the energy pool available for dissipation is either fixed or increases with a constant rate. The results showed that in single cells, limiting dissipation is marked by emergence of slow kinetics, large cell-to-cell variations of signaling kinetics, and arrest of activated or deactivated signaling states for prolonged time intervals, all of which severely disrupt the sensitive discrimination program in immune cells. Furthermore, the emergent kinetics in dissipation-limited situations displays dynamic facilitation (23) and ergodicity breaking (24), bearing an interesting similarity to that in facilitated models for glass formers (e.g., supercooled liquids) below the glass transition temperature (23). The presence of the glassy kinetics points to a fundamental disconnect between the signaling kinetics in single cells and the signaling kinetics obtained by averaging over a cell population. In addition, the results reveal, to my knowledge, a novel mechanism for emergence of glassy kinetics in nonequilibrium systems that is likely to generalize in a large variety of nonequilibrium systems where kinetic constraints in the dynamics are imposed by limiting dissipation.

Materials and Methods

MC simulations

A continuous-time MC method was used to simulate stochastic trajectories in the minimal model for multiple receptors and ligands. The construction of the master equation for ϕ_i(Q,t) for the dissipation-limited case shows that the propensities $\left(\left\{w_j^i\right\}\right)$ of the reactions that take the system over the dissipation limit (Q(t) = E)) should be set to zero. This result was used to construct a Gillespie (25)-like algorithm by calling two uniform random numbers (r₁ and r₂) in the unit interval for simulating the trajectories. In the simulations, a variable Q(t) keeps track of the entropy produced in the reservoir. The time for the next reaction (τ) in the system residing at state i at time t with a total entropy exchange Q(t) is given by τ = 1/a_totalln(1/r₁), where the total propensity, a_total, is ∑_j$w_j^i$. The next reaction (i→μ) is chosen by calculating μ satisfying the condition, ${\sum }_{j=1}^{\mu -1}{w}_j^i\le a_{\text{total}}{r}_2<{\sum }_{j=1}^{\mu }{w}_j^i$. Any propensity $\left(w_j^i\right)$ that results in Q(t) + ln$\left(w_j^i/w_j^{i^{\prime }}\right)$ > E is set to zero for the calculations of τ and μ in the above steps. Q(t) is updated to Q(t + τ) = Q(t) + ln$\left(w_{\mu }^i/w_i^{\mu }\right)$ after the transition i→μ is executed. When there is a supply of medium entropy at a constant rate (e_r) by an energy source, a stochastic variable q, decoupled from rest of the variables in the minimal model, is introduced. q increases by unity (q→q + 1) with a propensity $w_{q+1}^q$ = e_r, increasing the dissipation limit (i.e., E(t + τ)→E(t) + 1) by unity at every execution. $w_{q+1}^q$ is used along with other propensities in the model for the calculation of τ and μ. The condition Q(t) + ln$\left(w_j^i/w_j^{i^{\prime }}\right)$ > E(t) is used to set a reaction (i→j′) propensity that crosses the dissipation limit to zero. $w_{q+1}^q$ is not considered in evaluation of the above condition and is never set to zero.

Calculation of P_w(τ), P(t_p), P(t_x), C(n), and $\overline{{\delta }^2\left(t,T\right)}$

Waiting times (τ) in a time interval T(≫τ) were calculated for each stochastic trajectory in the continuous-time random-walk (CTRW) model, and P_w(τ) was calculated using all the τ values collected in a large ensemble of stochastic trajectories. A start time, t_start, was chosen. If the next reactions in a CTRW stochastic trajectory occurred at times t₁, t_2, t₃,…, then t_p, the persistence time, for the trajectory was defined as t_p = t₁ − t_start, and t_x, the exchange time, was calculated using t₂ − t₁, t₃ − t₂, and so on (26). t_start was chosen at times after the system reached the dissipation limit for the fixed dissipation limit. When the dissipation limit increased with a rate e_r, t_start was chosen either at t < τ_trans or t > τ_trans.Values of t_p and t_x were collected over a large number of stochastic trajectories (>10⁴) for the evaluation of P(t_p) and P(t_x). The reactions that take place after time t_start are indexed as 1, 2, …, n + 1 at times t₁, t_2, t₃, …, t_n+1, respectively. The waiting times τ₁ (t₂ − t₁), τ₂ (t₃ − t₂), .., τ_n (t_n+1 − t_n) were used to calculate C(n) using Eq. 8 for a large ensemble of stochastic trajectories. For the fixed dissipation limit, I set t_start = 0, and when the dissipation limit increases with a rate e_r, t_start was chosen either at t < τ_trans or t > τ_trans. A stochastic trajectory in the CTRW model, simulated for a long time T(∼2 × 10⁴ s), was assigned positions (r(t_i) = (m_i,n_i)) at regular time intervals of Δt (t = {t₁,…,t_N = T_N}). The time-averaged $\overline{{\delta }^2\left(t,T\right)}$ was calculated by replacing the integral in Eq. 10 with $\overline{{\delta }^2\left(n\Delta t,T_N\right)}=1/\left(N-n\right){\sum }_{i=1}^{N-n}{\left(\overrightarrow{r}\left(t_i+n\Delta t\right)-\overrightarrow{r}\left(t_i\right)\right)}^2\overline{{\delta }^2\left(t,T\right)}$, was calculated for different stochastic trajectories, as shown in Figs. 2 and 3.

Figure 2.

(A) Stochastic trajectories obtained from the MC simulation for the copy numbers of T^∗M (N_T^∗M) in the presence and absence of the fixed dissipation limit (E = 500). The parameters are set to values shown in the Materials and Methods section, with k_off = 0.001 s⁻¹. (B) Variation of P(t_p) and P(t_x) with their arguments for the dissipation-limited case. The solid line shows a fit to P(t_p) with a stretched exponential function (f(t) = 5.17exp(−at^β), where a = 3.418 and β = 0.5622). (Inset)The differences between P(t_p) and P(t_x) disappear when there is no restriction for energy dissipation. Both the distributions decay exponentially. The parameters are the same as in (A). (C) Variation of C(n) with n for the dissipation-limited and unrestricted cases. C(n) is scaled with C(0) to bring both the data on the same scale. The inset shows a close-up of the main figure at smaller values of n. C(n) reaches 1/3 of C(0) in n ≈ 10 when the dissipation is limited, whereas for the case with unlimited dissipation, C(n) falls well below C(0)/3 at n = 1. The parameters are the same as in (A). (D) Variation of $\overline{{\delta }^2\left(t,T\right)}$ with t for 20 different stochastic trajectories for the dissipation-limited and unlimited cases (inset). The parameters are the same as in (A), with T = 10⁶ s. The spread in $\overline{{\delta }^2\left(t,T\right)}$ for different configurations indicates ergodicity breaking, which disappears when the dissipation is unlimited (inset). More than 10⁴ trajectories were used for all of the calculations above. To see this figure in color, go online.

Figure 3.

(A) Kinetics of the total amount of medium entropy production, Q(t) and the total amount of medium entropy produced by the energy source, E(t), corresponding to a single stochastic trajectory or a single cell. The parameters are the same as those given in the Materials and Methods section, and, k_off = 0.001 s⁻¹ and e_r = 0.01 s⁻¹. For t < τ_trans, all the medium entropy generated by the energy source is consumed by the system, and for t > τ_trans, medium entropy E(t) accumulates for a timescale of τ_diss. The corresponding kinetics for N_T^∗M is shown in the inset. (Inset) The kinetics of N_T^∗M for the NKPR model is shown for comparison. (B) Variation of P(t_p) and P(t_x) with their arguments for data collected at a time t (20,000 s < τ_trans). The solid line shows a fit to P(t_p) with a stretched exponential function (f(t) = 14.4913exp(−at^β), where a = 10.543 and β = 0.836). (Inset) The differences between P(t_p) and P(t_x) disappear at a later time, t = 50,000 s ≫ τ_trans). The solid line shows a fit close to an exponential decay (∝exp(−x^1.045/0.0535). The parameters are the same as in (A). More than 10⁴ trajectories were used for all the calculations above. To see this figure in color, go online.

Model parameters

The simulation box in the MC simulations represents a region containing the plasma membrane (area = 1 μm²) and a thin layer (depth = 0.01 μm) of cytosol underneath the membrane. The size of the region was chosen such that the reaction timescales are larger than the diffusion timescales (27) (diffusion constant, ∼0.1–0.01 μm²/s); thus, the molecules in the box can be considered well mixed. The data shown in the main text were carried out for the following values of the parameters: k_on = 0.003 s⁻¹, k_p = 1.0 s⁻¹, k_d = 0.1 s⁻¹, k₁ = 10⁻⁸ s⁻¹, k_off is varied between 0.001 s⁻¹ and 10 s⁻¹, N_T0 (the number of T cell receptors (TCRs)) = N_M0 (the number of peptide-major-histocompatibility-complex (pMHC) molecules) = 100 molecules/μm². The values are based on their measured values published in the literature. Details are provided in Table S1 in the Supporting Material. All the simulations were started off with N_TM = N_T^∗M = 0.

Model

A minimal model based on the KPR model proposed by McKeithan (16) was developed to describe early-time signaling kinetics in a single T cell (Fig. 1 A). However, McKeithan’s KPR model and the later modifications of the model (2, 15, 21) do not consider restriction of energy limitation in the system. The novelty, to my knowledge, of the model constructed here is in its ability to study biochemical kinetics in various dissipation-limited situations. In the model, plasma-membrane-bound TCRs interact with antigens or pMHC molecules on antigen-presenting cells with an affinity characterized by a binding (k_on) and an unbinding rate (k_off). A single TCR (T) binds a pMHC molecule (M) to form a complex, TM, and TM then transitions to an activated state, T^∗M. The reaction, TM→T^∗M, represents kinase-mediated phosphorylation of the tyrosine residues in motifs of amino acids (also known as ITAMs) associated with TCRs (2). The activated state can become deactivated (e.g., TM→T^∗M) due to the action of phosphatases (2). Both the activation (occurs at a rate k_p) and deactivation (occurs at a rate k_d) transitions are assumed to be first-order reactions where action of kinases and phosphatases are accounted for implicitly. A key step proposed by McKeithan (16), which I will call the KPR step, leads to complete deactivation of the activated complex, T^∗M (occurs with a rate k_off^′), upon ligand unbinding, i.e, T^∗M→T + M. Unless mentioned, I will assume k_off′ = k_off. To keep the entropy calculations finite, a transition T + M→T^∗M (rate k₁) is assumed to occur at a timescale much larger than any biologically realistic timescale. I will designate the above model sans the KPR step as the non-KPR (NKPR) model hereafter. McKeithan (16) analyzed a general form of the above KPR model using deterministic mass-action kinetics and showed that the KPR step endows the model with a higher discriminatory power for selecting higher-affinity pathogen-derived peptide ligands from low-affinity naturally occurring ligands (self-ligands) in the host. A similar analysis for the above KPR model (Fig. 1 A) showed that for a range of parameters, the steady-state concentration of T^∗M varies as (1/k_off)² as opposed to 1/k_off in the NKPR model (Supporting Material).

Kinetics and dissipation in the minimal model

The biochemical kinetics of the copy numbers of the molecular species in the model is subject to intrinsic stochastic fluctuations arising due to the thermal noise (1). I will consider the molecules to be well mixed in a small volume (1 μm² (plasma membrane area) × 0.01 μm (depth in the cytosol)) in the membrane proximal region, which is a reasonable approximation. The stochastic kinetics of the biochemical reactions is described by the master equation (Eq. 1) in terms of the conditional probability p(i,t|i₀,0) (denoted as p_i(t) from now on for brevity), which is the probability for the system to be in state i at time t given it started at state i₀ at time t = 0. p_i(t) follows the kinetics below (1, 18, 20):

$$\frac{d{p}_i\left(t\right)}{dt}=\sum _{j\left(j\ne i\right)}\left[w_i^j{p}_j\left(t\right)-w_j^i{p}_i\left(t\right)\right],$$ (1)

where $w_j^i$ is the rate of the transition i→j. I follow a notation scheme where the system always transitions from the state in the superscript to that in the subscript. In the model, any state i is specified by a pair of integers, N_TM and N_T^∗M, denoting copy numbers of the species TM and T^∗M, respectively. The copy numbers of the other two species T and M are related to N_TM and N_T^∗M via the total numbers of TCRs (N_T0) and MHCs (N_M0) in the model, i.e., N_T0 = N_T + N_TM + N_T^∗M and N_M0 = N_M + N_TM + N_T^∗M. Since N_T0 and N_M0 do not change in the biochemical reactions, the stochastic kinetics in the model can be represented by a CTRW (28) model where the random walker moves on a two-dimensional square lattice with a lattice spacing of unity. A lattice point (n,m) in the model denotes the biochemical state with N_TM = n and N_T^∗M = m. When a reaction occurs, the random walker instantaneously steps to one of the four nearest-neighbor sites from its current site (n,m). The walker waits for a duration τ at the current site (n,m) before taking the next step, where the values of the waiting time, τ, are drawn from a continuous probability density function determined by Eq. 1. A particular stochastic trajectory in the CTRW model describes the kinetics of the molecular species in a single cell, and since I assume the total numbers of TCRs and pMHCs do not change from cell to cell, averaging over an ensemble of stochastic trajectories (denoted by the angular brackets, 〈⋅⋅⋅〉, hereafter) also implies averaging over a cell population. The average over the cell population is equivalent to an average over p(i,t|i₀,0) when the cell population contains a large number of single cells. The CTRW representation will be utilized later for analyzing stochastic trajectories from MC simulations.

The energy dissipation is characterized by the entropy production in the kinetics. The system entropy is defined as, S_sys = −∑_i p_i(t)ln[p_i(t)] (18, 19, 20, 29), where the sum over i also represents a sum over single cells in a cell population (or an ensemble of stochastic trajectories). S_sys follows the kinetics (18, 19, 20, 29)

$$\frac{d{S}_{\text{sys}}\left(t\right)}{dt}=\underset{\frac{d{S}_{\text{total}}}{dt}}{\underbracket{-\sum _{\begin{array}{l}i,j\\ j\ne i\end{array}}{w}_i^j{p}_j\text{ln}\left(\frac{w_j^i{p}_i}{w_i^j{p}_j}\right)}}-\underset{\frac{d{S}_{\text{med}}}{dt}}{\underbracket{\sum _{\begin{array}{l}i,j\\ j\ne i\end{array}}{w}_i^j{p}_j\text{ln}\left(\frac{w_i^j}{w_j^i}\right)}}=\frac{d{S}_{\text{total}}}{dt}-\frac{d{S}_{\text{med}}}{dt}.$$ (2)

According to Eq. 2, the entropy S_total never decreases (19, 29), i.e., dS_total/dt ≥ 0, and thus quantifies dissipation in the system. In the steady state, dS_sys/dt = 0, and consequently, dS_total/dt = dS_med/dt. dS_med/dt denotes the rate of entropy exchange between the system and the reservoir. Dissipative systems (e.g., the minimal model with the KPR step) produce entropy to the reservoir at a fixed rate (i.e., dS_med/dt ≈ ν >0) in the steady state to maintain a constant probability current (19, 20, 29). In contrast, dS_med/dt = 0 at the steady state in the NKPR model due to the vanishing steady-state probability current in the absence of the KPR step (Supporting Material). Thus, the steady-state kinetics in the NKPR state is dissipationless, i.e., dS_total/dt = 0. Following Seifert (19), it is possible to construct different entropies for single stochastic trajectories or kinetics in single cells that correspond to S_total, S_med, and S_sys, defined above. Due to the relevance of the medium entropy (S_med) in characterizing dissipation in a cell population, I focus on the entropy exchanges for single cells or single stochastic trajectories. For a sequence of N biochemical reaction events in a time interval t, the total amount of medium entropy that flows to the reservoir is given by (19)

$$Q\left(t\right)=\sum _{\alpha =1}^N{\left(\Delta s_{\text{m}}\right)}_{i_{\alpha }}^{j_{\alpha }}=\sum _{\alpha =1}^N\text{ln}\left(\frac{w_{i_{\alpha }}^{j_{\alpha }}}{w_{j_{\alpha }}^{i_{\alpha }}}\right),$$ (3)

where, the αth stochastic transition, j_α →i_α, occurring at time t_α, is associated with an entropy flow, ${\left(\Delta s_m\right)}_{i_{\alpha }}^{j_{\alpha }}=\text{ln}\left(w_{i_{\alpha }}^{j_{\alpha }}/w_{j_{\alpha }}^{i_{\alpha }}\right)$ (19), from the reservoir to the system. ${\left(\Delta s_m\right)}_i^j$ will be denoted by $\Delta s_i^j$ from now on. Q(t) in Eq. 3 is also a stochastic variable that varies between stochastic trajectories or single cells and will be used to quantify dissipation in a single trajectory or a single cell.

The joint probability distribution, ϕ(i,Q,t|i₀,0,0), (denoted as ϕ_i(Q,t) hereafter) describes the conditional probability of the system to be at state i at time t, after producing Q amount of medium entropy to the reservoir in the time interval t, starting at t = 0, at the initial state i₀ and a state of zero entropy exchange. ϕ_i(Q,t) follows the equation (29)

$$\frac{\partial {\varphi }_i\left(Q,t\right)}{\partial t}=\sum _{j\left(\ne i\right)}\left[w_i^j{\varphi }_j\left(Q-\Delta s_i^j,t\right)-w_j^i{\varphi }_i\left(Q,t\right)\right].$$ (4)

Equation 4 can be used to monitor the kinetics of entropy exchanges in individual cells in a cell population. I investigated dissipation-limited situations where the entropy exchange required to carry out the stochastic transition became restricted. I consider two scenarios: 1) the total amount of entropy (E) available for exchange with the reservoir is fixed. This represents a situation where the total amount of energy available for dissipation is fixed. 2) E(t) increases at a fixed rate, which is lower than that required to maintain the probability current in the steady state of Eq. 1. The above constraints are imposed in the kinetics in the following manner. The system is not allowed to make a transition j→i if that requires crossing of the limit E, i.e., ϕ(i,Q,t |j,Q − Q′,t − τ) = 0 when Q > E(t). However, if other reactions (say, j→k) at the state j satisfying Q ≤ E(t) are available, then the particular reaction j→i is replaced by one of those reactions. Thus, the dissipation limit, E, imposes a reflecting boundary condition (30) at Q = E in Eq. 4. It is possible to solve Eq. 4 under this boundary condition exactly semianalytically for simple cases when E is a constant, but for large numbers of receptors and ligands or a time-dependent E(t), such calculations become intractable. A continuous-time MC method, akin to the standard Gillespie method (25), was developed here to simulate stochastic trajectories in these cases.

An exactly solvable case

Consider a single TCR interacting with a pMHC molecule in the minimal model. The signaling kinetics then involves transitions between three different states representing the unbound TCR and pMHC (state T₁), the TCR-pMHC complex (state T₂), and the activated TCR-pMHC complex (state T₃). The biochemical reactions are described by

$$T_1\underset{k_{off}}{\overset{k_{on}}{\rightleftarrows }}{T}_2\underset{k_d}{\overset{k_p}{\rightleftarrows }}{T}_3;T_1\underset{k_{off}}{\overset{k_1}{\rightleftarrows }}{T}_3.$$

This simple example is amenable to analytical calculations and provides valuable insights into the kinetics in the dissipation-limited case. The probabilities p₁(t), p₂(t), and p₃(t), follow the equation

$$\frac{d{p}_i}{dt}=\sum _{j=1}^3{L}_{ij}{p}_j,$$ (5)

where

$$L=\left[\begin{array}{ccc}-k_1-k_{\text{on}}& k_{\text{off}}& k_{\text{off}}\\ k_{\text{on}}& -k_{\text{off}}-k_{\text{p}}& k_{\text{d}}\\ k_1& k_{\text{p}}& -k_{\text{d}}-k_{\text{off}}\end{array}\right].$$

Exact solution of Eq. 5. (details in the Supporting Material) shows that at the steady state, p₃ (t→∞) ∼ 1/(k_off)², for weak-affinity ligands (k_off ≫ k_on, k_off ≫ k_d), and k_p > k_d. The mean value and higher moments of Q are usually calculated by solving the kinetics for the moment-generating function, ${\psi }_i\left(\lambda ,t\right)=\int dQ{e}^{\lambda Q}{\varphi }_i\left(Q,t\right)$ (19, 29). ψ_i(λ,t) follows the equation (19, 29)

$$\frac{\partial {\psi }_i\left(\lambda ,t\right)}{\partial t}=\sum _j{H}_{ij}\left(\lambda \right){\psi }_j\left(\lambda ,t\right),$$ (6)

where $H_{ij}\left(\lambda \right)={\left(w_i^j\right)}^{1-\lambda }{\left(w_j^i\right)}^{\lambda }\left(1-{\delta }_{ij}\right)-{\delta }_{ij}{\sum }_{j^{\prime }\ne i}{w}_{j^{\prime }}^i.$ However, as shown in the Supporting Material, direct solution of Eq. 6 can be avoided and the moments of Q at all times can be recursively calculated analytically or semianalytically using the solutions of Eq. 5. The calculations (details in the Supporting Material) show that the average rate of dissipation (d〈Q(t)〉/dt) in the steady state is a constant and shows a peak at intermediate values of k_off $\left(\sim \sqrt{k_{\text{d}}{k}_{\text{on}}}\right)$. The ligand discrimination costs more energy at intermediate k_off values, because the system executes the KPR step more frequently compared to the low-affinity or high-affinity ligands. This also implies that the ligands with intermediate values of k_off will arrive at a dissipation limit faster than ligands with other affinities. Equation 4, with a reflective boundary condition at Q = E = const was analyzed with two goals in mind. 1) Find the general structure of the equation (equivalent of Eq. 4) that the system should satisfy under this condition. Such an equation can be further used to formulate a continuous-time MC method (or Gillespie’s method) to simulate stochastic trajectories in dissipation limited cases. 2) Explore whether any nontrivial behavior emerges even in this simple setup. Next, I outline the derivation of the equation followed by the joint probability distribution, ϕ_i(Q,t), with the reflective boundary condition at Q(t) = E for the simple example described above. The full derivation is shown in the Supporting Material. The kinetics of ϕ₁(Q,t), ϕ₂(Q,t) and ϕ₃(Q,t), according to Eq. 4, are given by

$$\frac{\partial {\varphi }_1\left(Q,t\right)}{\partial t}=k_{\text{off}}{\varphi }_2\left(Q+{\Delta }_1,t\right)+k_{\text{off}}{\varphi }_3\left(Q-{\Delta }_3,t\right)-\left(k_{\text{on}}+k_1\right){\varphi }_1\left(Q,t\right)$$

$$\frac{\partial {\varphi }_2\left(Q,t\right)}{\partial t}=k_{\text{on}}{\varphi }_1\left(Q-{\Delta }_1,t\right)+k_{\text{d}}{\varphi }_3\left(Q+{\Delta }_2,t\right)-\left(k_{\text{off}}+k_{\text{p}}\right){\varphi }_2\left(Q,t\right)$$

$$\frac{\partial {\varphi }_3\left(Q,t\right)}{\partial t}=k_1{\varphi }_1\left(Q+{\Delta }_3,t\right)+k_{\text{p}}{\varphi }_2\left(Q-{\Delta }_2,t\right)-\left(k_{\text{off}}+k_{\text{d}}\right){\varphi }_3\left(Q,t\right),$$ (7)

where ${\Delta }_1=\text{ln}\left(k_{\text{on}}\text{/}{k}_{\text{off}}\right),{\Delta }_2=\text{ln}\left(k_{\text{p}}\text{/}{k}_{\text{d}}\right)$ and ${\Delta }_3=\text{ln}\left(k_{\text{off}}\text{/}{k}_1\right)$. The reflecting boundary condition at Q = E demands that ϕ_i(Q > E,t) = 0 for all i values. Since the reflection boundary condition does not lead to loss of any stochastic trajectory, the equation for ϕ_i(Q = E,t) at the boundary, Q = E, is obtained by using the conservation of total probability (30), p₁(t) + p₂(t) + p₃(t) = 1, where p_i(t) = ∑_Q ϕ_i(Q,t). The resulting equation showed that imposing the boundary condition for a system at a state (j, Q) at time t, is the same as setting the transition rates $\left(\left\{w_{i^{\prime }}^j\right\}\right)$ to zero when those transitions {j→i′} lead to the crossing of the dissipation limit at Q = E. It is straightforward to construct a continuous-time MC method following Gillespie’s algorithm (25) to simulate stochastic trajectories in this situation (see Materials and Methods). An exact solution of Eq. 7 with reflective boundary conditions at two boundaries, Q = E₁ and Q = E₂, was obtained by calculation of the eigenvalues and eigenvectors of the linear system (details in the Supporting Material). The comparison between the exact solution and the MC simulations showed excellent agreement (Fig. 1 B).

Results

Arrested states arise when dissipation is limited in the example of a single TCR interacting with a single pMHC

Analysis of Eq. 7 for the case of a single TCR and a single pMHC with a fixed dissipation limit at Q = E shows the presence of arrested states in the kinetics, where the system becomes confined to a single state (e.g., state 1) or multiple states (e.g., states 1 and 2) for a very long time (∼1/k₁). The specific nature of the arrested state and the time when it occurs depend on the values of the rate constants and the dissipation limit, E. The physical origin of the arrested states is discussed below (Fig. 1 C). Suppose the system reaches the energy dissipation limit (Q = E) when it arrives at state 1 at time t. The value of t will particularly depend on how often the KPR step was executed in the stochastic trajectory before it reached the limit, because this step induces a nonzero probability current flow in the kinetics and its execution requires a much larger entropy flow (ln(k_off/k₁)) compared to the other transitions for a biologically relevant model (k₁ ≪ {k_off, k_on, k_p, k_d}). The possible transitions at state 1, 1→2 and 1→3, will need entropy flows $\Delta s_2^1$ = ln(k_on/k_off) and $\Delta s_3^1$ = ln(k₁/k_off), respectively. Since k₁ ≪ k_off, $\Delta s_3^1$ < 0, the system can receive entropy to the reservoir and move below the dissipation limit, E, by executing the reaction 1→3. However, this reaction (1→3) occurs within a timescale of 1/k₁, which can be much longer than any timescale of biological or physical interest. Different kinetic responses arise for high-affinity and moderate- and low-affinity ligands in this long time interval (∼1/k₁) (Fig. 1 B). In the case of high-affinity (k_off < k_on) ligands, since $\Delta s_2^1$ > 0, the transition 1→2 cannot occur without crossing the limit at Q = E. Thus, the system will remain at state 1 for a timescale of 1/k₁. In the case of moderate- and low-affinity (k_off > k_on) ligands, on the other hand, $\Delta s_2^1$ < 0, so the transition 1→2 could occur without crossing the limit at Q = E. However, after the system reaches state 2, the possible transitions, 2→3 and 2→1, are associated with entropic flows $\Delta s_3^2$ = ln(k_p/k_d) > 0 (since k_p > k_d) and $\Delta s_1^2$ = −$\Delta s_2^1$ = −ln(k_on/k_off), respectively. When $\Delta s_3^2$ ≤ $\Delta s_2^1$ or ln(k_p/k_d) ≤ ln(k_on/k_off) or k_off ≤ k_on(k_d/k_p), the transition 2→3 can occur without breaching the limit Q = E and the system stays mobile between states 1, 2, and 3, without executing the KPR step (3→1). However, when $\Delta s_3^2$ > $\Delta s_2^1$ or k_off > k_on(k_d/k_p), the entropy gain from the previous 1→2 transition is not sufficient to support the 2→3 transition but can support the 2→1 transition; as a result, the system becomes confined between states 1 and 2. The above-described properties of the kinetics are also present when the system reaches the dissipation limit precisely or there is a small gap (e.g., E − Q < min(|Δ₁|, |Δ₂|, |Δ₃|)) between the dissipation limit and Q.

How do the properties of the arrested and mobile states change when there are multiple ligand and receptor molecules? I investigated this question next using MC simulations of Eq. 4 (see Materials and Methods for details). Interestingly, the results showed that the kinetics in the KPR model in these situations is similar to that of tagged molecules in models of glass formers at low temperatures marked by temporal clustering of stochastic events describing transitions between mobile and immobile states or dynamic facilitation (23).

Kinetics with a fixed dissipation limit displays dynamic facilitation for multiple TCRs interacting with multiple pMHCs

First, I studied the kinetics in the presence of a fixed dissipation limit at Q = E = const. MC simulation of the stochastic trajectories in this condition showed three key differences compared to its counterpart without the dissipation limit (Fig. 2 A): 1) the kinetics slowed down substantially once the system reached the dissipation limit; 2) there were large copy-number fluctuations (Fig. 2 A and Fig. S1); and 3) stochastic events in the neighborhood of low- and high-activation states appeared to be bunched in time. Since similar features are also observed in kinetics of tagged molecules in models of glass formers below the glass transition temperature, I analyzed the above features further by calculating quantities that are frequently used in characterizing kinetics in glass formers (23). A CTRW representation is often used for analyzing the glassy kinetics of the tagged molecules in glass formers (31). I use a similar analogy for studying the stochastic trajectories in the minimal model.

Analysis of the kinetics using a CTRW representation

As described in the Model section, the stochastic kinetics in the minimal model can be represented by a CTRW model where the walker moves on a two-dimensional square lattice spanned by N_TM and N_T^∗M with a waiting-time distribution P_w(τ). The waiting time, τ, is the time the random walker stays put at any state before making the next jump. The CTRW representing kinetics of tagged probes in glass formers is marked by a decoupling between the diffusion and relaxation timescales (23, 26). The relaxation timescale is proportional to the persistence timescale (t_p), defined as the time during which an initial state does not change (23, 26). The diffusion timescale is related to the exchange time (t_x), the time interval between any two subsequent random steps in the CTRW model (23, 26). The distributions of the timescales t_p and t_x show different forms (P(t_p) ≠P(t_x)) for the tagged molecules in glass formers due to the disparity between the timescales (23, 26). The kinetics in glass formers also display ergodicity breaking, where the time average of an observable over a long time interval is not equal to the ensemble average (23, 24). In a CTRW model, this can be caused by a power-law variation of P_w(τ) (24) or when the waiting times in the subsequent steps become correlated (33). For example, the kinetics of potassium channels in the plasma membrane show a power-law waiting-time distribution (34), or the dynamics of financial markets show correlation in successive waiting times (24, 33). To probe the presence of ergodicity breaking and its underlying origin in the kinetics of the minimal model, I calculated P_w(τ) and the correlation (C(n)) between subsequent waiting times. C(n) is defined as

$$C\left(n\right)=\langle 1/\text{M}{\Sigma }_{m=1}^{\text{M}}{\tau }_m{\tau }_{n+m}\rangle -{\langle 1/\text{M}{\Sigma }_{m=1}^{\text{M}}{\tau }_m\rangle }^2,$$ (8)

where τ_n represents the waiting time at the nth step taken by the random walker. A finite C(n) for n≠0 would indicate temporally correlated movements (33) or dynamic facilitation. The ergodicity breaking is characterized by calculating the ensemble average (〈r²(t)〉), defined as (24)

$$\langle r^2\left(t\right)\rangle =\langle {\left(m-m_0\right)}^2\rangle +\langle {\left(n-n_0\right)}^2\rangle ,$$ (9)

where the random walker starts at the position $\overrightarrow{r}\left(0\right)\equiv \left(m_0,n_0\right)$ at time t = 0 and reaches at $\overrightarrow{r}\left(t\right)\equiv \left(m,n\right)$ at time t, and 〈…〉 denotes the average over an ensemble of trajectories. 〈r²(t)〉 is compared with the time-averaged mean-square distance for a single stochastic trajectory (24), defined as

$$\overline{{\delta }^2\left(t,T\right)}=1/\left(T-t\right)\underset{0}{\overset{T-t}{\int }}d\tau {\left(\overrightarrow{r}\left(t+\tau \right)-\overrightarrow{r}\left(\tau \right)\right)}^2.$$ (10)

The presence of (weak) ergodicity breaking implies, when T $\gg$ t (24),

$$\overline{{\delta }^2\left(t,T\right)}\ne \langle r^2\left(t\right)\rangle .$$ (11)

The extent of ergodicity breaking in the kinetics can be quantified further by the ergodicity-breaking parameter (24), EB, as

$$EB={\text{lim}}_{T\to \infty }\left[\langle {\xi }^2\rangle -{\langle \xi \rangle }^2\right],$$ (12)

where $\xi =\overline{{\delta }^2\left(t,T\right)}\text{/}\langle \overline{{\delta }^2\left(t,T\right)}\rangle$. EB = 0 for a kinetics with ergodicity (such as the standard Brownian motion (32)), and EB > 0 when ergodicty is broken in the kinetics (24).

Calculation of specific quantities in the CTRW

The calculation showed an exponentially decaying P_w(τ) in the absence of any dissipation limit (Fig. S2). This is expected, as in this case, the waiting time, τ, at any state in the CTRW is distributed exponentially with a mean value (μ) equal to the inverse of the sum of the propensities of the outgoing transitions (25). Thus, P_w(τ) for a time interval of t is given by the superposition of exponential distributions with appropriate weights g(μ), i.e., ${\sum }_{{\mu }_{\text{min}}}^{{\mu }_{\max }}g\left(\mu \right)e^{-\mu \tau }$. When g(μ) does not change with μ appreciably, the smallest μ (μ_min) makes the largest contribution to the above summation, producing an exponential form for P_w(τ). However, in the presence of the dissipation limit, the distribution displayed a much slower decay (non-Debye) than the exponential decay (Fig. S2). This can occur when g(μ) varies with μ with a particular form pertaining to hierarchically constrained dynamics (35). The slower decay of P_w(τ) in the dissipation-limited case is a manifestation of increased occurrences of longer waiting times characterizing the slow kinetics. However, the non-Debye exponential form of P_w(τ) alone does not establish dynamic facilitation or ergodicity breaking in the kinetics.

Calculations showed that P(t_p) = P(t_x) in the absence of the dissipation limit, demonstrating the equivalence between timescales t_p and t_x (Fig. 2 B, inset). Imposing the dissipation limit broke the equality (i.e., P(t_p) ≠ P(t_x)), and both P(t_p) and P(t_x) displayed non-Debye decays, and P(t_p) agreed well with a stretched exponential decay (∝exp(−at_p^β)) for over three decades (Fig. 2 B). t_p is associated with the relaxation timescale of the initial state, and in glass formers, it corresponds to relaxation of spatial structures (23), whereas t_x is associated with the diffusive timescale of the random walker. The emergence of the stretched exponential or non-Debye relaxation times in glassy systems is accompanied by hierarchical activation of the underlying microscopic processes (23, 35). When the system reaches the dissipation limit, certain reactions can take place only when the appropriate amount of entropy is received by a concerted execution of a series of reactions; this provides a source for hierarchical activation in the system. In the simulations, 〈t_p〉 is about three times larger than 〈t_x〉. Similar behavior (〈t_p〉 > 〈t_x〉) in glass formers indicates breakdown of the Stokes-Einstein relationship relating dissipative and diffusion timescales in liquids (23, 36). The nonequivalence of t_p and t_x, as in glass formers, points to the presence of dynamic facilitation or clustering of mesoscopic events in time (23, 36).

The correlation function, C(n), as defined in Eq. 8, further characterized the nature of the dynamic facilitation in the KPR model. Calculation of C(n) showed that waiting times separated by multiple events are more correlated in the dissipation-limited case compared to that with no dissipation limit (Fig. 2 C). C(n) decreased substantially within a single step when there was no dissipation limit (Fig. 2 C). Next, I investigated whether these correlations are able to generate ergodictiy breaking as found in the models of CTRW with correlated time steps. The calculations of <r²(t)> (Eq. 9) and $\overline{{\delta }^2\left(t,T\right)}$ (Eq. 10) showed that <r²(t)> ≠ $\overline{{\delta }^2\left(t,T\right)}$ in the KPR model with the dissipation limit, demonstrating a breakdown of ergodicity in the kinetics due to the confinement of stochastic trajectories in specific regions in the state space for very long times (∼1/k₁) (Figs. 2 D and S9). $\overline{{\delta }^2\left(t,T\right)}$ saturates at large t, as the values of N_TM and N_T^∗M are bounded by total numbers of T and M (see also Fig. S10). Removing the dissipation limit restored ergodicity (Figs. 2 D and S9), i.e., <r²(t)> = $\overline{{\delta }^2\left(t,T\right)}$. The ergodicity breaking is further quantified by calculating the ergodicity parameter, EB (Eq. 12), for the KPR model with and without the dissipation limit. In the presence of the limit, nonzero EB values were generated (Fig. S9), whereas, in the absence of any limit, EB became vanishingly small (Fig. S9).

Kinetics with a fixed rate of energy supply

This case was investigated by including a variable representing an energy source in the biochemical reactions that increased the medium entropy (or the available energy for dissipation) at a constant rate (e_r) (see Materials and Methods for details). The simulations were performed for the cases where the required rate of energy dissipation (ν) was larger than that available from the reservoir. Analysis of the kinetics revealed the presence of two dynamically distinct regions (Fig. 3 A). 1) For times 0 < t ≤ τ_trans, most of the medium entropy produced by the energy source flowed into the system to fuel the reactions. At the end of τ_trans, when the total medium entropy inflow into the system became comparable to the dissipation required to bring the initial state to the steady state of the NKPR model (Fig. S3), the kinetics moved into the second regime. 2) For t > τ_trans, the entropy produced by a single cell (or single trajectory) did not change appreciably over a timescale τ_diss, despite medium entropy being produced by the energy source. Beyond τ_diss, the KPR step is executed and the medium entropy produced in the reservoir changes abruptly by ∼ln(k_off/k₁). A possible mechanism underlying the above behavior is that when t ≤τ_trans, the medium entropy produced by the energy source is fully spent on carrying out the biochemical reactions, but since the KPR step requires the largest amount of entropy influx (∼ln(k_off/k₁)), it rarely takes place in this regime, and consequently, the system evolves as the dissipation-limited NKPR model. Toward the end of τ_trans, when the stochastic trajectories in the system are close to the dissipationless steady state of the NKPR model, the system does not draw much medium entropy from the energy source over a timescale of τ_diss. This results in accumulation of sufficient medium entropy to fuel the execution of the KPR step at the end of τ_diss (or τ_disse_r ≥ ln(k_off/k₁)). Thus, the time evolution for t > τ_trans can be intuitively thought of as successions of time segments of scale τ_diss, where the kinetics is similar to that of the dissipationless steady state of the NKPR model. Further analysis of the simulation results confirmed the above picture.

Calculation of P(t_p) and P(t_x) showed that for t < τ_trans, P(t_p) ≠P(t_x), suggesting similarities of the kinetics to that of the fixed-dissipation-limit case (Fig. 3 B). For t > τ_trans, I found P(t_p) ≈ P(t_x), and both the distributions decayed exponentially, as in unlimited-dissipation cases (Fig. 3 B, inset). Distributions of N_T^∗M and N_TM demonstrated that the system closely follows the steady state of the NKPR model for t > τ_trans (Fig. S4). The value of τ_trans is roughly related to e_r, the available medium entropy (Q₀) at t = 0, and the total medium entropy required to change the initial state to the steady state of the NKPR model (Q^NKPR_steady) as τ_trans ≈ (Q^NKPR_steady − Q₀)/e_r. C(n) and $\overline{{\delta }^2\left(t,T\right)}$ showed the emergence (or absence) of dynamic facilitation and ergodicity breaking for t < τ_trans (or t > τ_trans) (Figs. S5 and S6). The difference between 〈t_p〉 and 〈t_x〉 calculated at increasing values of e_r showed that for t < τ_trans, increasing e_r decreased the magnitude of the difference, which reaches zero as e_r increases to e_r ≥ν (Fig. S7). Similarly, dynamic facilitation and ergodicity breaking disappears at t < τ_trans for e_r ≥ ν. Thus, at t < τ_trans, increasing e_r appears to generate an effect qualitatively similar to that of increasing the temperature across the glass transition in glass formers.

Implications for ligand discrimination

The emergence of glassy kinetics in the dissipation-limited simulation negatively affects ligand discrimination. Arrested states slow down the kinetics in addition to making an undesired state (e.g., TM for low k_off) persist in single cells over a long timescale (∼1/k₁ for fixed dissipation limit or τ_diss for a fixed rate of entropy increase). Both these effects oppose a successful discrimination program. Without the dissipation limit, the biochemical kinetics reached the steady state in a short timescale (approximately minutes), where the cell population average of the activated species (T^∗M) decreased with the ligand affinity 〈N_T^∗M〉∼ 1/k_off², allowing the cells to discriminate between pathogenic (low k_off) and self-ligands (high k_off) with greater sensitivity (Fig. 4). Limiting dissipation qualitatively changed this pattern, where 〈N_T^∗M〉 displayed a nonmonotonic variation with k_off at short times (approximately minutes) (Fig. 4). In this case, decreasing ligand affinity leads to an increase in the activation, producing an outcome opposite to that required by a successful discrimination program. When energy for dissipation is supplied at a fixed rate, for timescales t < τ_trans, the response is similar to that with a fixed dissipation limit, and, at longer timescales (t > τ_trans), the system responds with a lower precision (〈N_T^∗M〉 ∼ 1/k_off) compared to the unrestricted KPR model. Thus, the response at long timescales in this case is similar to that of the less discriminatory NKPR model. However, depending on the initial (basal) signaling state of the single cells, τ_trans could be much longer than biologically relevant timescales (approximately minutes).

Figure 4.

Variation of N_T^∗M, averaged over a population of single cells (n = 10,000) at t = 5 min, with k_off. The data are shown for the cases of a fixed dissipation limit (E = 500), a fixed rate of medium entropy production (e_r = 0.01 s⁻¹), unlimited dissipation, and, the NKPR model. The dissipation-limited cases offer a poorer discrimination with a decreased range of variation of 〈N_T^∗M〉 and a nonmonotonic variation with k_off. To illustrate, T cells following the KPR model are able to discriminate between the ligand affinities (dashed and dotted vertical lines) by crossing the activation threshold (horizontal solid line) for the stronger ligand; however, limiting dissipation abrogates this discrimination. To see this figure in color, go online.

Moreover, in a dissipation-limited scenario, large cell-to-cell variations of copy numbers of activated species will hinder discrimination when multiple types of ligands are presented simultaneously to a T cell population. For example, a successful discrimination program requires that T cells should be able to recognize a small fraction (f_path) of pathogenic ligands (say, k_off = k_path) within a large population (f_self ≫ f_path and f_self + f_path = 1) of self-ligands (k_off = k_self). Therefore, the discrimination program should generate widely different distributions (or P(k_off, N_T^∗M)) of the active species (T^∗M) in a T cell population when ligands are presented with an input distribution, $P_{\text{ligand}}\left(k_{\text{off}}\right)=f_{\text{path}}{\delta }_{k_{\text{off}},k_{\text{path}}}+f_{\text{self}}{\delta }_{k_{\text{off}},k_{\text{self}}}$ or $P_{\text{ligand}}\left(k_{\text{off}}\right)={\delta }_{k_{\text{off}},k_{\text{self}}}$. f_path(f_self) denotes the fraction of pathogenic (self) ligands presented to the T cells. The large variation in N_T^∗M in the dissipation limited cases will produce a wide spread in P(k_off, N_T^∗M) (Fig. S8). Consequently, there will be a substantial overlap between the above input distributions, leading to a much poorer discrimination (Fig. S8) in the dissipation limited case.

Discussion

The analysis carried out here showed that restricting energy dissipation qualitatively changes signaling kinetics of high-precision responses functioning outside thermodynamic equilibrium. The changes are marked by the advent of slow kinetics, long-lived arrested states, dynamic facilitation, and, ergodicity breaking. The origin of this emergent behavior is purely dynamical and arises due to dynamical constraints imposed by limited dissipation. The appearance of the glassy kinetics rectifies the naïve intuition that in the presence of a dissipation limit the system will revert to the condition of its dissipationless counterpart (e.g., the steady-state kinetics without the KPR step). The results show, in contrast to the naïve intuition, that when the energy for dissipation is limited by a fixed amount, the kinetics becomes confined to specific biochemical states for long durations, and when energy for dissipation is supplied at a fixed rate, depending on the energy supply rate and the initial state of the system, the kinetics for a long time can behave similar to that with a fixed dissipation limit. Furthermore, the breakdown of ergodicity in the dissipation-limited cases points to a basic difficulty in deriving details regarding single-cell kinetics from cell population data.

The emergence of the glassy kinetics, characterized by long-lived states (activated or deactivated) and large cell-to-cell variations of copy numbers of signaling products, prove to be detrimental to the discrimination program involving KPR. KPR is a central concept used by biophysical models (2, 15, 21) describing experiments pertaining to ligand discrimination in immune cells, such as T cells. The presence of the KPR step in these models leads to complete or partial reversal of intermediate activated states breaking the detailed balance condition (8, 18, 20). As a result, these models work outside thermodynamic equilibrium and a constant probability current in the network in the steady state is sustained by a constant supply of energy. Therefore, the qualitative features (e.g., slow kinetics, large cell-to-cell variations, ergodicity breaking, and poor ligand discrimination) that arise due to limiting dissipation in the KPR scheme are likely to impact the detailed biophysical models of ligand discrimination when the available energy becomes restricted. A possible test of these results will involve single-cell experiments carried out in energy-limited conditions, possibly induced by manipulation of nutrient metabolism or signaling events regulating ATP production (9).

It is assumed in the minimal model that molecules are well mixed in the simulation volume describing a small region (1 μm² area × 0.01 μm depth) proximal to the plasma membrane. However, immune receptors and associated signaling molecules can be distributed inhomogeneously in larger regions of the plasma membrane (e.g., microclusters) (37, 38) and in the cytosol (39). The kinetics of single molecules in the spatially heterogeneous cellular environment displays ergodicity breaking in certain biological systems (24, 32, 34, 40). For example, spatial kinetics of single potassium channels in the plasma membrane was observed to follow a CTRW model with a power-law waiting-time distribution that led to weak ergodicity breaking in the kinetics (34). In contrast, in the minimal model, the biochemical reaction kinetics showed ergodicity breaking that arose due to finite correlations between successive waiting times. An interesting future direction would be to study the interplay between the diffusion and reaction kinetics, where ergodicity breaking in the two types of kinetics could be induced by spatial heterogeneity and dissipation limitation, respectively.

The framework considered here for quantifying dissipation does not explicitly include activation energy (41). Thus, if the system resides at the free-energy limit (i.e., Q = E) and a particular reaction (say, 1→2) receives medium entropy from the reservoir (e.g., $\Delta s_2^1$ = ln$\left(w_2^1/w_1^2\right)$ <0), the reaction is then assumed to occur. However, it is possible that the reaction also requires crossing of an activation barrier (41) and thus might not occur in this situation. This would impose a stricter restriction on the reactions that can potentially arise at the dissipation limit. Therefore, realistically, there could a larger number of arrested states and a greater degree of dynamic facilitation in the signaling kinetics.

The kinetics in the dissipation-limited cases in the KPR model demonstrates similarities with that in glass formers in terms of the appearance of slow kinetics and dynamic facilitation. However, the glassy kinetics in the two systems also shows a few important contrasts. For example, the shapes of P(t_p) and P(t_x) (23) are different in these models. In glass formers, the glassy kinetics arises when the temperature is lowered past the glass transition temperature and the system undergoes a phase transition in the space and time of stochastic trajectories (23, 42). In the KPR model, the notion of temperature or any phase transition is not evident. In simple networks, violating detailed balance reveals dynamical phase transitions between localized and delocalized states induced by increasing entropy production rate in the limit of large system sizes (43). Increasing the rate (e_r) of medium entropy supply in the minimal model produces changes in the kinetics like the temperature; however, further work is required to make this connection transparent or establish any presence of a phase transition in the KPR model.

Editor: Klipp Edda.