Systems Biology and Physiology: Notes on Models and Dynamics

A Systems Biology View on Physiology

• Prof. Stan Marée, Systems and Predictive Biology, Cardiff University
• Unraveling physiological mechanisms through systems thinking

Tonic Descending or Afferent Input

• Illustrates a neural circuit with excitatory and inhibitory interneurons affecting flexor and extensor motor neurons.

Nernst Equilibrium Potential

• Setup with inner and outer chambers separated by an artificial membrane.
• Inner chamber: 150 mM KCl
• Outer chamber: 5 mM KCl, 15 mM NaCl
• Selective permeability to $K^+$ leads to a Nernst equilibrium potential of $-92 mV$.
• Selective permeability to $Na^+$ leads to a Nernst equilibrium potential of $+62 mV$.
• Ions present : Potassium channel, Sodium channel, $Cl^−$, $Na^+$
• Outer chamber : 150 mM NaCl

Goldman Equation

• Ohm's Law: $I = V/R$
• The current across the membrane due to ion fluxes are combined in the Goldman equation: $I = g<em>N(V</em>N − V) + g<em>K(V</em>K − V) + g<em>R(V</em>R − V)$
• $g<em>N$, $g</em>K$, and $g_R$ are the conductances for sodium, potassium, and the rest-group respectively.
• Conductance $g = 1/R$
• $V<em>N$, $V</em>K$, and $V_R$ are the Nernst equilibrium potentials for sodium, potassium, and the rest-group.

Membrane as Capacitor

• Membrane functions as a capacitor with conductive parallel plates and a dielectric.
• Illustrates electrical charge, $Q^+$ and $Q^−$, and voltage $V_c$.
• Equation for a capacitor: $C \frac{dV}{dt} = \frac{I}{C}$, where $C$ is the capacity of the membrane.
• Change in membrane potential $V$ obeys: $\frac{dV}{dt} = \frac{1}{C} [g<em>N(V</em>N − V) + g<em>K(V</em>K − V) + g<em>R(V</em>R − V)]$

Ion Channel Dynamics

• Hodgkin and Huxley defined three variables using voltage-clamp experiments on the giant squid axon: $m$, $n$, and $h$, representing curve-fitted voltage-sensitive channel proteins.
• $g<em>N$ and $g</em>K$ are complex functions of the voltage $V$.
• $m$: opening of the sodium channel
• $h$: inhibition (closure) of the sodium channel
• $n$: opening of the potassium channel
• Equations for $m$, $h$, and $n$ are:
  • $\frac{dm}{dt} = 0.1(1 − m) \frac{V + 25}{e^{(V+25)/10} − 1} − 4me^{V/18} = α<em>m(1 − m) − β</em>mm$
  • $\frac{dh}{dt} = 0.07(1 − h)e^{V/20} − \frac{h}{e^{(V+30)/10} + 1} = α<em>h(1 − h) − β</em>hh$
  • $\frac{dn}{dt} = 0.01(1 − n) \frac{V + 10}{e^{(V+10)/10} − 1} − 0.125ne^{V/80} = α<em>n(1 − n) − β</em>nn$
• These equations are data fitting.
• Adding the channels to the equation:
  • $V_N = −115 mV$
  • $V_K = 12 mV$
  • $V_R = −10.5989 mV$
  • $\frac{dV}{dt} = \frac{1}{C} [g<em>N(V</em>N − V) + g<em>K(V</em>K − V) + g<em>R(V</em>R − V)]$
  • $\frac{dV}{dt} = \frac{1}{C} [120m^3h(V<em>N − V) + 36n^4(V</em>K − V) + 0.3(V_R − V)]$

Behavior in Time

• Graphs illustrating the behavior of voltage $V$, and the fraction of open channels $m$, $n$, and $h$ over time.
• Hodgkin and Huxley scaled their model such that the overall resting potential was 0 (instead of -70 mV).
• They expressed the voltage over the membrane as (outside minus inside) instead of the more common (inside minus outside).

Hodgkin-Huxley Model Critique

• The Hodgkin-Huxley model is unpleasantly complex.
• One can do little more than simulate the model on a computer to see its behavior in time.
• The output of the model is almost as `mysterious' as the output of biological experiments.

From Hodgkin-Huxley to FitzHugh-Nagumo

• FitzHugh (1961) stated that “The usefulness of an equation to an experimental physiologist depends on his understanding how it works”.
• FitzHugh (1961) and Nagumo (1962) simplified the equations to the `core' of excitable media
  • $V$ represents the voltage.
  • $W$ is a slow recovery variable.
  • $\epsilon$ has to be sufficiently small.
  • $\frac{dV}{dt} = −V(V − a)(V − 1) − W$
  • $\frac{dW}{dt} = \epsilon(V − bW)$

Timeplots and Phase Plane

• Timeplots showing the dynamics of $V$ and $W$ over time.
• Phase plane plots illustrating the relationship between $V$ and $W$.
• Illustrates trajectory, threshold, action potential, refractory period, hyperpolarisation, and resting potential.

Calcium Physiology

• Diagram illustrating Calcium fluxes and various processes involving Calcium.
• Various components of the diagram
  *Plasma membrane, Cytoplasm, ER/SR, Mitochondrion
• Ca++ sensitive processes such as Contraction, Proliferation, Fertilization, Learning and memory
• Crosstalk with Cyclic AMP, PDE, other signalling pathway's
• Various Calcium related enzymes such as Adenylyl cyclase, NOS, PKC, PYK2, Ins(1,4,5)P 3-kinase, Phosphorylase kinase, Mitochondrial enzymes
• Model consists of two ODEs, where c represents cytosolic $Ca^{2+}$ concentration (in μM) and n represents the proportion of non inactivated IP3 receptors.

Bell-Shaped Response to Ca2+

• $Ca^{2+}$ binds to the IP3R.
• IP3R has both an activation site and an inhibitory site.
• $Ca^{2+}$ flux out of ER depends in a complex way on $Ca^{2+}$.

Oscillatory Calcium Dynamics

• Graphs showing oscillatory calcium dynamics over time.

Understanding Oscillations

• Phase plane plots with $c$ (cytosolic $Ca^{2+}$ concentration) and $n$ (proportion of non-inactivated IP3 receptors).

Back to FHN

• More complex dynamics are expected.
• Time series and phase plane plots demonstrating various dynamics.

Richer Dynamics

• Illustrates pulse duration, pulse irregularity, and pulse height.

FitzHugh-Nagumo in Space

• Diffusion added to the V-equation:
  • $\frac{∂V}{∂t} = −V(V − a)(V − 1) − W + D (\frac{∂^2V}{∂x^2} + \frac{∂^2V}{∂y^2} )$
  • $\frac{∂W}{∂t} = \epsilon(V − bW)$
• Excitable medium

Spatial Calcium Patterns

• Examples of spatial calcium patterns in various biological systems.
• Larva (Pieris rapae)
• Arabidopsis eaten by Caterpillar
• Ciona (sea squirt) oocyte

Identifying Processes

• How to understand complex patterns.
• Nullclines and mesoscale spatial patterning.

Isomorphisms

Explanatory Power

• Example of heart fibrillations.

Models and Reality

• George Box: "All models are wrong, but some models are useful"
• Example comparing different map projections and their characteristics, drawing an analogy to models in general.
• Emphasis on understanding connections rather than specific players, drawing an analogy to the FitzHugh-Nagumo model.