Quantitative Physiology

• Physiological scales: molecules, cells, and organ systems.
• How equations describe physiological phenomena: variables, parameters, units, and assumptions.
• A simple example connecting a physiological question to a mathematical model.
• Introduction to AI coding tools and setup instructions.

Prepare for October 6: choose a coding tool and follow the setup instructions provided in class.

Understand: Trace an agentic loop. Explain function-call schemas, structured outputs, and the MCP host–client–server architecture. Identify where the model’s equations, parameters, and validation requirements enter an agent’s implementation task.

Use AI: Use a coding agent to create the passive membrane explorer described in the AI coding studio. Keep equations, solver, and interface in separate modules.

Check: Compare the numerical trajectory with the supplied analytic solution. Review the code diff and document a correction.

Milestone: Build the analytic-benchmark membrane explorer; reuse this workflow for every demo.

Understand: Explain how forward and reverse reaction terms change species concentrations while conserving total material. Interpret Kd = k₋/k₊ as a measure of binding affinity, and relate the sign of ΔG to reaction direction using dimensionless activities in Q.

Use AI: Build a reversible-binding simulator with adjustable initial conditions and rate constants. Display transient concentrations and equilibrium predictions.

Check: Confirm conserved totals and zero net reaction rate at equilibrium. Treat equilibrium and driven steady states distinctly.

Understand: Interpret each term in the full enzyme-reaction model and the conservation relation e + c = e₀. Explain how Vmax and Km shape the Michaelis–Menten curve and when the quasi-steady-state approximation is appropriate.

Use AI: Class activity: compare full and reduced enzyme models while changing substrate and enzyme concentrations.

Check: Verify mass balance, positivity, and low- and high-substrate asymptotes. Identify a regime where the reduction fails.

Milestone: Written 1 and Demo 1: initial model for your final project, including the project question and a validation check.

Understand: Membrane transport: Explain how concentration gradients, diffusivity, and membrane thickness determine flux. Interpret the diffusion equation as local mass conservation and explain carrier-state balances.

Molecular motors: Explain how step size and forward/backward rates determine mean drift and spreading. Connect random trajectories to the ensemble quantities v and D.

Use AI: Membrane transport & cellular homeostasis: Create a diffusion or carrier-transport explorer with explicit initial and boundary conditions.

Molecular motors & biased random walks: Simulate many motor trajectories and compare mean displacement and variance with theory. Use this as a class activity or a relevant extension of your project.

Check: Membrane transport & cellular homeostasis: Check flux direction, boundary conditions, mass balance for the chosen boundaries, and spatial/time-step convergence.

Molecular motors & biased random walks: At equal rates, verify zero drift with nonzero diffusion. Report uncertainty from a finite ensemble.

Milestone: Plan a relevant extension of your Demo 1 model for Demo 2.

Understand: Hill model: Interpret the force–velocity curve, parameters, isometric limit, and zero-load limit. Explain the physiological scope of this phenomenological relation.

Huxley model: Explain how attachment, detachment, and movement change the bound cross-bridge population. Interpret n, f, g, the velocity sign convention, and how individual bridge forces contribute to total force.

Use AI: Muscle mechanics: the Hill model: Build a force–velocity explorer. Compare parameter sets and compute mechanical power p·v.

Cross-bridge dynamics: the Huxley model: Explore cross-bridge dynamics as a class activity. For Demo 2, extend your existing project and check the new model behavior.

Check: Muscle mechanics: the Hill model: Recover v = 0 at p = p₀ and identify the valid shortening regime. Check force and velocity units.

Cross-bridge dynamics: the Huxley model: Check 0 ≤ n ≤ 1 and the zero-velocity equilibrium n = f/(f + g), where f + g > 0.

Milestone: Written 2 and Demo 2: extended final-project model with parameter exploration and validation.

Understand: Vascular networks: Explain how pressure differences and resistance determine flow, why radius has a fourth-power effect, and how series and parallel vessels redistribute flow.

Windkessel models: Interpret resistance, compliance, pressure, and inflow. Explain how the pressure-change term represents volume storage, under constant-parameter and zero venous reference-pressure assumptions.

Use AI: Class activity: use a worked resistance-network example and explore a Windkessel app with adjustable resistance, compliance, and pulse timing. For Demo 3, integrate and validate the components of your existing project.

Check: Circulation as a flow network: Conserve flow at each junction and verify the fourth-power radius dependence. State why real vessels can depart from this model.

Pulsatile flow & Windkessel models: With Q = 0, recover Pc(t) = Pc(0) exp[−t/(RvC)]. Verify units and periodic behavior after transients.

Milestone: Begin Demo 3: integrate the components of your final project and strengthen validation.

Understand: Distinguish dissolved oxygen concentration, partial pressure, saturation, and total oxygen content. Interpret α, P₅₀, and the Hill exponent h; explain how binding and cooperativity shape oxygen uptake and release.

Use AI: Create a binding-curve explorer and calculate illustrative oxygen-content changes across a pressure range.

Check: Recover Y → 0 and Y → 1 at the limits, with Y = 1/2 at P₅₀. Label illustrative parameter values.

Milestone: Written 3 and Demo 3: integrated final-project demo with validation and physiological interpretation.

Understand: Interpret the Nernst equation using ion concentrations, valence, temperature, and an inside-minus-outside voltage convention. Explain how capacitance, conductances, reversal potentials, and applied current determine membrane-voltage changes.

Use AI: Extend the Week 01 app to multiple conductances; show how each changes the resting voltage and time constant.

Check: For constant conductances and zero applied current, recover Vrest = ΣgiEi/Σgi and τ = C/Σgi.

Milestone: Revisit and explain the Week 01 membrane explorer; no new standalone demo milestone.

Understand: Explain α and β as opening and closing rates, n∞ as the steady-state open fraction, and τn as the response timescale. Interpret how voltage–recovery feedback and current signs produce rest, threshold behavior, and oscillations.

Use AI: Create a Morris–Lecar phase-plane explorer with trajectories and nullclines. Vary input current and distinguish rest from repetitive firing.

Check: Keep gates in [0,1], verify current signs, and compare behavior across solver tolerances.

Milestone: Working final-project demo and peer validation review.

Understand: Action potentials and propagation: Interpret the sodium, potassium, leak, and capacitive terms; explain activation, inactivation, and refractoriness. Explain how axial current coupling in the cable equation supports propagation.

Synapses: Interpret receptor activation and decay, synaptic conductance, and the driving force Esyn − Vpost. Explain how reversal potential shapes the postsynaptic response.

Use AI: Use an agent to implement a Hodgkin–Huxley stimulus-response example and add a prescribed synaptic input. Analyze the cable equation and coupled-cell formulation; full spatial simulation is optional. Peer-review final-project equations and numerical evidence.

Check: Action potentials & electrical propagation: Use a documented parameter set and voltage convention. Check gate bounds, refractory behavior, and time-step or spatial convergence.

Synapses, coupled cells & model critique: When transmitter T = 0, recover exponential decay of s. At Vpost = Esyn, verify zero synaptic current.

Milestone: Present the refined final-project demo developed through Demos 1–3. The single-cell example is a class activity; spatial propagation is an optional implementation.