First Product Formation Time (FPFT)

• FPFT is defined as the random time when a system first reaches any product-containing state, using absorbing-state formulations and first-passage concepts.
• It captures the full distribution of first product formation, providing insights into kinetic heterogeneity, extreme-event effects, and transport-controlled dynamics.
• Applications span chemical kinetics, gene expression, and enzymatic reactions, helping to elucidate mechanisms behind delayed injections, proofreading, and inhibition.

Searching arXiv for recent FPFT-related papers and foundational first-passage formulations. {"^^^^^^^^8query8 First Product Formation Time biochemical reaction networks first passage time product formation8^^^^^^^^", "^^^^^^^^8max_results8 ^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8query8^^^^, "^^^^^^^^8sort_by8 "^^^^^^^^8relevance8 {"^^^^^^^^8query8 Product Formation Time\"^^^^^^^^8 OR abs:\8^^^^^^^^"first product formation time\"", "^^^^^^^^8max_results8 ^^^^8max_results8query8^^^^, "^^^^^^^^8sort_by8 "^^^^^^^^8relevance8 {"^^^^^^^^8query8 "^^^^^^^^8max_results8 ^^^^8query8^^^^, "^^^^^^^^8sort_by8 "^^^^^^^^8relevance8 First Product Formation Time (FPFT) is the random time at which a stochastic system first reaches any state in which product has been formed. In chemical-reaction-network language, this is the first-passage time from the transient set PRESERVED_PLACEHOLDER_^^^^8query8^^^^ to the absorbing set PRESERVED_PLACEHOLDER_^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8^^^^; in transport-controlled settings it is the first arrival time to a reactive target, possibly followed by an additional reaction latency; and in many-searcher systems it coincides with the fastest successful arrival when product forms upon first encounter (&&&^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8^^^^&&&). Across these settings, FPFT is not merely a mean timescale but a full distribution whose survival, hazard, moments, and asymptotics encode mechanistic structure, heterogeneity, and extreme-event effects (&&&^^^^8query8^^^^&&&).

^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8^^^^. Definitions and conceptual scope

The core definition is absorbing-state based. In a stochastic reaction network, FPFT is the random time PRESERVED_PLACEHOLDER_^^^^8max_results8^^^^ at which the system first reaches any state with product count PRESERVED_PLACEHOLDER_^^^^8sort_by8^^^^. The absorbing set is therefore the set of all product-containing states, while the transient set contains all states with PRESERVED_PLACEHOLDER_^^^^8relevance8^^^^. Survival is the probability that no product has formed by time PRESERVED_PLACEHOLDER_^^^^8query8^^^^, and the FPFT density is the probability flux into the absorbing set at time PRESERVED_PLACEHOLDER_^^^^8ti:\8^^^^ (&&&^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8^^^^&&&).

This same object admits a transport interpretation. If one or more particles search for an immobile target and product forms immediately upon first arrival, FPFT equals the fastest first-passage time. With particle-specific entrance delays PRESERVED_PLACEHOLDER_^^^^8 OR abs:\8^^^^ and search times $\tau_k$, the fastest arrival is

$$T_N=\min\{\tau_1+\delta_1,\ldots,\tau_N+\delta_N\}.$$

If reaction is immediate on contact, then PRESERVED_PLACEHOLDER_^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8query8^^^^. If product formation requires an additional independent latency PRESERVED_PLACEHOLDER_^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8^^^^, then

PRESERVED_PLACEHOLDER_^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8max_results8^^^^

and the FPFT density is the convolution

PRESERVED_PLACEHOLDER_^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8sort_by8^^^^

when PRESERVED_PLACEHOLDER_^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8relevance8^^^^ has density PRESERVED_PLACEHOLDER_^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8query8^^^^ (&&&^^^^8query8^^^^&&&).

A related threshold-based usage appears in stochastic gene expression. There the general first-passage time PRESERVED_PLACEHOLDER_^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8ti:\8^^^^ becomes FPFT when the threshold equals one product molecule, PRESERVED_PLACEHOLDER_^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8 OR abs:\8^^^^. Under the burst model with constant transcription rate, the first product time is exponential, and under auto-regulation that depends only on protein count it remains exponential because before the first product appears the state is deterministically PRESERVED_PLACEHOLDER_^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation88^^^^ (&&&^^^^8query8^^^^&&&). This suggests that FPFT is often the simplest nontrivial threshold statistic, but not necessarily the most informative one for high-threshold events.

^^^^8max_results8^^^^. Absorbing-state formulation on Markov networks

For continuous-time Markov chains, one orders states so that transient states precede absorbing product states and partitions the generator as

PRESERVED_PLACEHOLDER_^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8^^^^9

After making PRESERVED_PLACEHOLDER_^^^^8max_results8query8^^^^ absorbing, the reduced dynamics on PRESERVED_PLACEHOLDER_^^^^8max_results8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8^^^^ is

PRESERVED_PLACEHOLDER_^^^^8max_results8max_results8^^^^

If the initial distribution PRESERVED_PLACEHOLDER_^^^^8max_results8sort_by8^^^^ is supported on PRESERVED_PLACEHOLDER_^^^^8max_results8relevance8^^^^, then

PRESERVED_PLACEHOLDER_^^^^8max_results8query8^^^^

and the hazard rate is

PRESERVED_PLACEHOLDER_^^^^8max_results8ti:\8^^^^

The mean and variance follow from resolvent identities,

PRESERVED_PLACEHOLDER_^^^^8max_results8 OR abs:\8^^^^

PRESERVED_PLACEHOLDER_^^^^8max_results88^^^^

and the Laplace transform of the density is

PRESERVED_PLACEHOLDER_^^^^8max_results8^^^^9

(&&&^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8^^^^&&&).

For discrete-time Markov chains with transient block PRESERVED_PLACEHOLDER_^^^^8sort_by8query8^^^^ and absorption block PRESERVED_PLACEHOLDER_^^^^8sort_by8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8^^^^,

PRESERVED_PLACEHOLDER_^^^^8sort_by8max_results8^^^^

the survival after PRESERVED_PLACEHOLDER_^^^^8sort_by8sort_by8^^^^ steps is

PRESERVED_PLACEHOLDER_^^^^8sort_by8relevance8^^^^

the FPFT mass function is

PRESERVED_PLACEHOLDER_^^^^8sort_by8query8^^^^

and the mean number of steps is

PRESERVED_PLACEHOLDER_^^^^8sort_by8ti:\8^^^^

(&&&^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8^^^^&&&).

A complementary exact discrete formulation is obtained by edge splitting. Each original edge is replaced by a cascade of PRESERVED_PLACEHOLDER_^^^^8sort_by8 OR abs:\8^^^^ unidirectional edges with rate PRESERVED_PLACEHOLDER_^^^^8sort_by88^^^^; as PRESERVED_PLACEHOLDER_^^^^8sort_by8^^^^9, the travel time along an original edge converges to a delta function centered at PRESERVED_PLACEHOLDER_^^^^8relevance8query8^^^^, so the continuous first-passage problem on the expanded graph approaches the discrete first-passage time problem. In that setting,

PRESERVED_PLACEHOLDER_^^^^8relevance8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8^^^^

and predecessor-resolved statistics are

PRESERVED_PLACEHOLDER_^^^^8relevance8max_results8^^^^

(Albert, 2024). For FPFT, this formulation is useful when product formation is naturally counted in reaction steps rather than physical time.

^^^^8sort_by8^^^^. Transport-controlled FPFT, fastest arrivals, and delayed injection

In diffusion-limited systems, FPFT may be determined by the earliest successful arrival among many searchers. For PRESERVED_PLACEHOLDER_^^^^8relevance8sort_by8^^^^ i.i.d. first-passage times with single-searcher survival PRESERVED_PLACEHOLDER_^^^^8relevance8relevance8^^^^ and density PRESERVED_PLACEHOLDER_^^^^8relevance8query8^^^^,

PRESERVED_PLACEHOLDER_^^^^8relevance8ti:\8^^^^

and the fastest-time hazard is PRESERVED_PLACEHOLDER_^^^^8relevance8 OR abs:\8^^^^. The mean obeys

PRESERVED_PLACEHOLDER_^^^^8relevance88^^^^

(Lawley, 2023). This order-statistics structure is the transport analogue of first product formation by the first successful searcher.

When particles enter the domain over an extended time window rather than simultaneously, the delayed single-particle survival and density become temporal mixtures,

PRESERVED_PLACEHOLDER_^^^^8relevance8^^^^9

so that for random injection

PRESERVED_PLACEHOLDER_^^^^8query8query8^^^^

For deterministic injection,

PRESERVED_PLACEHOLDER_^^^^8query8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8^^^^

with PRESERVED_PLACEHOLDER_^^^^8query8max_results8^^^^, so deterministic injection is stochastically faster than random injection for the same PRESERVED_PLACEHOLDER_^^^^8query8sort_by8^^^^ (&&&^^^^8query8^^^^&&&).

The large-PRESERVED_PLACEHOLDER_^^^^8query8relevance8^^^^ behavior depends on the short-time structure of both search and injection. If

PRESERVED_PLACEHOLDER_^^^^8query8query8^^^^

and

PRESERVED_PLACEHOLDER_^^^^8query8ti:\8^^^^

then the delayed survival has the unified form

PRESERVED_PLACEHOLDER_^^^^8query8 OR abs:\8^^^^

This yields

PRESERVED_PLACEHOLDER_^^^^8query88^^^^

Three regimes are distinguished. If PRESERVED_PLACEHOLDER_^^^^8query8^^^^9, the leading PRESERVED_PLACEHOLDER_^^^^8ti:\8query8^^^^ scale is unchanged but convergence is much slower. If PRESERVED_PLACEHOLDER_^^^^8ti:\8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8^^^^, PRESERVED_PLACEHOLDER_^^^^8ti:\8max_results8^^^^ is replaced by PRESERVED_PLACEHOLDER_^^^^8ti:\8sort_by8^^^^. If PRESERVED_PLACEHOLDER_^^^^8ti:\8relevance8^^^^, the leading decay becomes PRESERVED_PLACEHOLDER_^^^^8ti:\8query8^^^^, independent of PRESERVED_PLACEHOLDER_^^^^8ti:\8ti:\8^^^^ in leading order (&&&^^^^8query8^^^^&&&).

Extreme-value limits are likewise regime dependent. Lower tails of the form PRESERVED_PLACEHOLDER_^^^^8ti:\8 OR abs:\8^^^^ produce Gumbel limits after centering and scaling, whereas power-law lower tails PRESERVED_PLACEHOLDER_^^^^8ti:\88^^^^ produce Weibull limits. In diffusion with starting positions bounded away from the target,

PRESERVED_PLACEHOLDER_^^^^8ti:\8^^^^9

while uniform starts including arbitrarily near-target points can give PRESERVED_PLACEHOLDER_^^^^8 OR abs:\8query8^^^^ or PRESERVED_PLACEHOLDER_^^^^8 OR abs:\8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8^^^^ scaling, depending on boundary reactivity (Lawley, 2023). A related message from partially reactive intracellular search is that the full first-passage distribution is often broad, the MFPT can differ from the mode by orders of magnitude, and finite reactivity can generate long plateaus and strong mean–mode separation (&&&^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8sort_by8^^^^&&&). This suggests that FPFT is often controlled by rare-event structure rather than by a single characteristic timescale.

^^^^8relevance8^^^^. Exact FPFT distributions in chemical reaction networks

For monomolecular reaction networks, FPFT is often analytically tractable. In the simplest single-channel case PRESERVED_PLACEHOLDER_^^^^8 OR abs:\8max_results8^^^^ with rate PRESERVED_PLACEHOLDER_^^^^8 OR abs:\8sort_by8^^^^ and fixed initial count PRESERVED_PLACEHOLDER_^^^^8 OR abs:\8relevance8^^^^,

PRESERVED_PLACEHOLDER_^^^^8 OR abs:\8query8^^^^

With random initial count PRESERVED_PLACEHOLDER_^^^^8 OR abs:\8ti:\8^^^^,

PRESERVED_PLACEHOLDER_^^^^8 OR abs:\8 OR abs:\8^^^^

so the survival is the pgf of the initial distribution evaluated at PRESERVED_PLACEHOLDER_^^^^8 OR abs:\88^^^^. Competing monomolecular channels remain exactly reducible to linear-generator formulas, and arbitrary initial conditions enter linearly through convex combinations of delta-initial solutions (&&&^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8relevance8^^^^&&&).

For the nonlinear bimolecular reaction PRESERVED_PLACEHOLDER_^^^^8 OR abs:\8^^^^9, exact FPFT results require a different construction. One introduces an auxiliary species $\tau_k$^^^^8query8^^^^ produced only by the second-order event, so FPFT is

$\tau_k$^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8^^^^

For the class consisting of one second-order reaction $\tau_k$^^^^8max_results8^^^^ together with arbitrary zero- or first-order upstream reactions, the exact survival under Poisson-product initial conditions is

$\tau_k$^^^^8sort_by8^^^^

where $\tau_k$^^^^8relevance8^^^^ is coupled to stochastic mean processes obeying linear SDEs,

$\tau_k$^^^^8query8^^^^

$\tau_k$^^^^8ti:\8^^^^

The FPFT density and hazard are then

$\tau_k$^^^^8 OR abs:\8^^^^

and $\tau_k$8 matches the standard conditional-propensity identity $\tau_k$9 (&&&^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8query8^^^^&&&).

The ^^^^8max_results8query8max_results8query8^^^^ extension to arbitrary initial conditions gives an exact operator construction. For fixed-count initial condition $$T_N=\min\{\tau_1+\delta_1,\ldots,\tau_N+\delta_N\}.$$^^^^8query8^^^^,

$$T_N=\min\{\tau_1+\delta_1,\ldots,\tau_N+\delta_N\}.$$^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8^^^^

and for arbitrary discrete initial distribution $$T_N=\min\{\tau_1+\delta_1,\ldots,\tau_N+\delta_N\}.$$^^^^8max_results8^^^^,

$$T_N=\min\{\tau_1+\delta_1,\ldots,\tau_N+\delta_N\}.$$^^^^8sort_by8^^^^

This removes the earlier restriction to Poisson initial conditions and makes non-Poisson initial heterogeneity an explicit determinant of the full FPFT law rather than just its first moment (&&&^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8relevance8^^^^&&&).

From a computational standpoint, both exact second-order approaches emphasize moment-based evaluation rather than state-space enumeration. In the SDE representation, one computes moments of $$T_N=\min\{\tau_1+\delta_1,\ldots,\tau_N+\delta_N\}.$$^^^^8relevance8^^^^, constructs a Padé approximant to $$T_N=\min\{\tau_1+\delta_1,\ldots,\tau_N+\delta_N\}.$$^^^^8query8^^^^, and evaluates $$T_N=\min\{\tau_1+\delta_1,\ldots,\tau_N+\delta_N\}.$$^^^^8ti:\8^^^^ (&&&^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8query8^^^^&&&). In the arbitrary-initial-condition framework, one either applies the differential operator to the Poisson-product solution or computes survival by the reduced generator on the transient set (&&&^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8relevance8^^^^&&&).

^^^^8query8^^^^. Enzymatic FPFT, inhibition, and proofreading

In stochastic Michaelis–Menten kinetics with reversible inhibitors, FPFT is the time to the first product molecule under absorbing boundary $$T_N=\min\{\tau_1+\delta_1,\ldots,\tau_N+\delta_N\}.$$^^^^8 OR abs:\8^^^^. The paper reformulates the master equation in Fock space,

$$T_N=\min\{\tau_1+\delta_1,\ldots,\tau_N+\delta_N\}.$$8

with product-forming configurations treated as absorbing. The survival and density are

$$T_N=\min\{\tau_1+\delta_1,\ldots,\tau_N+\delta_N\}.$$9

For many-copy partial inhibition,

PRESERVED_PLACEHOLDER_^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8query8query8^^^^

(&&&^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8^^^^9&&&).

A central result is the emergence of an intermediate timescale in inhibited Michaelis–Menten kinetics. Without inhibitors, first-passage observables show a fast initial timescale and a slow long-time exponential tail. With competitive, uncompetitive, or noncompetitive inhibition, additional inhibitor-binding pathways introduce subleading eigenmodes, so the FPFT density becomes a finite mixture of exponentials,

PRESERVED_PLACEHOLDER_^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8query8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8^^^^

and exhibits a distinct intermediate exponential segment between the short-time onset and the long-time tail (&&&^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8^^^^9&&&). The same study shows that in partial inhibition the inhibitor can act effectively as an activator when PRESERVED_PLACEHOLDER_^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8query8max_results8^^^^, is kinetically equivalent to the uninhibited case when PRESERVED_PLACEHOLDER_^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8query8sort_by8^^^^, and hinders product formation when PRESERVED_PLACEHOLDER_^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8query8relevance8^^^^. This is a direct reminder that longer or more branched reaction pathways do not necessarily imply slower first product formation.

In kinetic proofreading, FPFT is the time to first product after binding and processing. In the convolved model,

PRESERVED_PLACEHOLDER_^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8query8query8^^^^

with activation delay PRESERVED_PLACEHOLDER_^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8query8ti:\8^^^^. The first-passage discrimination strategy yields exponential gains in accuracy with proofreading time but at a speed cost. In DNA replication,

PRESERVED_PLACEHOLDER_^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8query8 OR abs:\8^^^^

while the mean first-passage time grows exponentially with PRESERVED_PLACEHOLDER_^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8query88^^^^ (&&&^^^^8max_results8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8^^^^&&&).

By contrast, product-counting strategies do not necessarily improve with longer proofreading. The same work shows that product-based channel capacity has an approximately PRESERVED_PLACEHOLDER_^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8query8^^^^9-independent optimal PRESERVED_PLACEHOLDER_^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8query8^^^^, and that thresholding product counts decomposes the product-based strategy into a sequence of first-passage problems through times PRESERVED_PLACEHOLDER_^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8^^^^ (&&&^^^^8max_results8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8^^^^&&&). A plausible implication is that whether proofreading improves “FPFT performance” depends on the decision variable: first product time, first activation time, and product count can rank mechanisms differently.

^^^^8ti:\8^^^^. Statistical interpretation, inference, and limitations

A recurrent theme across FPFT theory is that full distributions matter more than mean times alone. In intracellular diffusion to partially reactive targets, the FPT distribution is often broad, the most probable time depends strongly on the starting position and only weakly on target size and reactivity, and the MFPT can exceed the mode by orders of magnitude (&&&^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8sort_by8^^^^&&&). In many-searcher systems with extended injection, Gumbel asymptotics may require extremely large PRESERVED_PLACEHOLDER_^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8max_results8^^^^, and for practical regimes from few tens to few thousands of particles the body of the distribution and the variance can be poorly captured by asymptotic formulas (&&&^^^^8query8^^^^&&&). In gene-expression threshold models, by contrast, FPFT at threshold PRESERVED_PLACEHOLDER_^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8sort_by8^^^^ is exponential with PRESERVED_PLACEHOLDER_^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8relevance8^^^^, and auto-regulation depending on protein count cannot affect FPFT variability because regulation cannot act before the first product appears (&&&^^^^8query8^^^^&&&).

For inference and computation, the workflow depends on the physical mechanism. In reaction networks one identifies the transient set PRESERVED_PLACEHOLDER_^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8query8^^^^ and absorbing set PRESERVED_PLACEHOLDER_^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8ti:\8^^^^, builds the reduced generator PRESERVED_PLACEHOLDER_^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8 OR abs:\8^^^^ and product flux block PRESERVED_PLACEHOLDER_^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8arXiv First Product Formation Time biochemical reaction networks first passage time product formation88^^^^, and computes

PRESERVED_PLACEHOLDER_^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8^^^^9

or their discrete-time analogues (&&&^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8^^^^&&&). In transport-mediated FPFT with delayed injection, one first determines PRESERVED_PLACEHOLDER_^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8max_results8query8^^^^ and PRESERVED_PLACEHOLDER_^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8max_results8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8^^^^ for single-particle transport, then convolves with the entrance-time profile PRESERVED_PLACEHOLDER_^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8max_results8max_results8^^^^ to obtain PRESERVED_PLACEHOLDER_^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8max_results8sort_by8^^^^, PRESERVED_PLACEHOLDER_^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8max_results8relevance8^^^^, and finally PRESERVED_PLACEHOLDER_^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8max_results8query8^^^^ (&&&^^^^8query8^^^^&&&). In second-order biochemical networks one may instead compute moment hierarchies for PRESERVED_PLACEHOLDER_^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8max_results8ti:\8^^^^ and reconstruct PRESERVED_PLACEHOLDER_^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8max_results8 OR abs:\8^^^^ by Padé approximation (&&&^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8query8^^^^&&&).

The principal limitations are equally consistent across formulations. Common assumptions are independence of particles or searchers, well-mixed or Markovian dynamics, and time-homogeneous transport after injection. Large-PRESERVED_PLACEHOLDER_^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8max_results88^^^^ asymptotics are dominated by short-time behavior and may fail when Brownian short-time structure is physically invalid, when finite-speed effects matter, or when practical sample sizes do not reach the asymptotic regime (&&&^^^^8query8^^^^&&&). Exact second-order chemical-network results presently cover one PRESERVED_PLACEHOLDER_^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8max_results8^^^^9-type reaction with zero- or first-order context, not multiple second-order reactions or reversible bimolecular schemes (&&&^^^^8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8query8^^^^&&&). Product-counting and first-passage strategies can disagree on what constitutes “better” discrimination in proofreading systems (&&&^^^^8max_results8arXiv First Product Formation Time biochemical reaction networks first passage time product formation8^^^^&&&). These caveats do not weaken the FPFT framework; they define its domain of validity and show why mechanistic specification of the absorbing event is essential.

In that sense, FPFT is best understood as a family of first-passage observables indexed by the definition of “first product”: first visit to a product-containing state, first successful arrival at a reactive target, first passage of a copy-number threshold, or first activation within a branched kinetic scheme. The common mathematical structure is absorbing probability flux. The substantive differences arise from transport mode, injection statistics, network nonlinearity, initial-condition heterogeneity, and the specific experimental decision rule used to declare that product formation has occurred.