ReceptorLigand Binding

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Receptor-Ligand Binding
Basic reference Keener and Sneyd, Mathematical
Physiology
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A nice cell picture
Membrane Proteins http//trc.ucdavis.edu/biosci10
v/bis10v/media/ch03/membrane_proteins_v2.html
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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.

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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.

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Example VEGF Receptors
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Possible Cellular Responses
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Reversible enzymes
Of course, all enzymes HAVE to be reversible, so
its naughty to put no back reaction from P to
C. Should use
I leave it as an exercise to calculate that
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Allosteric modulation
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.
(Inhibition in this case, but it doesnt have to
be)
substrate binding
X
inhibitor binding at a different site
Z
Y
this state can form no product
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Equilibrium approximation
Could change these rate constants, also.
Inhibition decreases the Vmax in this model
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Cellular Release and Uptake

• Molecules are taken up and released by cells in
  different ways
• glucose is transported inside cells by
  facilitated diffusion
• other molecules must be carried into or out of
  the cell via receptor-mediated endocytosis or
  exocytosis
• Endocytosis is the process by which cells
  internalize molecules via the inward budding of
  plasma membrane vesicles containing proteins with
  receptor sites specific to the molecules being
  internalized.
• Exocytosis is the process by which a cell
  directs the contents of secretory vesicles out of
  the cell membrane. These membrane-bound vesicles
  contain soluble proteins to be secreted to the
  extracellular environment, as well as membrane
  proteins and lipids that are sent to become
  components of the cell membrane.

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Facilitated Diffusion and Receptor-Mediated
Endocytosis
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One of the main pathways of internalization and
re-insertion of the so called G-protein-coupled
receptors
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Example Endocytic trafficking of VEGF Receptors
in angiogenesis
Dr Harry Mellor Medical School, UoB
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VEGFR2 in unstimulated endothelial cells
HUVEC
HMVEC
VE-cadherin VEGFR2
Alex Gampel
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VEGFR2 is constantly internalised
and recycled
Matt Jones/ Jim Norman Alice Scott
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Redistribution of VEGFR2 on VEGF stimulation
30min VEGF F-actin VEGFR2
tubulin VEGFR2
Lara Moss
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VEGFR2 is rapidly turned over
VEGFR2
DVEGFR2
Alice Scott
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(No Transcript)
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Mathematical modelling of VEGFR2 traffic
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Goal

• Model the process of molecule uptake
• Schematic Diagram

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STEP 1

• Reaction Diagram
• Reaction diagrams can be converted to a system of
  odes that describe the rates of change of the
  concentration of the reactants

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The Law of Mass Action

• To go from molecules to concentration we use the
  Law of Mass Action
• 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 kis are the proportionality
  constants

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Model Variables
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The Model Equations
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Notes

• The p equation is decoupled
• We only need to consider 3 equations
• The total number of receptors is conserved
• We only need to consider 2 differential equations
  (the c and n equations), together with

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Reduced Model

• Note Because this is a system of two equations,
  we can use the traditional stability and phase
  plane analysis, but lets do something different
  first.

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Quasi-Steady State Assumption

• The concentration of the substrate-bound enzyme
  (and hence also the unbound enzyme) 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

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Quasi-Steady State Approximation

• The QSSA is written as

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Michaelis-Menten Kinetics

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

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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, lets
  nondimensionalize.

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Nondimensionalization

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

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Nondimensional Equations

• Now we see that assuming dc/dt 0 is equivalent
  to assuming that e ltlt 1, which means r0 ltlt n0.

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Validity of QSSA

• So, on timescales of the order 1/(k1r0) (i.e.
  long timescales), receptor-mediated molecule
  uptake can be approximated by

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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 cant always be true, recall that we started
  with c(0) 0
• Lets see how the solutions behave on short
  timescales.

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Nondimensionalize

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

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On Short Timescales

• We can now predict how receptors fill up!

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On Short Timescales

• Now if e r0/n0 0, we have
• We can now predict how receptors fill up

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Short Timescale Solutions

• So v rises quickly to a maximum on short
  timescales.

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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.

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QSSA vs Full Model Behavior
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Traditional Analysis

• The only steady state is u 0, v 0 and it is
  stable
• Eventually all of the ligand is consumed and
  internalized and all of the receptors are empty

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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

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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

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Full Reaction Diagram For a Homodimeric Receptor
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Simplified Reaction Diagram For a Homodimeric
Receptor

• 2N R C R 2P

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Model Equations
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Reduced Model Equations

• p-equation is decoupled and r r0 c

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QSSA Equation
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Sigmoidal Kinetics
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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

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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

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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

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Model Equations
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Model Reduction

• p-equation is decoupled and r r0 c1 c2

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I leave it as an exercise to calculate that

• QSSA gives

• Define the velocity of the reaction, V dp/dt