--- title: First Product Formation Time (FPFT) url: https://www.emergentmind.com/topics/first-product-formation-time-fpft type: topic --- # First Product Formation Time (FPFT) Searching arXiv for recent FPFT-related papers and foundational first-passage formulations. {"query":"arXiv First Product Formation Time biochemical reaction networks first passage time product formation", "max_results": 10, "sort_by": "relevance"} {"query":"ti:\"First Product Formation Time\" OR abs:\"first product formation time\"", "max_results": 20, "sort_by": "relevance"} {"query":"2503.19105", "max_results": 5, "sort_by": "relevance"} 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 \(N=\{x:n_P=0\}\) to the absorbing set \(A=\{x:n_P\ge 1\}\); 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 [2110.02216]. 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 [2503.19105]. ## 1. Definitions and conceptual scope The core definition is absorbing-state based. In a stochastic reaction network, FPFT is the random time \(T\) at which the system first reaches any state with product count \(n_P \ge 1\). The absorbing set is therefore the set of all product-containing states, while the transient set contains all states with \(n_P=0\). Survival is the probability that no product has formed by time \(t\), and the FPFT density is the probability flux into the absorbing set at time \(t\) [2110.02216]. 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 \(\delta_k\) 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 \(\mathrm{FPFT}=T_N\). If product formation requires an additional independent latency \(\tau_{\mathrm{react}}\), then \[ \mathrm{FPFT}=T_N+\tau_{\mathrm{react}}, \] and the FPFT density is the convolution \[ p_{\mathrm{FPFT}}(t)=\int_0^t H_N(s)\,g(t-s)\,ds \] when \(\tau_{\mathrm{react}}\) has density \(g\) [2503.19105]. A related threshold-based usage appears in stochastic gene expression. There the general first-passage time \(T_N:=\inf\{t\ge 0:X(t)\ge N\}\) becomes FPFT when the threshold equals one product molecule, \(N=1\). 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 \(X(t)=0\) [1405.3226]. This suggests that FPFT is often the simplest nontrivial threshold statistic, but not necessarily the most informative one for high-threshold events. ## 2. 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 \[ K=\begin{pmatrix} K_{NN} & K_{NA}\\ K_{AN} & K_{AA} \end{pmatrix}. \] After making \(A\) absorbing, the reduced dynamics on \(N\) is \[ \frac{d}{dt}\vec{P}^{\,*}(t)=K_{NN}\vec{P}^{\,*}(t),\qquad \vec{P}^{\,*}(t)=e^{K_{NN}t}\vec{P}^{\,*}(0). \] If the initial distribution \(p_0\) is supported on \(N\), then \[ S(t)=p_0^\top e^{K_{NN}t}\mathbf{1}, \qquad f(t)=p_0^\top e^{K_{NN}t}K_{NA}\mathbf{1}=-\frac{d}{dt}S(t), \] and the hazard rate is \[ h(t)=\frac{f(t)}{S(t)}. \] The mean and variance follow from resolvent identities, \[ \mathbb{E}[T]=p_0^\top(-K_{NN})^{-1}\mathbf{1}, \] \[ \mathbb{E}[T^2]=2\,p_0^\top(-K_{NN})^{-2}\mathbf{1}, \qquad \mathrm{Var}(T)=2\,p_0^\top(-K_{NN})^{-2}\mathbf{1} -\bigl(p_0^\top(-K_{NN})^{-1}\mathbf{1}\bigr)^2, \] and the Laplace transform of the density is \[ \tilde f(s)=p_0^\top (sI-K_{NN})^{-1}K_{NA}\mathbf{1} \] [2110.02216]. For discrete-time Markov chains with transient block \(Q\) and absorption block \(R\), \[ \mathcal{K}= \begin{pmatrix} Q & R\\ \cdot & \cdot \end{pmatrix}, \] the survival after \(n\) steps is \[ S_n=p_0^\top Q^n\mathbf{1}, \] the FPFT mass function is \[ f_n=p_0^\top Q^{n-1}R\mathbf{1}=S_{n-1}-S_n, \] and the mean number of steps is \[ \mathbb{E}[N]=p_0^\top (I-Q)^{-1}\mathbf{1} \] [2110.02216]. A complementary exact discrete formulation is obtained by edge splitting. Each original edge is replaced by a cascade of \(h\) unidirectional edges with rate \(h\); as \(h\to\infty\), the travel time along an original edge converges to a delta function centered at \(1\), so the continuous first-passage problem on the expanded graph approaches the discrete first-passage time problem. In that setting, \[ \Pr[T=n]=\mathbf{e}_s^\top Q_d^{\,n-1}R_d, \qquad G(z)=z\,\mathbf{e}_s^\top (I-zQ_d)^{-1}R_d, \] and predecessor-resolved statistics are \[ \Pr[T=n,\text{prev}=i]=\bigl(\mathbf{e}_s^\top Q_d^{\,n-1}\bigr)_i\,P_{ij} \] [2403.14149]. For FPFT, this formulation is useful when product formation is naturally counted in reaction steps rather than physical time. ## 3. 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 \(N\) i.i.d. first-passage times with single-searcher survival \(S_1(t)\) and density \(f_1(t)\), \[ S_{\min}(t)=[S_1(t)]^N,\qquad f_{\min}(t)=Nf_1(t)[S_1(t)]^{N-1}, \] and the fastest-time hazard is \(h_{\min}(t)=Nh_1(t)\). The mean obeys \[ \mathbb{E}[T_N]=\int_0^\infty [S_1(t)]^N\,dt \] [2310.02157]. 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, \[ S_\psi(t)=\int_0^\infty \psi(t')S(t-t')\,dt', \qquad H_\psi(t)=\int_0^t \psi(t')h(t-t')\,dt', \] so that for random injection \[ S_N(t)=[S_\psi(t)]^N,\qquad H_N(t)=N\,H_\psi(t)\,[S_\psi(t)]^{N-1}. \] For deterministic injection, \[ \bar S_\psi(t)=\exp\!\left(\int_0^t \psi(t')\ln S(t-t')\,dt'\right), \qquad \bar H_N(t)=-\frac{d}{dt}[\bar S_\psi(t)]^N, \] with \(\bar S_\psi(t)\le S_\psi(t)\), so deterministic injection is stochastically faster than random injection for the same \(\psi\) [2503.19105]. The large-\(N\) behavior depends on the short-time structure of both search and injection. If \[ 1-S(t)\sim A\,t^\alpha e^{-C/t} \] and \[ \psi(t)\approx a\,t^{\nu-1}e^{-(c/t)^\mu}, \] then the delayed survival has the unified form \[ 1-S_\psi(t)\approx \bar A\,t^{\bar\alpha}e^{-(\bar C/t)^{\bar\mu}}. \] This yields \[ \mathbb{E}[T_N]\approx \frac{\bar C}{(\ln N)^{1/\bar\mu}} \left( 1+\frac{\frac{\bar\alpha}{\bar\mu}\ln\ln N -\ln\!\bigl(\bar A\,\bar C^{\bar\alpha}e^{\gamma}\bigr)} {\bar\mu\,\ln N} \right). \] Three regimes are distinguished. If \(\mu<1\), the leading \(C/\ln N\) scale is unchanged but convergence is much slower. If \(\mu=1\), \(C\) is replaced by \((\sqrt C+\sqrt c)^2\). If \(\mu>1\), the leading decay becomes \(c(\ln N)^{-1/\mu}\), independent of \(C\) in leading order [2503.19105]. Extreme-value limits are likewise regime dependent. Lower tails of the form \(A t^p e^{-C/t}\) produce Gumbel limits after centering and scaling, whereas power-law lower tails \(A t^p\) produce Weibull limits. In diffusion with starting positions bounded away from the target, \[ \mathbb{E}[T_N]\sim \frac{L^2}{4D\ln N}, \] while uniform starts including arbitrarily near-target points can give \(N^{-2}\) or \(N^{-1}\) scaling, depending on boundary reactivity [2310.02157]. 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 [1811.11612]. This suggests that FPFT is often controlled by rare-event structure rather than by a single characteristic timescale. ## 4. Exact FPFT distributions in chemical reaction networks For monomolecular reaction networks, FPFT is often analytically tractable. In the simplest single-channel case \(A\to C\) with rate \(k\) and fixed initial count \(n\), \[ S(t)=e^{-knt},\qquad f(t)=kn\,e^{-knt},\qquad \mathbb{E}[T]=\frac{1}{kn}. \] With random initial count \(N_A(0)\), \[ S(t)=\sum_{n=0}^\infty P(N_A(0)=n)e^{-knt} =G_{N_A(0)}(e^{-kt}), \] so the survival is the pgf of the initial distribution evaluated at \(e^{-kt}\). Competing monomolecular channels remain exactly reducible to linear-generator formulas, and arbitrary initial conditions enter linearly through convex combinations of delta-initial solutions [2503.04477]. For the nonlinear bimolecular reaction \(A+B\to C\), exact FPFT results require a different construction. One introduces an auxiliary species \(S_0\) produced only by the second-order event, so FPFT is \[ T\equiv \inf\{t>0:x_0(t)\ge 1\}, \qquad S(t)=P(x_0(t)=0). \] For the class consisting of one second-order reaction \(S_1+S_2\to S_0\) together with arbitrary zero- or first-order upstream reactions, the exact survival under Poisson-product initial conditions is \[ S(t)=\langle e^{\lambda_S(t)}\rangle, \] where \(\lambda_S(t)\) is coupled to stochastic mean processes obeying linear SDEs, \[ d\lambda=(\mathcal M_1\lambda+\mathcal M_2)\,dt+\mathcal N_1\lambda\,dW_t^1+\mathcal N_2\lambda\,dW_t^2, \] \[ d\lambda_S=(a_S\lambda_1-a_S\lambda_2)\,dW_t^1+(i a_S\lambda_1+i a_S\lambda_2)\,dW_t^2, \qquad d\lambda_0=a_0\lambda_1\lambda_2\,dt. \] The FPFT density and hazard are then \[ f(t)=-\frac{d}{dt}S(t), \qquad h(t)=-\frac{d}{dt}\log S(t), \] and \(h(t)\) matches the standard conditional-propensity identity \(E[a_0(t)X_1(t)X_2(t)\mid x_0(t)=0]\) [2409.02698]. The 2025 extension to arbitrary initial conditions gives an exact operator construction. For fixed-count initial condition \(\mathbf Z_0\), \[ S(t)= \Big(\nabla_{\boldsymbol{\theta}+1}\Big)^{\mathbf Z_0} \left\langle e^{\lambda_S(t;\boldsymbol{\theta})}\right\rangle\Big|_{\boldsymbol{\theta}=0}, \] and for arbitrary discrete initial distribution \(d_{\mathbf y}\), \[ S(t)= \sum_{\mathbf y} d_{\mathbf y} \Big(\nabla_{\boldsymbol{\theta}+1}\Big)^{\mathbf y} \left\langle e^{\lambda_S(t;\boldsymbol{\theta})}\right\rangle\Big|_{\boldsymbol{\theta}=0}. \] 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 [2503.04477]. 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 \(\lambda_S\), constructs a Padé approximant to \(H(s,t)=\langle e^{s\lambda_S(t)}\rangle\), and evaluates \(S(t)\approx H(1,t)\) [2409.02698]. 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 [2503.04477]. ## 5. 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 \(n_P\ge 1\). The paper reformulates the master equation in Fock space, \[ \frac{\partial}{\partial t}\ket{\Psi(t)}=-\boldsymbol H\ket{\Psi(t)}, \qquad \ket{\Psi(t)}=e^{-\boldsymbol H t}\ket{\Psi(0)}, \] with product-forming configurations treated as absorbing. The survival and density are \[ S(t)=\sum_{\eta\neq\{\eta_{\rm FP}\}}P_\eta(t), \qquad f(t)=\sum_{\eta\neq\{\eta_{\rm FP}\}}\sum_{\eta'=\{\eta_{\rm FP}\}} T_{\eta\to\eta'}P_\eta(t). \] For many-copy partial inhibition, \[ f(t)=k_2\langle n_{C_1}(t)\rangle_{\eta\neq\{\eta_{\rm FP}\}} +k_6\langle n_{C_3}(t)\rangle_{\eta\neq\{\eta_{\rm FP}\}} \] [2508.11645]. 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, \[ f(t)=\sum_{i=1}^{N'}\alpha_i e^{-\bar\lambda_i t}, \qquad \sum_{i=1}^{N'}\alpha_i=0, \] and exhibits a distinct intermediate exponential segment between the short-time onset and the long-time tail [2508.11645]. The same study shows that in partial inhibition the inhibitor can act effectively as an activator when \(k_6>k_2\), is kinetically equivalent to the uninhibited case when \(k_6=k_2\), and hinders product formation when \(k_6