Optimal design for CSTR's in series performing enzymatic lactose hydrolysis

Analytical expressions are derived for the optimal design (based on minimum overall reactors volume) of a series of N CSTR's performing enzymatic lactose hydrolysis. It is assumed that lactose hydrolysis obeys Michaelis-Menten kinetics with competitive product (galactose) inhibition and no enzyme deactivation occurs. The optimum design of a cascade of ideally mixed reactors are compared with equal size reactors and with plug flow reactor required for a given overall degree of lactose conversion. The effect of operating parameters such as temperature, lactose initial (feed) concentration and conversion, enzyme and product initial concentration on the optimal overall holding time are also investigated. Optimization results for a series of N CSTR's up to five are obtained and compared with plug flow reactor.     

To what extent is a constant volume design worse than a minimum volume design for a series of CSTR's?

The balance equations for substrate in a cascade of CSTR's undergoing an enzyme-catalyzed reaction following Michaelis-Menten kinetics are developed in dimensionless form. Analytical expressions relating the intermediate concentrations are independently obtained for the cases of minimum overall volume and constant volume. The fractional deviations between the overall volumes following these two design criteria are calculated and presented for several values of the relevant parameters. For situations of practical interest, the fractional deviation is below 10%. Increasing values of the Michaelis-Menten parameter, K _m(or decreasing values of the number of reactors in the cascade, N) lead to lower values of the maximum deviation; this maximum deviation is attained at lower conversions of substrate when K _mis increased or N decreased.List of Symbols C _{S, i}mol.m^–3 concentration of substrate at the outlet of the i-th reactor - C _* ^{S, i} normalized concentration of substrate at the outlet of the i-th reactor - C _* ^{S, i, eq} normalized concentration of substrate at the outlet of the i-th reactor using the design criterion of constant volume - C _* ^{S, i, opt} normalized concentration of substrate at the outlet of the i-th reactor using the design criterion of minimum overall volume - C _{S, 0} mol.m^–3 concentration of substrate at the inlet to the first reactor - Da _i Damköhler number for the i-th reactor - Da _eq constant Damköhler number for each reactor of the cascade - Da _{tot, eq} overall Damköhler number for the cascade assuming equal-sized reactors - Da _{tot, min} minimum overall Damköhler number for the cascade - Er fractional deviation between the overall volumes using the two different design criteria - K _mmol. m^–3 Michaelis-Menten constant - K _* ^M dimensionless Michaelis-Menten constant - N number of reactors of the cascade - Q m³. s^–1 volumetric flow rate - V _im³ volume of the i-th reactor - v _max mol. m^–3. s^–1 reaction rate under saturation conditions of the enzyme with substrate - V _{tot, opt} m³ minimum overall volume of the cascade - V _{tot, eq} m³ overall volume of the cascade assuming equal-sized reactors     

Optimum design of a series of CSTRs performing reversible Michaelis-Menten kinetics: a rigorous mathematical study

The optimum design of a given number of CSTRs in series performing reversible Michaelis-Menten kinetics in the liquid phase assuming constant activity of the enzyme is studied. In this study, the presence of product in the feed stream to the first reactor, as well as the effect of the product intermediate concentrations in the downstream reactors on the reaction rate are investigated. For a given number of N CSTRs required to perform a certain degree of substrate conversion and under steady state operation and constant volumetric flow rate, the reactor optimization problem is posed as a constrained nonlinear programming problem (NLP). The reactor optimization is based on the minimum overall residence time (volume) of N reactors in series. When all the reactors in series operate isothermally, the constrained NLP is solved as an unconstrained NLP. And an analytical expression for the optimum overall residence time is obtained. Also, the necessary and sufficient conditions for the minimum overall residence time of N CSTRs are derived analytically. In the presence of product in the feed stream, the reversible Michaelis-Menten kinetics shows competitive product inhibition. And this is, because of the increase in the apparent rate constant K^' _m that results in a reduction of the overall reaction rate. The optimum total residence time is found to increase as the ratio (‚₀) of product to substrate concentrations in the feed stream increases. The isomerization of glucose to fructose, which follows a reversible Michaelis-Menten kinetics, is chosen as a model for the numerical examples.     

Optimal design of a number of membrane reactors in series using Michaelis-Menten one-substrate reversible kinetics

Summary Exact analytical expressions are derived for the optimal design (minimum overall reaction volume) of N perfectly mixed membrane reactors in series carrying out an enzyme catalysed Michaelis-Menten, one-substrate/one-product reversible reaction. The equations enable the direct calculation of the smallest total reactor volume needed for a given overall conversion degree. Results show that when substrate rejection is present, membrane reactors perform better compared with continuous stirred tank reactors.     

Aryl acylamidase activity of human serum albumin with o-nitrotrifluoroacetanilide as the substrate

Albumin is generally regarded as an inert protein with no enzyme activity. However, albumin has esterase activity as well as aryl acylamidase activity. A new acetanilide substrate, o-nitrotrifluoroacetanilide (o-NTFNAC), which is more reactive than the classical o-nitroacetanilide, made it possible to determine the catalytic parameters for hydrolysis by fatty-acid free human serum albumin. Owing to the low enzymatic activity of albumin, kinetic studies were performed at high albumin concentration (0.075 mM). The albumin behavior with this substrate was Michaelis-Menten like. Kinetic analysis was performed according to the formalism used for catalysis at high enzyme concentration. This approach provided values for the turnover and dissociation constant of the albumin-substrate complex: k_cat = 0.13 ± 0.02 min ? ¹ and K_s = 0.67 ± 0.04 mM. MALDI-TOF experiments showed that unlike the ester substrate p-nitrophenyl acetate, o-NTFNAC does not form a stable adduct (acetylated enzyme). Kinetic analysis and MALDI-TOF experiments demonstrated that hydrolysis of o-NTFNAC by albumin is fully rate-limited by the acylation step (k_cat = k₂). Though the aryl acylamidase activity of albumin is low (k_cat/K_s = 195 M^{? 1}min^{? 1}), because of its high concentration in human plasma (0.6–1 mM), albumin may participate in hydrolysis of aryl acylamides through second-order kinetics. This suggests that albumin may have a role in the metabolism of endogenous and exogenous aromatic amides, including drugs and xenobiotics.     

Optimal Design of a Series of GSTR's Performing Reversible Reactions Catalyzed by Soluble Enzymes: A Theoretical Study

The condition for the minimum overall reactor volume of a given number of CSTR's in series is theoretically determined for a reversible, single reactant-single product (Uni-Uni) enzyme catalyzed reaction. The reactor network is assumed to operate in steady-state, isothermal conditions with a single phase and a constant activity of biocatalyst. The method is based on a mathematical analysis of the discrete substrate concentration profile along the CSTR's assuming complete micromixing. The algebraic equations describing the critical loci are obtained for the general case, the mathematical proof that these equations define a minimum is presented, and an exact solution arising from an asymptotic situation is found. An approximate analytical method of optimization based on the aforementioned critical behavior is reported and its validity and usefulness discussed. The formulae introduced can be used in more general situations as tools for getting the approximate range where the optimal overall volume of the series of CSTR's lies. Hence, the reasoning developed is important for the preliminary CSTR design and relevant in the initial steps of the more involved methods of numerical optimization. Finally, the enzymatic conversion of fumarate to L-malate is examined as a model system in order to assess the usefulness and applicability of the analysis developed.     

Optimal temperature and concentration profiles in a cascade of CSTR's performing Michaelis-Menten reactions with first order enzyme deactivation

A necessary condition is found for the intermediate temperatures and substrate concentrations in a series of CSTR's performing an enzyme-catalyzed reaction which leads to the minimum overall volume of the cascade for given initial and final temperatures and substrate concentrations. The reaction is assumed to occur in a single phase under steady state conditions. The common case of Michaelis-Menten kinetics coupled with first order deactivation of the enzyme is considered. This analysis shows that intermediate stream temperatures play as important a role as intermediate substrate concentrations when optimizing in the presence of nonisothermal conditions. The general procedure is applied to a practical example involving a series of two reactors with reasonable values for the relevant five operating parameters. These parameters are defined as dimensionless ratios involving activation energies (or enthalpy changes of reaction), preexponential factors, and initial temperature and substrate concentration. For negligible rate of deactivation, the qptimality condition corresponds to having the ratio of any two consecutive concentrations as a single-parameter increasing function of the previous ratio of consecutive concentrations.List of Symbols C _E,0 mol.m^–3 Initial concentration of active enzyme - C _E,i mol.m^–3 Concentration of active enzyme at the outlet of the i-th reactor - C _S,0 mol.m^–3 Initial concentration of substrate - C _S,i mol.m^–3 Concentration of substrate at the outlet of the i-th reactor - Da _i Damköhler number associated with the i-th reactor ((V _i.k_v,0.C_E,0)/(Q.C_S,0)) - Da _min Minimum value of the overall Damköhler number - Da _tot Overall Damköhler number - E _d J.mol^–1 Activation energy of the step of deactivation of the enzyme - E _m J.mol^–1 Standard enthalpy change of the step of binding of substrate to the enzyme - E _v J.mol^–1 Activation energy of the step of enzymatic transformation of substrate - i Integer variable - j Dummy integer variable - k Dummy integer variable - k _d,i s^–1 Kinetic constant associated with the deactivation of enzyme in the i-th reactor (k _d,o·exp{–E _d/(R.T _i}) - k _d,0 s^–1 Preexponential factor of the kinetic constant associated with the deactivation of the enzyme - K _m,i mol.m^–3 Equilibrium constant associated with the binding of substrate to the enzyme in the i-th reactor, (k _m,o·exp{–E _m}(R.T _i}) - K _m,0 mol.m^–3 Preexponential factor of the Michaelis-Menten constant associated with the binding of substrate to the enzyme - k _v,i s^–1 Kinetic constant associated with the transformation of the substrate by the enzyme in the i-th reactor (k _v,o·exp{–E _v/(R.T _i})) - k _v,0 s^–1 Preexponential factor of the kinetic constant associated with the transformation of the substrate by the enzyme - N Number of reactors in the series - Q m³.s^–1 Volumetric flow rate of reacting liquid through the reactor network - R J.K^–1.mol^–1 Ideal gas constant - T _i K Absolute temperature at the outlet of the i-th reactor - T ₀ K Initial absolute temperature - V _i m³ Volume of the i-th reactor - v _max mol.m^–3.s^–1 Maximum rate of reaction under saturation conditions of substrate - x _i Normalized concentration of substrate (C_S,i/C_{S, 0}) - x _i,opt Optimum value of the normalized concentration of substrate - y _i Dimensionless temperature (exp{–T ₀/T _i}) - y _i,opt Optimum value of the dimensionless temperature Greek Symbols Dimensionless preexponential factor associated with the Michaelis-Menten constant (K _m,0/C_s,0) - Dimensionless activation energy of the step of enzymatic transformation of substrate (E _v/R.T₀)) - Dimensionless standard enthalpy change of the step of binding of substrate to the enzyme (E _m/(R.T₀)) - Dimensionless activation energy of the step of deactivation of the enzyme (E _d/(R.T₀)) - Dimensionless deactivation preexponential factor ((k _d,0.C_S,0)/(k_v,0.C_E,0)     

COMPARISON OF HALF-SATURATION CONSTANTS FOR GROWTH AND NITRATE UPTAKE OF MARINE PHYTOPLANKTON 2

Growth rates and rates of nitrate uptake by N-depleted cells were measured for an oceanic diatom, Chaetoceros gracilis, and a neritic diatom, Asterionella japonica, as functions of nitrate concentration of the medium. Both growth and N-uptake rates appeared to be hyperbolic with nitrate concentration and could be fit to an equation of Michaelis-Menten form: where v is rate, V^m. is the maximum rate, S is nitrate concentration, and K_sis the half-saturation constant. K_svalues for uptake and growth were similar if not identical for each species. Uptake experiments can provide a presumptive measure of K_sfor growth, thought to be an ecologically significant characteristic of a species.     

Inducibility of Rat Liver Drug-Metabolizing Enzymes as an Index of Food-Screeing for Safety Assessment

D-Lactate dehydrogenase (D-LDH) from Pediococcus pentosaceus ATCC 25745 was found to produce D-3-phenyllactic acid from phenylpyruvate. The optimum pH and temperature for enzyme activity were pH 5.5 and 45 °C. The Michaelis-Menten constant (K _m), turnover number (k _cat), and catalytic efficiency (k _cat?K _m) values for the substrate phenylpyruvate were estimated to be 1.73 mmol/L, 173 s^?1, and 100 (mmol/L)^?1 s^?1 respectively.     

On the optimum distribution of enzyme feed in a cascade of CSTR's performing an enzyme-catalyzed reaction with deactivation

A search for the optimum fractional distribution of an enzyme-rich stream to the various reactors of a cascade of CSTR's was implemented. A theoretical analysis, laid out in dimensionless form and based on the assumptions that the system is operated under steady state conditions, the enzyme undergoes first order deactivation, and the reaction catalyzed by the enzyme follows Michaelis-Menten kinetics, is reported. The objective function utilised is the minimisation of the overall volume of the cascade, and analytical expressions are obtained for the concentration of active enzyme and substrate in the outlet stream from each reactor. It is found that the best option is to add the whole enzyme-rich stream to the first reactor in the cascade irrespective of the operating and kinetic parameters of the system.     

Oxidative dehydrogenation of an iron-tetraazacyclotetradecatriene complex

The oxidative dehydrogenation of the bis(N- methyl imidazole)(meso-5,5,7,12,12,14-hexamethyl- 1 4,8 11-tetraazacyclotetradeca-1,3,8-triene)iron(II) complex, forming the corresponding 1 3,8,10-tetraene product was investigated by cyclic voltammetry, spectroelectrochemistry and stopped-flow kinetics with [Fe(CN)₆]³⁻, at 25 °C, I = 0.50 M and pH 7–10. The results led to a mechanism consistent with a reversible one electron transfer process generating iron(III) species which lose a proton (pK_a = 9.77) and undergo induced electron transfer in the presence of the hexacyanoferrate(III) ion (k = 4.2 × 10⁵ M⁻¹ s⁻¹). The intermediate precursor complex (λ_max = 665 nm) formed at this step, converted to the tetraene product according to a first order kinetics, with k = 0.12 s⁻¹.     

Delayed-rectifying potassium channels in mouse peritoneal macrophages: Pharmacological analysis

The effects of 4-aminopyridine, verapamil, and 4-bromophenacylbromide (4-BPB) on the kinetics of delayed outward-rectifying potassium currents (I _K) were investigated in cultured mouse peritoneal macrophages using a classical whole-cell patch-clamp technique. The outwardI _K was completely blocked by 4-aminopyridine at 1.0 mM concentration. Verapamil at the same concentration also blockedI _K completely. Lower concentration (50 µM) of verapamil demonstrated only partial blocking action, which was almost fully reversible, and markedly increased the rate ofI _K inactivation. The main effect of 4-BPB on the outwardI _K was a significant acceleration ofI _K activation and inactivation kinetics. It is suggested that this modulation results from a direct effect of 4-BPB on potassium channels or relates to the arachidonic acid cascade.Neirofiziologiya/Neurophysiology, Vol. 26, No. 1, pp. 49–53, January–February, 1994.     

Effects of variation of ion and methylation of carrier on the rate constants of macrotetralide-mediated ion transport in lipid bilayers

Summary The effects of methylation on the rate constants of carrier-mediated ion transport have been studied on monooleindecane bilayers with K⁺, Rb⁺, NH ₄ ⁺ , and TI⁺ ions, using the series of homologue carriers, nonactin, monactin, dinactin, trinactin, and tetranactin, each member of the series differing from the previous one by only one methyl group. Measurements of the amplitude and time constant of the current relaxation after a voltage jump over a large domain of voltage and permeant ion concentration, together with a computer curve-fitting procedure, have allowed us, without the help of steady-state current-voltage data, to deduce and compare the values of the various rate constants for ion transport: formation (k _Ri) and dissociation (k _Di) of the ion-carrier complex at the interface, translocation across the membrane interior of the carrier (k _s) and the complex (k _is). With the additional information from steady-state low-voltage conductance measurements, we have obtained the value of the aqueous phase-membrane and torus-membrane partition coefficient of the carrier ({ie191-1} and {ie191-2}). From nonactin to tetranactin with the NH ₄ ⁺ ion,k _is, and {ie191-3} are found to increase by factors of 5 and 3, respectively,k _Di and {ie191-4} to decrease respectively by factors 8 and 2, whilek _Ri andk _s are practically invariant. Nearly identical results are found for K⁺, Rb⁺, and Tl⁺ ions.k _Ri,k _s andk _is are quite invariant from one ion to the other except for Tl⁺ wherek _Ri is about five times larger. On the other hand,k _Di depends strongly on the ion, indicating that dissociation is the determining step of the ionic selectivity of a given carrier. The systematic variations in the values of the rate constants with increasing methylation are interpreted in terms of modifications of energy barriers induced by the carrier increasing size. Within this framework, we have been able to establish and verify a fundamental relationship between the variations ofk _is andk _Di with methylation.     

Inhibition by nisin of K+ uptake in a sensitivePediococcus sp. strain

Summary A nisin-sensitive strain ofPediococcus sp possessed an uptake system for K⁺ which was apparently dependent on metabolic energy and ATPase activity. K⁺ uptake rate was dependent on the glucose and K⁺ concentrations and showed approximately Michaelis-Menten kinetics with respect to both of these variables with K_t values of 1.2 mM and 599 μM respectively. The presence of nisin inhibited K⁺ uptake with the percentage inhibition proportional to the nisin activity,. Total inhibition occurred at between 4.5 and 5.0 IU ml⁻¹ and the MIC was approximately 0.6 IU ml⁻¹.     

Phosphonate inhibitors of West Nile virus NS2B/NS3 protease

West Nile virus (WNV) is a member of the flavivirus genus belonging to the Flaviviridae family. The viral serine protease NS2B/NS3 has been considered an attractive target for the development of anti-WNV agents. Although several NS2B/NS3 protease inhibitors have been described so far, most of them are reversible inhibitors. Herein, we present a series of α-aminoalkylphosphonate diphenyl esters and their peptidyl derivatives as potent inhibitors of the NS2B/NS3 protease. The most potent inhibitor identified was Cbz-Lys-Arg-(4-GuPhe)^P(OPh)₂ displaying K_i and k₂/K_i values of 0.4 µM and 28 265 M^?1s^?1, respectively, with no significant inhibition of trypsin, cathepsin G, and HAT protease.     

Pseudo-enzymatic hydrolysis of 4-nitrophenyl acetate by human serum albumin: pH-dependence of rates of individual steps

Human serum albumin (HSA) displays esterase activity reflecting multiple irreversible chemical modifications rather than turnover. Here, kinetics of the pseudo-enzymatic hydrolysis of 4-nitrophenyl acetate (NphOAc) are reported. Under conditions where [HSA] ? 5×[NphOAc] and [NphOAc] ? 5×[HSA], the HSA-catalyzed hydrolysis of NphOAc is a first-order process for more than 95% of its course. From the dependence of the apparent rate constants k_app and k_obs on [HSA] and [NphOAc], respectively, values of K_s, k₊₂, and k₊₂/K_s were determined. Values of K_s, k₊₂, and k₊₂/K_s obtained at [HSA] ? 5×[NphOAc] and [NphOAc] ? 5×[HSA] are in good agreement, the deacylation step being rate limiting in catalysis. The pH-dependence of k₊₂/K_s, k₊₂, and K_s reflects the acidic pK_a shift of the Tyr411 catalytic residue from 9.0 ± 0.1 in the substrate-free HSA to 8.1 ± 0.1 in the HSA:NphOAc complex. Accordingly, diazepam inhibits competitively the HSA-catalyzed hydrolysis of NphOAc by binding to Tyr411.     

A Simple Construction Procedure for Eesolvable Incomplete Block Designs for any Number of Treatments

A simple, straightforward procedure, which requires no special tables or generators, is presented for constructing resolvable incomplete block designs for v=pk, v=p²k, …, treatments, for k≥p, in incomplete blocks of size k. Also, it is shown, how to obtain incomplete block designs for any v in blocks of size k and k+1. The procedure allows construction of balanced incomplete block designs for p = k a prime number. For p = n not a prime number, incomplete block designs can be obtained by the procedure, but are not balanced. However, for p_s being the smallest prime factor of n, p_s + 1 for v = n², p_s²+ p_s + 1 for v = n³, …, arrangements can be obtained for which the occurrence of any treatment pair in the blocks is either zero or one. This is called a zero-one concurrence design. Procedures are described for obtaining additional zero-one concurrence arrangements. It is shown that the efficiency of these designs is maximum. Both intra-block and inter-block analyses are described.     

A kinetic model and simulation of starch saccharification and simultaneous ethanol fermentation by amyloglucosidase and Zymomonas mobilis

A mathematical model is described for the simultaneous saccharification and ethanol fermentation (SSF) of sago starch using amyloglucosidase (AMG) and Zymomonas mobilis. By introducing the degree of polymerization (DP) of oligosaccharides produced from sago starch treated with -amylase, a series of Michaelis-Menten equations were obtained. After determining kinetic parameters from the results of simple experiments carried out at various substrate and enzyme concentrations and from the subsite mapping theory, this model was adapted to simulate the SSF process. The results of simulation for SSF are in good agreement with experimental results.List of Symbols g/g rate coefficient of production - _max 1/h maximum specific growth rate - E %, v/w AMG concentration - G ₁ mmol/l glucose concentration - G _c mmol/l glucose concentration consumed - G _f mmol/l glucose concentration formed - G _n mmol/l n-mer maltooligosaccharide concentration - K _i g/l ethanol inhibition constant for ethanol production - K _g mmol/l glucose inhibition constant for glucose production - K _p mmol/l glucose limitation constant for ethanol production - K _x mmol/l glucose limitation constant for cell growth - K _m,n mmol/l Michaelis-Menten constant for n-mer oligosaccharide - k _e %, v/w enzyme limitation constant - k _es proportional constant - k _{max, n} 1/s maximal velocity for n-mer digestion - k _s g/l substrate limitation constant - m _s g/g maintenance energy - MW _n g/mol molecular weight of n-mer oligosaccharide - P g/l ethanol concentration - P ₀ g/l initial ethanol concentration - P _m g/l maximal ethanol concentration - Q _pm g/(g · h) maximum specific ethanol production rate - S _n mmol/h branched n-mer oligosaccharide concentration - S ₀ g/l initial starch concentration - S _sta g/l starch concentration - S _tot g/l total sugar concentration - V _{max, n} 1/h maximum digestion rate of n-mer oligosaccharide - V ₀ g/(l · h) initial glucose formation rate - X g/l cell mass - X ₀ g/l initial cell mass - Y _p/s g/g ethanol yield - Y _x/s g/g cell mass yield     

Batch kinetic studies of glucose isomerization to fructose by Arthrobacter species

Glucose can be isomerized to fructose by the catalytic action of the enzyme, glucose isomerase. This enzyme is synthesized by a variety of micro-organisms, predominantly by bacteria. Arthrobacter species cells are grown in a medium standardized specifically to synthesize the enzyme and are then used to isomerize glucose under conditions of no further cell growth. Effect of metal ions on the isomerization is studied and it is found that magnesium promoted the reaction, sodium had no effect and calcium and manganese inhibited the reaction. Rate of reaction per unit of catalyst is found to be constant. Michaelis-Menten model modified for the reversibility of the reaction is suitable to describe the isomerization kinetics and the kinetic parameters are determined and reported.List of Symbols k ₁ rate constant (Glucose to intermediate complex) - k _–1 rate constant (Intermediate complex to glucose) - k ₂ rate constant (Intermediate complex to fructose) - k _–2 rate constant (Fructose to intermediate complex) - v _mf maximum reaction velocity of the forward (GF) reaction - v _mb maximum reaction velocity of the reverse (FG) reaction - K _f Michaelis-Menten constant for the forward (GF) reaction - K _b Michaelis-Menten constant for the reverse (FG) reaction - K _eq equilibrium constant - r _G rate of glucose consumption     

A Model for the Unusual Kinetics of Thermal Denaturation of Rubredoxin

The thermal denaturation of the simple, redox-active iron protein rubredoxin is characterized by a slow, irreversible decay of the characteristic red color of the iron center at elevated temperatures in the presence of oxygen at pH 7.8. The denaturation rate is essentially constant and the time period for complete bleaching is nearly independent of protein concentration. These two characteristics of the kinetics can be fit by a simple self-catalyzed kinetics model consisting of the combination of a first-order decay and catalysis by some product of that decay, i.e., dP/dt=k ₁[A]+(k ₂[P][A])/(K _m+[A]), where A is native rubredoxin, P, is unspecified product, k ₁ is a first-order rate constant, and k ₂ and K _m are the catalytic constants. In order for the second term to be of this simple form over the full course of a decay, the model must include the condition that the reaction is effectively irreversible. This model has properties which suggest other biological roles in regulation (changes in k ₁ or k ₂ can dramatically modulate the kinetics), in timing (titer-independent fixed reaction time), and in self-activation reactions. At one extreme (k₁ k₂) the kinetics becomes exponential, but at the other extreme (k₂ k₁) they show a dramatic and rapid terminal increase after a lag period. Some obvious possible roles in the kinetics of programmed cell death, prion disease, and protease autoactivation are discussed.