Modern Physics · Advanced Topics

Biophysics

Physics provides the quantitative framework for understanding biological systems — from the mechanics of molecular motors to the statistical physics of protein folding, from the electrophysiology of neurons to the optics of the eye. Life operates at the boundary between order and thermal chaos.

PrerequisitesStatisticalmechanics(Ch.S)⋅Fluidmechanics(Ch.FM)⋅Irreversibleprocesses(Ch.IRStatistical mechanics (Ch. S) \cdot Fluid mechanics (Ch. FM) \cdot Irreversible processes (Ch. IR ⋅Probability(Ch.PR\cdot Probability (Ch. PR
Learning Goals
  • Apply the worm-like chain model to compute the force-extension curve of DNA and identify the entropic spring regime.
  • Compare diffusion timescales with active motor transport to explain why neurons require kinesin-based transport.
  • Explain the funnel energy landscape and Kramers rate theory as resolutions to the Levinthal paradox for protein folding.
  • Derive the resting membrane potential from the Goldman equation and describe the Hodgkin-Huxley action potential mechanism.
  • Interpret the Hill equation for cooperativebindingandexplainhowcooperativityenableshemoglobintoloadandunloadO2cooperative binding and explain how cooperativity enables hemoglobin to load and unload O_{2}

BP.1 Forces at the Molecular Scale

Biological molecules operate in the thermal energy scale k_BT ≈ 4.1 pN·nm (at 310 K). Forces in biology:

Thermal forces: kT/ℓ ≈ 4 pN for ℓ = 1 nm. Dominates at nanoscale.Chemical bonds: covalent ~1 nN (GPa range), hydrogen bonds ~5–50 pN.Motor forces: kinesin 5–7 pN, myosin ~3 pN (optical trap measurements).DNA mechanics: persistence length ℓ_p = 50 nm (double-stranded DNA). The worm-like chain (WLC) model:

F=kBT/(2ℓp)×[x/L+1/(4(1−x/L)2)−1/4](WLCforce−extension,DNA)F = k_{BT}/(2 ℓ_p) \times [x/L + 1/(4(1-x/L)^{2}) - 1/4] \qquad (WLC force-extension, DNA)(BP.1)

where x is extension and L is contour length. At small extension: F ≈ k_BT x/(ℓ_p L) (entropic spring). At near full extension: F ≈ k_BT/(4ℓ_p (1−x/L)²) (diverges as x→L).

BP.2 Molecular Motors

Molecular motors convert chemical energy (ATP hydrolysis, ΔG ≈ −54 kJ/mol = −23 k_BT) to mechanical work. Key systems:

Kinesin: moves along microtubules toward the plus end (8 nm steps). Two heads alternately bind/release: hand-over-hand mechanism. Stall force: F_stall ≈ 7 pN. Velocity: v ≈ 800 nm/s at zero load. Efficiency: η = F×d/(ΔGATP) ≈ 7×8/(23×4.1) ≈ 60%.

ATP synthase: rotary motor. The F₀ motor (driven by proton gradient) rotates the γ-subunit, mechanically coupling to F₁ (synthesizes ATP). Speed: up to 100 rotations/s. Generates ~3 ATP per revolution (3 catalytic sites). Efficiency near 100% (thermodynamic limit). Flagellar motor: similar rotary mechanism, powers swimming.

v=v0(1−F/Fstall)(linearforce−velocityformotornearstall)v = v_{0}(1 - F/F_{stall}) \qquad (linear force-velocity for motor near stall)(BP.2)

The fluctuation-dissipation theorem applies to motors: the randomness (diffusion-like wandering) and the mean drift are both thermally driven. Efficiency is limited by the Carnot-like second law for chemical machines: η ≤ 1 − T_diss/T.

Example BP.1 — Diffusion vs. Active Transport in a Cell

Comparediffusiontimeforaprotein(D=10\mum2/s)vs.kinesintransportacrossa1maxoCompare diffusion time for a protein (D = 10 \mum^{2}/s) vs. kinesin transport across a 1 m axon(v=1\mum/sn (v = 1 \mum/s.

Diffusion time:From⟨x2⟩=2Dt:tdiff=L2/(2D)=(1m)2/(2×10×10−12m2/s)=1/(2×10−11)=5×1010s≈16From ⟨x^{2}⟩ = 2Dt: t_{diff} = L^{2}/(2D) = (1 m)^{2}/(2\times10\times10^{-12} m^{2}/s) = 1/(2\times10^{-11}) = 5\times10^{10} s \approx 1600 years.
Active transport:tmotor=L/v=1m/(10−6m/s)=106s≈11.6dayst_{motor} = L/v = 1 m/(10^{-6} m/s) = 10^{6} s \approx 11.6 days.
Comparison:Motortransportis 50,000×fasterthandiffusionfor1m.ThisiswhyneuronshaveanelaMotor transport is ~50,000\times faster than diffusion for 1 m. This is why neurons have an elaborate axonal transport system — diffusion simply fails at macroscopic distances.
Small cells:Fora1\mumcell:tdiff=(10−6)2/(2×10−11)=5×10−2s=50ms.Diffusionadequate!ThecrFor a 1 \mum cell: t_{diff} = (10^{-6})^{2}/(2\times10^{-11}) = 5\times10^{-2} s = 50 ms. Diffusion adequate! The crossoverlengthfortransport:L∗∼(2Dτ)whereτisacharacteristictime.Motorsbecomeossover length for transport: L* ∼ \sqrt(2D\tau) where \tau is a characteristic time. Motors become necessary when L ≫ L*.

BP.3 Protein Folding

A protein of N amino acids in its native state has a unique 3D structure determined by its sequence (Anfinsen's principle). The folding energy landscape:

Levinthal paradox: if a 100-residue protein sampled all conformations at 10⁹ s⁻¹: τ ∼ 10^(300)/10⁹ ≈ 10^(291) years — longer than the age of the universe! Yet proteins fold in μs–ms. Resolution: the energy landscape is funnel-shaped (not random) — folding is directed by the overall gradient toward the native state.

Two-state folding: small proteins fold in a highly cooperative, all-or-none transition. Rate: k_fold ∝ e^(−ΔG‡/(k_BT)) (Kramers theory). Chymotrypsin inhibitor 2 (CI2): folds in 10 μs. Folded proteins marginally stable: ΔG_stab ≈ −50 kJ/mol ≈ −20 k_BT — only a few hydrogen bonds above random-coil.

Misfolding diseases:Alzheimer's (amyloid β), Parkinson's (α-synuclein), prion diseases — all involve proteins aggregating into ordered β-sheet fibers. The free energy of the fiber can be lower than the native state when concentration is high enough.

BP.4 Membrane Biophysics

Cell membranes are lipid bilayers (~4 nm thick). Elasticity described by the Helfrich Hamiltonian:

E=∫dA[12κ(2H)2+κGK+σ](Helfrichmembraneenergy)E = \int dA [\frac{1}{2} \kappa (2H)^{2} + \kappa_G K + \sigma] \qquad (Helfrich membrane energy)(BP.3)

where H is mean curvature, K is Gaussian curvature, κ ≈ 10–100 k_BT is the bending rigidity, κ_G is saddle-splay modulus, and σ is surface tension. Thermal fluctuations cause the membrane to undulate (Helfrich fluctuations). The Gauss-Bonnet theorem: ∫ K dA = 4π(1−g) (genus g) — topology constrains the total Gaussian curvature.

Ion channels and action potentials: membrane potential V_m across a cell (inside −70 mV). An action potential propagates along an axon via the Hodgkin-Huxley model: coupled ODEs for V_m and gating variables n, m, h of Na⁺/K⁺ channels. Nobel 1963. Propagation speed: v ∝ √(D/τ_RC) where D = λ²/(RC) is the cable diffusivity (λ = space constant, τ_RC = RC = time constant).

BP.5 Statistical Physics of Biological Networks

Gene regulatory networks, protein-protein interaction networks, and neural networks all share statistical properties. The Hill equation describes cooperative binding:

θ=[L]n/(Kdn+[L]n)(Hillequation,n=Hillcoefficient)\theta = [L]^n / (K_{d}^n + [L]^n) \qquad (Hill equation, n = Hill coefficient)(BP.4)

n = 1: Michaelis-Menten (no cooperativity). n > 1: cooperative (sigmoidal switch). Hemoglobin: n = 2.8 (cooperative O₂ binding). This cooperativity allows hemoglobin to load in the lungs (pO₂ = 100 mmHg) and unload in tissues (pO₂ = 40 mmHg).

Definition BP.1 — Common Traps
  • Thermal energy is comparable to molecular energies: kBTsetsthescaleforbiologicalfluctuationsk_{BT} sets the scale for biological fluctuations
  • Low Reynolds number changes intuition: microscopic swimmers cannot coast.
  • Binding curves reflect ensembles: fractional occupancy is a probability, not one molecule half-bound.
  • Free energy drives direction: favorable processes can still have kinetic barriers.
Exercises — BP.1–BP.5 Biophysics
1.
ApplytheWLCmodeltoDNA(ℓp=50nm,L=3400nm)andcalculatetheforceat50Apply the WLC model to DNA (ℓ_p = 50 nm, L = 3400 nm) and calculate the force at 50% and 90% extension. Sketch the force-extension curve.
pN
Straightforward
2.
Use the Goldman equation to calculate the resting membrane potential for a neuron with given ionic concentrations. How does the Na-K pump maintain this potential?
mV
Intermediate
3.Explain why Feynman's ratchet cannot extract work from a single heat bath, but biological motors can extract work from ATP. What principle distinguishes the two cases?
Intermediate
4.Describe the Hodgkin-Huxley model for the action potential. What is the role of each ionic current? How does myelination increase conduction velocity?
Challenging
Key Takeaways
  • ThermalenergykBT≈4pN\cdotnmsetsthescaleforbiologicalforcesandfluctuationsThermal energy k_{BT} \approx 4 pN\cdotnm sets the scale for biological forces and fluctuations.
  • WLCmodel:DNAaselasticrodwithpersistencelengthℓp=50nm.EntropicspringatlowWLC model: DNA as elastic rod with persistence length ℓ_p = 50 nm. Entropic spring at low extension.
  • MolecularmotorsconvertATP(23kBT)tomechanicalwork.Kinesin:7pNstall,800nm/sMolecular motors convert ATP (23 k_{BT}) to mechanical work. Kinesin: 7 pN stall, 800 nm/s.
  • Protein folding: funnel energy landscape resolves Levinthal paradox. Two-state kinetics.
  • Helfrichmembrane:κ≈10−100kBTbendingrigidity.ThermalfluctuationscauseundulationHelfrich membrane: \kappa \approx 10-100 k_{BT} bending rigidity. Thermal fluctuations cause undulations.
  • Hodgkin−Huxley:Na+andK+channelgatingequationsdescribeactionpotentialpropagationHodgkin-Huxley: Na^{+} and K^{+} channel gating equations describe action potential propagation.