Itc1
7.1 Scattering and molar mass #
7.1.1 Dynamic light scattering #
Brownian motion makes the intensity of scattered light fluctuate. The decay of the intensity correlation gives the diffusion coefficient D, and the Stokes–Einstein relation converts it to a hydrodynamic radius:
R_h = k_B T / (6 π η D)
k_B is Boltzmann’s constant, T absolute temperature, and η solvent viscosity. R_h is the radius of a sphere that would diffuse as fast. A rod and a sphere with the same R_h are not the same geometric size. A wrong viscosity, or a wrong temperature in the software, moves R_h even though the particle has not changed.
For particles much smaller than the wavelength, intensity grows with the square of mass, that is with the sixth power of radius. A trace of aggregate can dominate the photons. The radius quoted first is an intensity-weighted average, and a number-weighted distribution is an inversion that can look comfortably monodisperse while the signal was the aggregate. Dynamic light scattering is a fast warning that large particles are present. It is not a molar mass and not a proof of homogeneity. Static scattering, using average intensity rather than fluctuation, is instead proportional to c M and can give a molar mass and a radius of gyration.
7.1.2 SEC-MALS #
Size-exclusion chromatography followed by multi-angle light scattering and a concentration detector keeps the separation and drops the globular calibration. Scattered intensity is proportional to c M (dn/dc)². With c measured independently, M follows without a shape assumption. For a simple protein in water, dn/dc is about 0.185 mL g⁻¹. Glycoproteins, nucleic acids, and protein–detergent complexes differ, and an error in dn/dc enters as its square.
Co-eluting species produce a weight-average mass that can look like a new oligomer. A weak complex can dissociate on dilution down the column and be weighed as the species that actually eluted. SEC-MALS answers “what is the mass in this peak?”. Sedimentation equilibrium asks a related question without that dilution, and the two disagree usefully when binding is weak.
7.2 Calorimetry #
7.2.1 Isothermal titration calorimetry #
One partner is injected in aliquots into the other at constant temperature, and the heat of each injection is recorded. For a simple binding reaction the isotherm can return the stoichiometry n, from the molar ratio at the equivalence point; the enthalpy ΔH, from the heat per mole of complex after the blank; and K_d, from the curvature. A nearly rectangular step means binding is tight relative to the concentrations used: n and ΔH are well determined and K_d is only bounded. A very shallow slope means binding is too weak for those concentrations, and heat and affinity trade off. The Wiseman parameter c = n [M]_t / K_d, with [M]_t the concentration of sites, expresses the window. Values of c in a middle range, often discussed as roughly 10 to a few hundred, let one isotherm carry all three. Outside it the experiment is still valid and simply lacks the information.
ΔG° = R T ln(K_d / c°) with the standard state c° = 1 mol L⁻¹, so a sub-millimolar K_d gives a negative ΔG°. Entropy follows from ΔG = ΔH − TΔS. The enthalpy is measured. A van’t Hoff analysis that assumes ΔH constant will disagree if the heat capacity change is large. The disagreement is physical.
The heat of diluting ligand into buffer must be subtracted, and the syringe and cell solutions should differ only in the binding partners. If binding takes up or releases protons, the buffer’s ionisation enthalpy is inside the observed heat: ΔH_obs = ΔH_bind + n_H ΔH_buffer. Repeating the titration in buffers of different ionisation enthalpy, phosphate and Tris being the usual pair, separates the binding enthalpy from the linked protons. A non-integer n may be real stoichiometry or a fraction of incompetent protein. The isotherm does not decide.
7.2.2 Differential scanning calorimetry #
Sample and reference are heated together and the difference in heat capacity is recorded. Unfolding is a peak of excess heat capacity. Its midpoint is T_m and its area is the calorimetric enthalpy ΔH_cal. The unfolded chain usually has the higher heat capacity, because non-polar groups meet water, so the baseline steps upward.
The van’t Hoff enthalpy, inferred from how sharp the peak is, equals ΔH_cal if the molecule unfolds in one step with no intermediates. A larger calorimetric enthalpy means a broader transition: intermediates or independent domains. That comparison is the reason to measure the heat rather than to assume two states from a CD melt. If a reheating scan does not retrace the first, the transition is not at equilibrium, usually because the unfolded chain aggregates, and a free energy taken from it is not thermodynamic. A T_m that moves with scan rate is the kinetic signature. Ligand bound to the native state raises T_m. The shift shows binding. It is not itself a K_d. Titration calorimetry is the direct binding measurement.
7.3 Surface plasmon resonance #
7.3.1 Rates from a sensorgram #
At a particular angle, light excites a surface plasmon in a thin metal film, usually gold, and reflectance drops. The angle depends on the refractive index within a few hundred nanometres of the surface, so bound macromolecules move it. On the dextran-coated chips widely used for this work, one response unit is about 1 pg of protein per square millimetre. The record is bound mass against time. It is not an optical spectrum, which is why the method is not in Chapter 3.
One partner is immobilised and the other flows past. Association reports the on-rate constant k_a together with the analyte concentration. Dissociation, with buffer alone, reports the off-rate constant k_d. For a one-to-one reaction
K_d = k_d / k_a
A plateau plotted against concentration gives K_d without a kinetic model. Random coupling through lysines can kill a fraction of sites, so the maximum response is lower than the coupled mass predicts. The K_d of a true one-to-one reaction can still be right. A mass stoichiometry read from that maximum is not.
7.3.2 Mass transport and rebinding #
If molecules bind as fast as they arrive, the rising curve measures delivery by flow and diffusion, not k_a. High surface density makes the surface a better sink and makes this worse. A molecular k_a does not change with flow rate. A transport-limited rate does.
During dissociation, analyte can rebind a neighbouring site before it escapes. The observed off-rate is then slower than the microscopic k_d, and the complex looks too tight. A true k_d is independent of how much ligand was immobilised. A k_d that slows as the surface is loaded more heavily is rebinding until proved otherwise. Avidity is the chemical version of the same trap: a bivalent antibody stays down after one arm lets go, and a one-to-one fit does not return the solution K_d of a single site.
A mismatch in bulk refractive index, common when the analyte solution contains dimethyl sulphoxide, produces a step that is not binding and is subtracted with a reference channel. After a careful experiment the number is still a surface number. Calorimetry or a spectroscopic titration in free solution is what shows that the surface did not invent it.
7.4 Key terms, summary, and self-check #
Key terms. Hydrodynamic radius; z-average; dn/dc; SEC-MALS; Wiseman c parameter; linked protonation; T_m; calorimetric enthalpy; van’t Hoff enthalpy; sensorgram; mass transport; rebinding; avidity.
Summary. Dynamic light scattering reports a hydrodynamic radius and is dominated by the largest particles. SEC-MALS reports the mass of an eluting peak if dn/dc is right and the peak is one species. One ITC isotherm can give n, ΔH, and K_d inside a concentration window, after dilution heats and buffer ionisation are dealt with. DSC gives T_m, the unfolding enthalpy, and a two-state test. Surface plasmon resonance gives k_a and k_d unless mass transport, rebinding, or avidity has replaced them with properties of the surface.
Self-check.
1. Why can a few percent by mass of aggregate dominate a dynamic-light-scattering signal?
2. What must still be known for a SEC-MALS mass, and what is reported if two proteins co-elute?
3. An ITC isotherm is almost rectangular. Which of n, ΔH, and K_d are well determined, and why is a dilution blank still required?
4. Why can a change of buffer change ΔH when K_d does not change?
5. Dissociation in an SPR experiment slows as more ligand is immobilised. What artefact is that, and why is the fitted K_d not the solution value?
—
—
—