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Metabolic Pathways

Field: Enzymology

A linked series of chemical reactions occurring within a cell.

Sequence of Expressions

Definition

Flux

Let M={M1,M2,…,MN}\mathbf{M} = \{M_1, M_2, \dots, M_N\} be the set of NN metabolites in a system, and let v={v1,v2,…,vE}\mathbf{v} = \{v_1, v_2, \dots, v_E\} be the set of EE enzymatic reactions. The flux J∈RE\mathbf{J} \in \mathbb{R}^E is defined by the steady-state mass balance equation for each metabolite MiM_i: \begin{equation} \label{eq:mass_balance} \frac{d[M_i]}{dt} = \sum_{j=1}^{E} \nu_{ij} v_j = 0 \end{equation}, where νij\nu_{ij} is the stoichiometric coefficient of MiM_i in reaction jj, and vjv_j is the flux through reaction jj. The flux vector J\mathbf{J} must satisfy the stoichiometry matrix N∈RN×E\mathbf{N} \in \mathbb{R}^{N \times E} such that NJ=0\mathbf{N} \mathbf{J} = \mathbf{0}.
Consider two reactions, R1R_1 and R2R_2, where R1R_1 provides the energy to drive R2R_2. Let ΔG1\Delta G_1 and ΔG2\Delta G_2 be the standard free energy changes. The coupling requires that the energy released by R1R_1 is sufficient to overcome the free energy barrier of R2R_2. The overall free energy change ΔGcoupled\Delta G_{\text{coupled}} is the sum of the individual changes: \begin{equation} \Delta G_{\text{coupled}} = \Delta G_1 + \Delta G_2 \end{equation}. For the coupling to be thermodynamically favorable, the energy released by the exergonic reaction (e.g., R1R_1) must satisfy ΔG1 (exergonic)≥∣ΔG2∣ (endergonic)\Delta G_1 \text{ (exergonic)} \ge |\Delta G_2| \text{ (endergonic)}. This ensures the net reaction proceeds spontaneously.
Let vv be the initial reaction rate, [S][S] be the substrate concentration, VmaxV_{max} be the maximum reaction rate, and KmK_m be the Michaelis constant. The rate is defined by the function:\nv=Vmax[S]Km+[S]v = \frac{V_{max} [S]}{K_m + [S]}
Let vv be the reaction rate, [S][S] be the substrate concentration, VmaxV_{max} be the maximum rate, KmK_m be the apparent dissociation constant, and nn be the Hill coefficient (n≠1n \neq 1). The rate is modeled by:\nv=Vmax[S]nKmn+[S]nv = V_{max} \frac{[S]^n}{K_m^n + [S]^n}
Consider a metabolic pathway composed of EE sequential reactions, R1→R2→⋯→RER_1 \to R_2 \to \dots \to R_E, with individual maximum reaction rates Vmax,iV_{\text{max}, i}. The overall maximum flux JoverallJ_{\text{overall}} is constrained by the minimum capacity of the steps. Formally, the system flux JJ is bounded by the minimum of the individual reaction capacities: \begin{equation} J \le \min_{i=1}^{E} \left( V_{\text{max}, i} \right) \end{equation}. If the system operates under quasi-steady-state conditions, the rate-limiting step RkR_k is defined such that Vmax,k=min⁡i=1E(Vmax,i)V_{\text{max}, k} = \min_{i=1}^{E} \left( V_{\text{max}, i} \right), thereby determining the system's maximum throughput.
Let vcatv_{\text{cat}} be the maximum catalytic rate of an enzyme EE acting on substrate SS. If an inhibitor II is present, the observed reaction rate vobsv_{\text{obs}} is modified. For a general reversible inhibition mechanism, the rate equation is given by: \begin{equation} v_{\text{obs}} = \frac{V_{\text{max}} [S]}{K_m \left( 1 + \frac{[I]}{K_i} \right) + [S]} \end{equation}. Here, KmK_m is the Michaelis constant, and KiK_i is the inhibition constant. The inhibition factor α=(1+[I]Ki)\alpha = \left( 1 + \frac{[I]}{K_i} \right) quantifies the reduction in apparent enzyme activity due to the binding of II to the enzyme-substrate complex or free enzyme.
Consider the reaction E+S⇌ES→E+PE + S \rightleftharpoons ES \rightarrow E + P. The initial rate vv is defined by the rate law:\nv=kcat[E]0[S]KM+[S]v = \frac{k_{cat} [E]_0 [S]}{K_M + [S]} \nwhere [E]0[E]_0 is the total enzyme concentration, kcatk_{cat} is the turnover number, and KMK_M is the apparent dissociation constant.
Consider an enzyme EE regulated by an effector AA. The modified rate vAv_A can be expressed by modifying the apparent KmK_m or VmaxV_{max} based on the effector binding equilibrium. For an activator AA, the rate may be approximated as:\nvA=Vmax[S]Km1+[A]/KA1+[A]/KA1+[S]Km11+[A]/KAv_A = \frac{V_{max} \frac{[S]}{K_m} \frac{1 + [A]/K_A}{1 + [A]/K_A}}{1 + \frac{[S]}{K_m} \frac{1}{1 + [A]/K_A}} \nwhere KAK_A is the dissociation constant for the effector AA.
For an intermediate species II in a metabolic pathway, the Steady-State Approximation (SSA) dictates that the net rate of change of its concentration is zero:\nd[I]dt=0\frac{d[I]}{dt} = 0 \nThis implies that the rate of formation of II equals the rate of consumption of II at steady state.
For a metabolic pathway consisting of a series of reactions R1→R2→⋯→RkR_1 \to R_2 \to \dots \to R_k, the overall spontaneity and feasibility are governed by the change in Gibbs free energy (ΔG\Delta G). The total free energy change ΔGtotal\Delta G_{\text{total}} must be negative for the pathway to proceed spontaneously: \begin{equation} \Delta G_{\text{total}} = \sum_{i=1}^{k} \Delta G_i < 0 \end{equation}. Furthermore, the pathway must satisfy the conservation of energy and entropy, implying that the total change in entropy ΔStotal\Delta S_{\text{total}} must be positive, consistent with the Second Law of Thermodynamics: \begin{equation} \Delta S_{\text{total}} = \frac{\Delta G_{\text{total}}}{T} > 0 \end{equation}.