6. Accumulation of biomass (X) in a batch systems is derived from degradation of pyruvate (U) an intermediate of glucose (S) degradation through glycolysis. The two-step process is defined by a system of two differential equations, and cells die following a first-order death rate law. The system in batch is governed by the kinetics listed below: dS .S S-S X dt S K + S + K du dt = kmuU K+U dX =kx dt where Scalculated substrate concertation at any time t (h) (Eq. 1) (Eq. 2) (Eq. 3) U calculated intermediate (pyruvate) concentration at any time t (h) X= calculated cell concentration at any time t (h) Kms = maximum substrate utilisation rate coefficient (h) kmu maximum pyruvate utilisation rate coefficient (h) Ks half velocity concentration of substrate (mg/L) Ku half velocity concentration of intermediate (mg/L) K substrate inhibition constant (mg/L) Xo measured initial cell concentration (use 2000 mg/L) So data) measured initial substrate concentration (use 2000 mg/L, from current Rc cellular substrate utilisation capacity (mg/mg) kd = cell death rate coefficient (h) 16 (a) From substrate utilisation rate (Eq. 1) optimise the parameters Kms, Ks, Rc, and Ki. Using parameter sensitivity analysis, check whether the term S/Ki can be safely removed from this analysis. Measured data for all state variable is presented in Table 3 below. ( 30 pts) (b) Using initial stoichiometric coefficients of a = -1, b = 1 for S and U, simulate and optimise the coefficients kmu and Ku for the resultant pyruvate profile against the data provided in Table 3 below. ( 30 pts) (c) Using an initial stoichiometric coefficient Y = 0.5 for the accumulation of cells X from Eq. 2, and the cell degradation rate -dx/dt = kaX, optimise the parameters Yand ka and determine the cell accumulation profile. ( 30 pts) (d) Perform the parameter optimisation of the full system and present the data graphically together with the final programme report (*.prn file report). ( 45 pts) Table 3: Metabolic pathway data for utilisation of glucose to create biomass Time [h] Glucose [mg/L] Intermidate [mg/L] Biomass, X [mg/L] 4 O21682100220005 2000 20 5 1800 150 30 1650 550 80 1200 880 120 1000 980 320 800 1100 550 650 950 800 14 480 900 920 380 720 980 260 540 970 160 168 880 32 110 205 780 40 75 120 640 48 35 75 520 20 45 440 15 33 360