1. Modelling biochemical reactions using the law of mass action; chemical kinetics Basic reference: Keener and Sneyd, Mathematical Physiology

2. Law of mass action Given a basic reaction A + B C k1k1k1k1 k -1 we assume that the rate of forward reaction is linearly proportional to the concentrations of A and B, and the back reaction is linearly proportional to the concentration of C.

3. Equilibrium Equilibrium is reached when the net rate of reaction is zero. Thus or This equilibrium constant tells us the extent of the reaction, NOT its speed. change in Gibb’s free energy

4. Enzymes  Enzymes are catalysts, that speed up the rate of a reaction, without changing the extent of the reaction.  They are (in general) large proteins and are highly specific, i.e., usually each enzyme speeds up only one single biochemical reaction.  They are highly regulated by a pile of things. Phosphorylation, calcium, ATP, their own products, etc, resulting in extremely complex webs of intracellular biochemical reactions.

5. Basic problem of enzyme kinetics Suppose an enzyme were to react with a substrate, giving a product. S + E P + E If we simply applied the law of mass action to this reaction, the rate of reaction would be a linearly increasing function of [S]. As [S] gets very big, so would the reaction rate. This doesn’t happen. In reality, the reaction rate saturates.

6. Michaelis and Menten In 1913, Michaelis and Menten proposed the following mechanism for a saturating reaction rate S + E k1k1k1k1 k -1 C k2k2k2k2 P + E Complex. product  Easy to use mass action to derive the equations.  There are conservation constraints.

7. Equilibrium approximation And thus, since Thus reaction velocity i.e. the substrate is in instantaneous equilibrium with the complex

8. Pseudo-steady state approximation (Quasi-Steady State Assumption) And thus, since Thus reaction velocity Looks very similar to previous, but is actually quite different! i.e. the rate of formation and breakdown of the complex are equal at all times

9. Basic saturating velocity s V V max KmKmKmKm V max /2

10. Cooperativity S + E k1k1k1k1 k -1 C1C1C1C1 k2k2k2k2 P + E S + C 1 k3k3k3k3 k -3 C2C2C2C2 k4k4k4k4 P + E Enzyme can bind two substrates molecules at different binding sites. or E C1C1C1C1 C2C2C2C2 E E SS S S P P Autocatalytic reactions are chemical reactions in which at least one of the products is also a reactant.

11. Pseudo-steady assumption Note the quadratic behaviour

12. Independent and identical binding sites E C1C1C1C1 C2C2C2C2 E E SS S S P P 2k + k+k+k+k+ 2k - k-k-k-k- Just twice the single binding rate, as expected

13. Hill equation In the limit as the binding of the second S becomes infinitely fast, we get a nice reduction. Hill equation, with Hill coefficient of 2. This equation is used all the time to describe a cooperative reaction. Mostly use of this equation is just a heuristic kludge. VERY special assumptions!

14. Positive/negative cooperativity  Usually, the binding of the first S changes the rate at which the second S binds.  If the binding rate of the second S is increased, it’s called positive cooperativity  If the binding rate of the second S is decreased, it’s called negative cooperativity.

15. Reversible enzymes Of course, all enzymes HAVE to be reversible, so it’s naughty to put no back reaction from P to C. Should use S + E k1k1k1k1 k -1 C k2k2k2k2 P + E k -2 I leave it as an exercise to calculate that

16. Allosteric modulation substrate binding inhibitor binding at a different site this state can form no product (Inhibition in this case, but it doesn’t have to be) X Y Z Allosteric modulation is the regulation of an enzyme or other protein by binding an effector molecule at the protein's allosteric site (that is, a site other than the protein's active site). The term allostery comes from the Greek allos, "other," and stereos, "solid (object)," in reference to the fact that the regulatory site of an allosteric protein is physically distinct from its active site. Allosteric regulations are natural example of control loops, such as feedback from downstream products or feed forward from upstream substrates.

17. Equilibrium approximation X YZ Could change these rate constants, also. Inhibition decreases the V max in this model

18. Receptor-Ligand Binding

19. Definitions  Receptor –a protein molecule, embedded in either the plasma membrane or cytoplasm of a cell, to which a mobile signaling (or "signal") molecule may attach.  Ligand –a signal triggering substance that is able to bind to and form a complex with a biomolecule to serve a biological purpose –May be a peptide (such as a neurotransmitter), a hormone, a pharmaceutical drug, or a toxin.

20. Why Receptor Ligand Binding is Important?  Individual cells must be able to interact with a complex variety of molecules, derived from not only the outside environment but also generated within the cell itself. Protein-ligand binding has an important role in the function of living organisms and is one method that the cell uses to interact with these wide variety of molecules. When such binding occurs, the receptor undergoes conformational changes, which ordinarily initiates a cellular response.

21. Facilitated Diffusion and G-protein coupled receptors

22. Example  Model the process of molecule uptake  Schematic Diagram

23.  Reaction Diagram  Reaction diagrams can be converted to a system of odes that describe the rates of change of the concentration of the reactants STEP 1 Extracellular Molecule Free Receptor N-R Complex Intracellular Molecule + + k2k2k2k2NRCR P + k1k1k1k1 k-1k-1k-1k-1 +

24. The Law of Mass Action  To use the Law of Mass Action we go from molecules to concentration and keep in mind that: –When two or more reactants are involved in a reaction step, the rate of the reaction is proportional to the product of the concentrations of the reactants. –Convention: k i ’s are the proportionality constants

25. Model Variables VariableDefinitionUnits r[R]#/cell n[N]Moles/volume c[C]#/cell p[P]moles/volume

26. The Model Equations + + k2k2k2k2NRCR P + k1k1k1k1 k-1k-1k-1k-1 +

27.  The p equation is decoupled –We only need to consider 3 equations  The total number of receptors could be conserved –We only need to consider 2 differential equations (the c and n equations), together with: Notes

28. Reduced Model  Note: Because this is a system of two equations, we can use stability and phase plane analysis, but let’s do something different first.

29. Quasi-Steady State Assumption  The concentration of the ligand-bound receptor (enzyme) and hence also the unbound receptor change much more slowly than those of the product and substrate.  Rationale: –Small molecules like glucose are found in higher concentrations than the receptors are –If this is true, then receptors are working at maximal capacity  Therefore the occupancy rate is virtually constant

30. Quasi-Steady State Approximation  The QSSA is written as:

31. Michaelis-Menten Kinetics  A simple substitution shows that we have derived the Michaelis-Mention kinetic form that is widely applied in modelling biochemical reactions.

32. Problem with QSSA  By assuming that dc/dt = 0, we changed the nature of the model from 2 ODEs to one ODE and one algebraic expression. There must be consequences for doing this.  To see which timescales QSSA is valid on, let’s nondimensionalize.

33. Nondimensionalization  The ligand and complexes are scaled by their initial conditions. Time is scaled by receptor density multiplied by the association rate.

34. Nondimensional Equations  Now we see that assuming dc/dt = 0 is equivalent to assuming that  << 1, which means r 0 << n 0.

35. Validity of QSSA  So, on timescales of the order 1/(k 1 r 0 ) (i.e. long timescales), receptor-mediated molecule uptake can be approximated by:

36. Behaviour of Solutions  u is a decreasing function of time and v decreases if u decreases;  Therefore, on this timescale ( = long times), both the ligand and complex concentrations are decreasing;  This can’t always be true, recall that we started with c(0) = 0;  Let’s see how the solutions behave on short timescales.

37. Nondimensionalize  The ligand and complexes are scaled by their initial conditions. Time is scaled by ligand concentration multiplied by the association rate.

38. On Short Timescales  We can now predict how receptors fill up!

39. On Short Timescales  Now if  = r 0 /n 0 ~ 0, we have  We can now predict how receptors fill up

40. Short Timescale Solutions So v rises quickly to a maximum on short timescales.

41. Complete Behaviour  Initially, v rapidly rises which means receptor complex density quickly increases;  Eventually, the ligand is depleted and the the density of bound complexes follows it;  The behaviour of the system can be completely determined by solving approximate equations on two different timescales.

42. QSSA vs Full Model Behavior

43. Definitions  Dimer: a molecule which consists of two similar (but not necessarily identical) subunits  Homodimer: A dimeric protein made of paired identical subunits  Heterodimer: a dimer in which the two subunits are different  Both receptors and ligands can be homodimers or heterodimers  Dimeric ligands can dimerize (bring together) monomeric receptors

44. Homodimeric Receptor-Ligand Binding  Consider the following schematic diagram  Draw a reaction diagram that corresponds to this situation  Write down a system of equations that models this situation

45. Full Reaction Diagram For a Homodimeric Receptor + + k1k1k1k1 k-1k-1k-1k-1 + 1111 k2k2k2k2 k-2k-2k-2k-2 + 2222

46. Simplified Reaction Diagram For a Homodimeric Receptor 2N + R C R + 2P + k1k1k1k1 k-1k-1k-1k-1 + k1k1k1k1 k-1k-1k-1k-1 k2k2k2k2 k2k2k2k2

47. Model Equations 2N + R C R + 2P k1k1k1k1 k-1k-1k-1k-1 k2k2k2k2

48. Reduced Model Equations  p-equation is decoupled and r = r 0 – c

49. QSSA Equation

50. Sigmoidal Kinetics n k max

51. Full Reaction Diagram For a Homodimeric Receptor  Under what conditions does this reduce to the simplified, sigmoid kinetics?  Note: if the receptor sites act independently and identically, then k 1 = 2k 2 and 2k -1 = k -2 + + k1k1k1k1 k-1k-1k-1k-1 + 1111 k2k2k2k2 k-2k-2k-2k-2 + 2222

52. Model Equations

53. Model Reduction  p-equation is decoupled and r = r 0 – c 1 – c 2

54. QSSA

55. Cooperative Reactions  In other words, once a single ligand has bound, a second binds more readily. This is called a cooperative reaction. –Intermediate stages are short-lived and can almost be neglected –Example hemoglobin can bind up to four oxygen molecules

56. Generalization Generalization  In general, for highly cooperative reactions, if “a” ligand molecules can bind to a receptor; the following holds as a good approximation for the rate of change of the ligand:

57. Competitive Binding  Consider the following reaction diagram that corresponds to the competitive binding of two ligands to the same receptor  Write down a system of equations that models this situation + k2k2k2k2+ k1k1k1k1 k-1k-1k-1k-1 + k3k3k3k3 k-3k-3k-3k-3

58. Model Equations

59. Model Reduction  p-equation is decoupled and r = r 0 – c 1 – c 2

60. I leave it as an exercise to calculate that:  Define the velocity of the reaction, V = dp/dt  QSSA gives: