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Modeling biochemical reactions Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4500, Spring 2016 M. Macauley (Clemson) Modeling biochemical reactions Math 4500, Spring 2016 1 / 10 Overview In biochemistry, 2+ species, or “reactants” can react if they come toegether and collide. Alternatively, one species can degrade. More is needed, though: correct orientation, enough energy, etc. Examples CH4 2O2 ÝÑ CO2 2H2 O (burning of methane) OH ÝÑ H2 O unfolded protein ÝÑ folded protein 2SO2 O2 Ý âÝÝá Ý 2SO3 H ÝÑ O2 O 2O3 ÝÑ 3O2 O3 M. Macauley (Clemson) Modeling biochemical reactions Math 4500, Spring 2016 2 / 10 Mass-action kinetics Classification of reactions: A ÝÑ P: A B A B “uni-molecular” ÝÑ P: “bi-molecular” C ÝÑ P: “tri-molecular” Law of mass-action kinetics A reaction rate is proportional to the probability of collision of reactants involved. Assume this probability is proportional to the concentration of each reactant R, denoted rR s. ODE model A ÝÑ P: k A B k ÝÑ P: A B k Ý âá Ý P: k 1 2 M. Macauley (Clemson) d rP s k rAs dt d rP s k rAsrB s dt d rP s k1 rAsrB s k2 rP s dt Modeling biochemical reactions Math 4500, Spring 2016 3 / 10 Mass-action kinetics Enzymes are proteins that catalyze reactions (up to 1012 -fold!) An example Consider the following chemical reaction E S k k Ý E âá Ý ES ÝÑ k 1 3 P 2 E enzyme, S substrate, ES enzyme-substrate complex, and P product. $ d rES s ' k1 rE srS s pk2 k3 qrES s ' & dt ' ddtrP s k rES s ' %E rE s rES s, 3 0 E0 initial enzyme concentration Assumptions E0 is constant. Enzyme-substrate complex reaches equilibrium much earlier than the product d rES s does, so 0. dt M. Macauley (Clemson) Modeling biochemical reactions Math 4500, Spring 2016 4 / 10 Mass-action kinetics Goal Write the differential equation Since d rES s dt d rP s dt k3 rES s in terms of rS s, not rES s. 0, we can simplify the ODE for rES s: d rES s k1 rE srS s pk2 k3 qrES s 0 . dt Upon solving for rE s, we get rE s pk2 k1krS3 qrs ES s . Plugging this into E0 rE s rES s and solving for rES s: rES s k Ek 0 rS s . rS s k 2 3 1 Alas, we can write d rP s dt M. Macauley (Clemson) k3 rES s k2 k3 E0 rS s k3 rS s k1 KVmmax rrSSss . Modeling biochemical reactions Math 4500, Spring 2016 5 / 10 Michaelis–Menten equation Recall the following chemical reaction: E S k Ýká E â Ý ES ÝÑ k 1 3 P 2 E enzyme, S substrate, ES enzyme-substrate complex, and P product. Definition The Michaelis–Menten equation is one of the best-known models of enzyme kinetics. d rP s dt KVmmax rrSSss , loooomoooon f where Vmax k3 E0 , and Km k2 k1 k3 prS sq Remarks The “reaction rate”, f prS sq, is a strictly increasing function of rS s. lim f prS sq Vmax , rS sÑ8 (biologically, the maximum reaction rate) f pKm q 12 Vmax . The reaction rate f prS sq is proportional to E0 . M. Macauley (Clemson) Modeling biochemical reactions Math 4500, Spring 2016 6 / 10 Michaelis–Menten equation Recall the following chemical reaction: E S k k Ý E âá Ý ES ÝÑ k 2 3 P 1 E enzyme, S substrate, ES enzyme-substrate complex, and P product. Further assumptions Substrate concentration is conserved: S0 E0 ! S0 , so rES s ! rS s and rP s. Together, this means S0 rS s rES s rP s. rS s rP s. Taking dtd of both sides yields rP s Vmax rS s . d rS s ddt km r S s dt Usually, Vmax , Km , and S0 are known quanities. This is now something we can easily solve, graph, analyze, etc. M. Macauley (Clemson) Modeling biochemical reactions Math 4500, Spring 2016 7 / 10 Multi-molecule binding Consider a reaction where n molecules of a substrate S react with an enzyme E : E nS k k Ý E âá Ý ESn ÝÑ k 1 3 P 2 The enzyme-substrate complex here is ESn . By mass-action kinetics, $ d rES s ' k rE srS s pk ' & dt ' ddtrP s k rES s ' % n n 1 3 2 k3 qrESn s n rE s rESn s, E0 initial enzyme concentration As before, assume rESn s reaches equilibrium much quicker than rP s and rS s: d rESn s 0 ùñ rE s pk2 k1krS3 qrsnESn s . dt Plugging this into E0 rE s rESn s and solving for rESn s yields n n d rP s ùñ KVmmax rrSSssn . rESn s k Ek 0 rS s n dt rS s k E0 2 3 1 M. Macauley (Clemson) Modeling biochemical reactions Math 4500, Spring 2016 8 / 10 Multi-molecule binding Hill equation Given the chemical reaction k Ýká E P â Ý ESn ÝÑ k we derived the following ODE involving rP s and rS s: d rP s Vmax rS sn k2 k3 , where Vmax k3 E0 , and Km n dt K r S s k1 m looooomooooon E nS 2 3 1 f prS sq This is called the Hill equation with Hill coefficient n. Remarks The “reaction rate”, f prS sq, is a strictly increasing function of rS s. lim f prS sq V max rS sÑ8 1{n f pKm q 12 Vmax . , (biologically, the maximum reaction rate) The reaction rate f prS sq is proportional to E0 . n 1 is just the Michaelis–Menden equation. M. Macauley (Clemson) Modeling biochemical reactions Math 4500, Spring 2016 9 / 10 Hill equations The following shows several “Hill functions” y M. Macauley (Clemson) 1t Modeling biochemical reactions n tn , for various values of n. Math 4500, Spring 2016 10 / 10