Fig 1.
Schematic of the KPR process and different strategies of interpreting the output.
(A) The complex of enzyme and substrate ES alone cannot produce the final product P. It has to undergo a number of proofreading steps, represented here by phosphorylation (Pi), before the activated state E*S can produce the final product. (B) The FPT-based discrimination strategy, simply reaching the activated state E*S is interpreted as the output. Xa(t) = 0 if the system has not reached the activated state by time t, and Xa(t) = 1 otherwise. At t = 0, the reaction starts with the enzyme in the free state E. The dashed vertical line represents the termination of entire process, e.g., the time T at which the T cell and antigen presenting cell (APC) separate. Xa(t) = 1 for some t < T is interpreted as a positive response. (C) The product-based discrimination strategy. In this strategy, the number of product molecules P(T) produced within a given time T is interpreted as the output. Due to the intrinsic noise, the number of product molecules is a random variable and their distribution is shown for correct and incorrect ligands.
Fig 2.
Reaction schemes of different descriptions of the simple kinetic proofreading process.
(A) The conventional description of KPR explicitly incorporating multiple proofreading steps. (B) A reduced representation of KPR in which multiple driven steps are lumped in a single proofreading step. Additionally, the state E*S can be taken to be a final “product” state E + P, in which case, there is no disassembly (marked by dashed lines) and . However, one may be interested in products P that are constitutively produced (at rate kp) by the E*S state (gray). This latter scenario can describe, e.g., TCR-mediated T cell activation. (C) For comparison, we show the classical Michaelis-Menten reaction scheme in which the “product” state E + P can be identified as an activated complex E*S. The Michaelis-Menten kinetics implicitly assumes values of τ are exponentially distributed. An internal proofreading process leads to a nonexponentially distributed τ and equivalence to the scheme in (B) and describes the FPT-based DNA replication setting (without any additional constitutive product formation). In our setting, substrate concentrations are held fixed, and k1 will be defined as a first-order reaction rate with physical units of 1/time.
Fig 3.
The simplified model of KPR in DNA replication with only one enzyme.
Here, the enzyme (e.g., DNA polymerase) has three states, namely, free (E), bound to correct substrate (ES), and bound to incorrect substrate (ES′). The transition rates between these states are specified in the model. When the enzyme is bound to substrate, it produces the product (P or P′) after a waiting time τ.
Table 1.
Notation and mathematical symbols.
Fig 4.
Statistics of the FPT-based strategy in the TCR recognition scenario.
(A) Accuracy as a function of processing time τ and contact duration T, evaluated using Eq (10). (B) The maximal accuracy
(squares) as a function of processing time τ, evaluated using Eq (12). The asymptotic behavior of
in the τ → ∞ limit, evaluated using Eq (13), is shown by the dashed curve. In (A,B), we set
, k−1 = 1, and
.
Fig 5.
The channel capacity as a function of cell-cell contact time T for first-passage-time-based (FPT-based) signaling and product-based signaling.
The channel capacity is evaluated between the input ξ indicating correct (1) or incorrect (0) substrate and the output Xa or P(T). We assumed ,
,
, τ = 3, and
for a slow product formation rate.
Fig 6.
The channel capacities in product-based (blue squares) and first activation time-based (red dots) discrimination as a function of processing time τ for various cell contact times T.
We evaluate the channel capacity using stochastic simulations (Gillespie algorithm) of the model in Eq (4) with parameters ,
,
, and kp = 1. The production rate was set higher for easier simulation of the product concentration.
Fig 7.
The channel capacity between the input ξ and the output P(T) or Xth as a function of cell-cell contact time T.
Here, ,
,
, τ = 3, and kp = 1.
Fig 8.
The channel capacity between the input ξ and the output Xa, P(T) or Xth as a function of processing time τ.
We assumed ,
,
, T = 1000, and kp = 1. 10,000 independent Gillespie simulations are conducted for each τ.
Fig 9.
Comparison between the simulated channel capacity C(ξ; P(T)) and the corresponding estimate using Eq (19).
(A) C(ξ; P(T)) as a function of cell-cell contact time T; (B) C(ξ; P(T)) as a function of processing time τ. Here, we took ,
,
, and kp = 1. τ = 3 in (A) and T = 1000 in (B).
Fig 10.
(A) Illustration of discrimination using a dynamic threshold as a function of time T since initial contact. The blue trajectories represent the number of products P with correct substrates. The red trajectories represent that of incorrect substrates. (B) A dynamic-threshold-based discrimination strategy maintains a high channel capacity when the total contact time T is uniformly distributed between 0 and Tmax. Filled bars represent the mutual information between input ξ and output Xth with a fixed contact time T and patterned bars represent the mutual information with a uniformly distributed contact time T between 0 and Tmax. The green bars indicate the maximal mutual information over all possible contact times T ≤ Tmax and all possible static thresholds Pth. The input ξ is assumed to be uniformly distributed on {0, 1}. We assumed
,
,
, τ = 3, and kp = 1. Tmax is set to 106. To filter out noisy transients, we additionally mandate that when the dynamic threshold
is smaller than 10 products, no response is initiated.