A Systems Biology View on Physiology: Unravelling Physiological Mechanisms Through Systems Thinking

Introduction

This section explores physiological mechanisms from a systems biology perspective, emphasizing the use of mathematical models to understand complex biological processes. It will cover:

• The Nernst equilibrium potential.

• The Goldman equation for membrane potential, incorporating multiple ion fluxes.

• The cell membrane's function as a capacitor.

• The Hodgkin-Huxley model for describing action potentials based on voltage-sensitive ion channels.

• The simplification of the Hodgkin-Huxley model into the Fitzhugh-Nagumo model to capture the core dynamics of excitable media.

• The application of these models to understand properties like threshold, action potential, refractory period, and resting potential.

• Systems approaches to understanding calcium (Ca2+) physiology and oscillations.

• The concept of spatial patterns in physiological systems and the explanatory power of simplified models.

Membrane Potential and Ion Fluxes

Nernst Equilibrium Potential:

• The Nernst equation calculates the equilibrium potential for a specific ion, which is the electrical potential that exactly balances the concentration gradient of that ion across a selectively permeable membrane.

• For a membrane selectively permeable to K+:

  • If the inner chamber has 150 mM KCl and the outer chamber has 5 mM KCl, the Nernst potential for K+ is approximately -92 mV.  

• For a membrane selectively permeable to Na+:

  • If the inner chamber has 15 mM NaCl and the outer chamber has 150 mM NaCl, the Nernst potential for Na+ is approximately +62 mV.  

Ohm's Law and Membrane Current:

• Ohm's law states that current (I) is equal to voltage (V) divided by resistance (R): I=V/R.  

• Conductance (g) is the inverse of resistance (g=1/R).  

The Goldman Equation (Goldman-Hodgkin-Katz Voltage Equation):

• This equation describes the membrane potential (V) based on the combined currents due to the fluxes of multiple ions across the membrane.

• The total ionic current (I) across the membrane can be expressed as the sum of currents for each major ion (e.g., Sodium (Na+), Potassium (K+), and a "rest-group" or leak current, often Cl−): I=gNa​(VNa​−V)+gK​(VK​−V)+gR​(VR​−V)  

  • gNa​, gK​, gR​ are the conductances for sodium, potassium, and the rest-group ions, respectively.  

  • VNa​, VK​, VR​ are the Nernst equilibrium potentials for these ions.  

  • (Vion​−V) represents the driving force for each ion.

• At steady state, when the net current across the membrane is zero (I=0), the Goldman equation can be rearranged to solve for the resting membrane potential, V.

Membrane as a Capacitor:

• The cell membrane, with its lipid bilayer (dielectric) separating conductive intracellular and extracellular solutions (parallel plates), functions as a capacitor, storing electrical charge.  

• The equation describing a capacitor is: dV/dt=I/C.  

  • dV/dt is the rate of change of membrane potential.

  • I is the net current flowing across the membrane.

  • C is the capacitance of the membrane.

• Combining this with the Goldman equation for ionic current, the change in membrane potential V over time obeys: dV/dt=(1/C)[gNa​(VNa​−V)+gK​(VK​−V)+gR​(VR​−V)].  

The Hodgkin-Huxley Model of the Action Potential

Developed from voltage-clamp experiments on the squid giant axon, the Hodgkin-Huxley model describes how changes in ion conductances generate action potentials.

• Voltage-Sensitive Ion Channels: The model incorporates voltage-sensitive "gating" variables (m, n, and h) that represent the probability of channel proteins being in a certain state (open or closed/inactivated). These variables were empirically fitted to experimental data.  

  • m: Represents the activation of sodium channels.

  • h: Represents the inactivation (closure) of sodium channels.

  • n: Represents the activation of potassium channels.

• Kinetics of Gating Variables: The rates of change of these variables are described by differential equations, which are functions of the membrane voltage (V). These equations include voltage-dependent rate constants (α and β) for the opening and closing of the channels.  

  • dm/dt=αm​(1−m)−βm​m

  • dh/dt=αh​(1−h)−βh​h

  • dn/dt=αn​(1−n)−βn​n

• Full Hodgkin-Huxley Equation for Membrane Potential: The equation for dV/dt is expanded to include these voltage-dependent conductances: dV/dt=(1/C)[gˉ​Na​m3h(VNa​−V)+gˉ​K​n4(VK​−V)+gL​(VL​−V)] (The slide uses gN​ for gˉ​Na​, gK​ for gˉ​K​, gR​ for gL​, and specific numerical values for maximal conductances: 120m3h for sodium, 36n4 for potassium, and 0.3 for leak. It also provides values for VN​=−115mV, VK​=12mV, VR​=−10.5989mV. Note: these Nernst potential values might reflect a different convention for voltage polarity or specific experimental conditions, as VNa​ is typically positive and VK​ negative. The slides later mention Hodgkin and Huxley scaled their model such that resting potential was 0 and expressed voltage as outside minus inside.)  

• Behavior in Time: Simulations of the Hodgkin-Huxley model can reproduce the characteristic shape of an action potential, showing the rapid depolarization (due to Na+ influx mediated by m-gate opening), repolarization (due to Na+ channel inactivation via h-gate and K+ efflux via n-gate opening), and afterhyperpolarization. The model also shows the time course of the gating variables (m, h, n) during an action potential.  

• Limitations:

  • The Hodgkin-Huxley model is mathematically complex.  

  • Its behavior often requires computer simulation to understand, making it almost as "mysterious" as the biological experiments themselves for gaining intuitive understanding.  

  • The equations are primarily data-fitting descriptions rather than derived from first principles of channel structure and function.  

Simplification: The Fitzhugh-Nagumo Model

Recognizing the complexity of the Hodgkin-Huxley model, FitzHugh (1961) and Nagumo (1962) aimed to simplify the equations to capture the "core" behavior of excitable media.  

• Model Equations: dV/dt=−V(V−a)(V−1)−W dW/dt=ϵ(V−bW)  

  • V represents the membrane voltage (fast variable).

  • W is a slow recovery variable (representing, for example, K+ channel activation or Na+ channel inactivation).

  • 'a' is a parameter related to the threshold (often between 0 and 1).

  • ϵ is a small parameter ensuring W is slower than V.

  • 'b' is a parameter influencing the resting state.

• Dynamics and Phase Plane Analysis:

  • Timeplots: Simulations show that the Fitzhugh-Nagumo model can generate action potential-like spikes for V, with W showing slower recovery dynamics.  

  • Phase Plane: Plotting W against V provides a powerful way to visualize the system's dynamics. The trajectories in the phase plane show how V and W co-evolve.  

    • Nullclines: These are curves where dV/dt=0 (V-nullcline, often N-shaped) or dW/dt=0 (W-nullcline, often linear). Their intersection points represent steady states (fixed points) of the system.

  • Properties Explained by the Model: The simplified model can qualitatively explain key features of neuronal excitability:  

    • Threshold: A stimulus must push V beyond a certain point to trigger a large excursion (action potential).

    • Action Potential: The rapid rise and fall of V.

    • Refractory Period: After an action potential, W is elevated, making it harder to trigger another immediate action potential.

    • Hyperpolarization/Undershoot: V can dip below the resting potential before returning to rest.

    • Resting Potential: The stable steady state to which the system returns in the absence of stimulation.

• Richer Dynamics: The Fitzhugh-Nagumo model can also exhibit more complex dynamics, such as oscillations or bursting, depending on parameter values.  

Fitzhugh-Nagumo in Space (Reaction-Diffusion Systems):

• By adding a diffusion term for V (representing electrical coupling between adjacent membrane patches), the model can describe the propagation of excitation in space: ∂V/∂t=−V(V−a)(V−1)−W+D(∂2V/∂x2+∂2V/∂y2) ∂W/∂t=ϵ(V−bW)  

• This allows for the study of phenomena like propagating action potentials (nerve impulses) and spiral waves in excitable media.  

Systems Approach to Calcium (Ca2+) Physiology

Intracellular Ca2+ signaling is complex, involving multiple channels, pumps, buffers, and Ca2+-sensitive processes.

• Key Components of Ca2+ Signaling:  

  • Plasma membrane Ca2+ channels (e.g., voltage-gated, ligand-gated, store-operated).

  • Receptors (GPCRs, Receptor Tyrosine Kinases - RTKs) coupled to Ca2+ mobilizing pathways (e.g., PLCβ, PLCγ producing IP3​).

  • IP3​ Receptors (InsP3​R) and Ryanodine Receptors (RyR) on the Endoplasmic/Sarcoplasmic Reticulum (ER/SR) release stored Ca2+.

  • Other Ca2+ mobilizing messengers like cADPR (cyclic ADP ribose) and NAADP (nicotinic acid adenine dinucleotide phosphate).

  • Pumps: PMCA (Plasma Membrane Ca2+-ATPase) and SERCA (Sarcoplasmic/Endoplasmic Reticulum Ca2+-ATPase) remove Ca2+ from the cytoplasm.

  • Exchangers: Na+/Ca2+ exchanger (NCX) in the plasma membrane and mitochondria.

  • Mitochondria can take up and release Ca2+.

  • Ca2+ buffers (e.g., calbindin, parvalbumin) and Ca2+-binding proteins (e.g., calmodulin, troponin C, synaptotagmin) mediate downstream effects.

• Ca2+ Fluxes: Cytosolic Ca2+ concentration is determined by the balance of influx (Jin​ from outside, Jchannel​ from ER/SR), efflux (Jpump​ to ER/SR, Jpm​ to outside), and leak (Jleak​ from ER/SR).  

• Modeling Ca2+ Oscillations:

  • Simplified models can capture the oscillatory behavior of cytosolic Ca2+. One such core model might involve two ordinary differential equations (ODEs):  

    • dc/dt=Kflux​nPO1​−(Ve​c2/(Ke2​+c2))

    • dn/dt=g(PO2​−n)

    • 'c' represents cytosolic Ca2+ concentration.

    • 'n' represents the proportion of non-inactivated IP3​ receptors.

  • Bell-Shaped Response of IP3​R to Ca2+: The IP3​ receptor (IP3​R) exhibits complex regulation by Ca2+. It has both an activation site and an inhibitory site for Ca2+. This results in a bell-shaped curve for Ca2+ flux out of the ER as a function of cytosolic Ca2+ concentration, where flux is maximal at intermediate Ca2+ levels and inhibited at very low or very high Ca2+ levels.  

  • Oscillatory Dynamics: This complex feedback allows for the generation of Ca2+ oscillations, which can be visualized in time plots (density vs. time) or phase plane plots (n vs. c). The oscillations arise from the interplay between Ca2+ release from the ER and the subsequent inactivation/reactivation of the IP3​R.  

  • IP3​ pulses can lead to diverse cellular responses, including switching between baseline dynamics and pulses, and modifications in pulse duration, irregularity, and height.  

Spatial Patterns and Explanatory Power of Models

• Mathematical models, including reaction-diffusion versions of the Fitzhugh-Nagumo model, can generate complex spatial patterns of activity, such as propagating waves and spirals.  

• These patterns are observed in biological systems:

  • Spatial calcium waves in Arabidopsis leaves when eaten by a caterpillar.  

  • Calcium waves in Ciona (sea squirt) oocytes during fertilization.  

• Isomorphisms and Explanatory Power: Simplified models like Fitzhugh-Nagumo, by focusing on the essential interactions and connections rather than all specific molecular players, can reveal "isomorphisms" – fundamental similarities in the dynamic behavior of seemingly different biological systems (e.g., nerve impulses and heart fibrillations). This provides significant explanatory power.  

• The famous quote by George Box, "All models are wrong, but some models are useful," underscores the value of simplified, abstract models in gaining understanding of complex systems.