Kinetics & Analysis

Binding Kinetics and Mass Transport

The 1:1 model derived properly, then the transport problem that quietly invalidates it.

28 min read 4 sections 17 sources cited (9 verified)

The 1:1 model, derived#

Start from the simplest chemistry that could produce a sensorgram: one analyte molecule A binds one surface site B to give one complex AB, with no cooperativity and no other species. Everything below follows from writing that down carefully.

The reaction and its rate law:

A + B ⇌ AB d[AB]/dt = k_a[A][B] − k_d[AB]

(5.1)

where

k_aassociation rate constant, M⁻¹ s⁻¹

k_ddissociation rate constant, s⁻¹

Now translate into what the instrument sees. The response R is proportional to complex on the surface, so R ∝ [AB]. Total site capacity corresponds to R_max, so free sites correspond to (R_max − R). And [A] is the analyte concentration C, which we hold constant by flowing fresh sample. Substituting:

dR/dt = k_a · C · (R_max − R) − k_d · R

(5.2)

where

Rresponse at time t (RU)

R_maxresponse at full occupancy (RU)

Cbulk analyte concentration (M), assumed constant

This equation, and the assumption behind it that C at the surface equals C in the bulk, is the entire foundation of routine SPR kinetics. The second half of this guide is about what happens when that assumption fails.

Solving it

Group the terms in R:

dR/dt = k_aC·R_max − (k_aC + k_d)·R

(5.3)

This is a linear first-order ODE with constant coefficients, so its solution is an exponential approach to the value that makes dR/dt zero (O’Shannessy et al., 1993; Morton et al., 1995). Setting the right side to zero gives that value directly:

R_eq = k_aC·R_max / (k_aC + k_d) = R_max · C / (C + K_D)

(5.4)

where

K_Dequilibrium dissociation constant, k_d/k_a, in M

The second form drops out by dividing top and bottom by k_a. This is the Langmuir isotherm — a rectangular hyperbola, half-maximal at C = K_D.

and the approach to it is exponential with rate constant equal to the coefficient of R:

R(t) = R_eq(1 − e^−k_obst), k_obs = k_aC + k_d

(5.5)

The dissociation phase

Set C = 0 in the rate equation and the first term vanishes:

dR/dt = −k_dR ⟹ R(t) = R₀e^−k_dt

(5.6)

A single exponential, containing k_d and nothing else. This is why the dissociation phase is so valuable and why cutting it short is such a common and expensive mistake.

What the numbers should look like#

Quantity	Typical range	Interpretation of extremes
k_a	10³ – 10⁷ M⁻¹ s⁻¹	Above ~10⁷ exceeds the practical diffusion limit for proteins without electrostatic steering — suspect transport or an un-subtracted bulk shift (Squires et al., 2008)
k_d	10⁻⁵ – 10⁻¹ s⁻¹	Below 10⁻⁵ cannot be resolved in a normal dissociation window (Drake et al., 2004); above 10⁻¹ is faster than most instruments sample cleanly
K_D	pM – mM	Sub-pM from kinetics is almost always an avidity artefact rather than an intrinsic affinity (Nieba et al., 1996)
t_½	seconds – hours	Directly ln2/k_d; this is the number a pharmacologist actually wants

Plausibility ranges. Values well outside these should prompt investigation before publication, not after.

Set k_a to 10⁸ and watch the association phase collapse to a near-vertical step — which is exactly what an un-subtracted bulk shift looks like.

Mass transport: the assumption that fails#

The rate equation above assumes the analyte concentration at the surface equals the concentration in the bulk. In a flow cell, it does not. Flow is laminar, so there is no turbulent mixing; the fluid velocity goes to zero at the wall, and analyte crosses the last few micrometres by diffusion alone. If binding consumes analyte faster than diffusion resupplies it, the surface sits in a depleted zone and the actual driving concentration is lower than the one you think you injected.

Glaser showed this numerically for antibody–antigen systems in 1993 (Glaser, 1993); Myszka and colleagues demonstrated the distortion on a real antigen–antibody pair (Myszka et al., 1997); Schuck developed the full treatment for ligand distributed in a polymer matrix (Schuck, 1996); and Schuck and Minton reduced it to the two-compartment form that most analysis software implements (Schuck & Minton, 1996).

The two-compartment model

Treat the flow cell as two compartments: the bulk, at the concentration you injected, and a thin surface compartment where binding happens. Analyte moves between them at a rate governed by a transport coefficient k_t:

dC_surf/dt = k_t(C_bulk − C_surf) − [ k_aC_surf(R_max − R) − k_dR ]

(5.7)

where

C_surfanalyte concentration in the surface compartment

C_bulkthe concentration you injected

k_ttransport coefficient, set by the flow cell geometry and the analyte’s diffusion coefficient

The surface compartment is thin, so it reaches quasi-steady state almost immediately. Setting dC_surf/dt = 0 and solving gives the expression the simulator on this page actually integrates:

C_surf = [ k_tC_bulk + k_dR ] / [ k_t + k_a(R_max − R) ]

(5.8)

Read the limits. When k_t is very large, C_surf → C_bulk and equation 5.2 is recovered. When k_t is small compared with k_a(R_max − R), C_surf collapses towards zero and the observed rate becomes k_t·C_bulk — independent of k_a entirely. Between those limits you are in partial transport limitation, which is the regime most real experiments occupy.

Raise R_max and lower k_t until the regime indicator turns red. Note that the association goes straight and the dissociation acquires a tail — both from the same cause.

The diagnostic signature

Association looks linear, because the rate is set by delivery and delivery is constant (Schuck & Minton, 1996).
Dissociation starts fast then drags, because analyte released from a site rebinds to a neighbouring free site before diffusing out of the matrix (Schuck, 1996).
Fitted k_a depends on ligand density. Halve the immobilisation level; if k_a rises, transport was limiting. This is the definitive test.
Fitted k_a depends on flow rate. A weaker test, because k_m scales only as the cube root of flow (the transport expression above), but a real one (Vijayendran et al., 1999).

Turning the problem into a method: CFCA#

Full mass transport limitation is a disaster for kinetics and a gift for concentration measurement. Under complete limitation the initial binding rate depends only on the analyte’s diffusion coefficient, the flow cell geometry and its concentration — the chemistry has dropped out entirely.

That means the initial rate gives an absolute concentration without a standard curve of the same molecule. Karlsson and colleagues set out both the kinetic and the concentration uses of the technology early (Karlsson et al., 1994); Christensen laid out the theory of the transport-limited concentration measurement (Christensen, 1997); the approach is now standard as calibration-free concentration analysis (Pol et al., 2016).

The practical requirements are the reverse of kinetic design: high ligand density, low flow rate, and confirmation that you are genuinely transport-limited — which is done by measuring at two flow rates and checking the rates differ by the expected F^1/3 ratio (Sigmundsson et al., 2002). If they do not, you are not fully limited and the absolute number is not valid.

Sources cited on this page

Listed alphabetically. Each badge records whether the bibliographic record was confirmed against Crossref. unverified marks a real, deliberately chosen source whose volume and page numbers we have not yet machine-checked — it is not a comment on the science.

Christensen, 1997L. L. H. Christensen (1997). Theoretical analysis of protein concentration determination using biosensor technology under conditions of partial mass transport limitation. Analytical Biochemistry 249, 153–164. doi:10.1006/abio.1997.2182 verified
The theoretical basis for using transport-limited initial rate as a concentration readout.
Drake et al., 2004A. W. Drake, D. G. Myszka, S. L. Klakamp (2004). Characterizing high-affinity antigen/antibody complexes by kinetic- and equilibrium-based methods. Analytical Biochemistry 328, 35–43. doi:10.1016/j.ab.2004.01.031 unverified
What can and cannot be measured when the off-rate is very slow.
Glaser, 1993R. W. Glaser (1993). Antigen–antibody binding and mass transport by convection and diffusion to a surface: a two-dimensional computer model of binding and dissociation kinetics. Analytical Biochemistry 213, 152–161. doi:10.1006/abio.1993.1399 verified
Early numerical demonstration that transport to the surface, not chemistry, can set the observed rate.
Karlsson et al., 1994R. Karlsson, H. Roos, L. Fägerstam, B. Persson (1994). Kinetic and concentration analysis using BIA technology. Methods 6, 99–110. doi:10.1006/meth.1994.1013 verified
Karlsson & Fält, 1997R. Karlsson, A. Fält (1997). Experimental design for kinetic analysis of protein–protein interactions with surface plasmon resonance biosensors. Journal of Immunological Methods 200, 121–133. doi:10.1016/S0022-1759(96)00195-0 verified
Where the low-density / high-flow-rate / analyte-range design rules come from.
Morton et al., 1995T. A. Morton, D. G. Myszka, I. M. Chaiken (1995). Interpreting complex binding kinetics from optical biosensors: a comparison of analysis by linearization, the integrated rate equation, and numerical integration. Analytical Biochemistry 227, 176–185. doi:10.1006/abio.1995.1268 verified
Introduced numerical integration to biosensor analysis, which is why arbitrary interaction models became fittable.
Myszka et al., 1997D. G. Myszka, T. A. Morton, M. L. Doyle, I. M. Chaiken (1997). Kinetic analysis of a protein antigen–antibody interaction limited by mass transport on an optical biosensor. Biophysical Chemistry 64, 127–137. unverified
Myszka et al., 1998D. G. Myszka, X. He, M. Dembo, T. A. Morton, B. Goldstein (1998). Extending the range of rate constants available from BIACORE: interpreting mass transport-influenced binding data. Biophysical Journal 75, 583–594. doi:10.1016/S0006-3495(98)77549-6 verified
Shows transport can be fitted rather than merely avoided, and defines the transport coefficient kₜ.
Nieba et al., 1996L. Nieba, A. Krebber, A. Plückthun (1996). Competition BIAcore for measuring true affinities: large differences from values determined from binding kinetics. Analytical Biochemistry 234, 155–165. doi:10.1006/abio.1996.0067 unverified
A direct demonstration that surface-measured kinetic constants can diverge substantially from solution affinities, and a solution-competition format that avoids the problem.
O’Shannessy et al., 1993D. J. O’Shannessy, M. Brigham-Burke, K. K. Soneson, P. Hensley, I. Brooks (1993). Determination of rate and equilibrium binding constants for macromolecular interactions using surface plasmon resonance: use of nonlinear least squares analysis methods. Analytical Biochemistry 212, 457–468. doi:10.1006/abio.1993.1355 verified
The case for fitting the sensorgram directly rather than linearising it.
O’Shannessy & Winzor, 1996D. J. O’Shannessy, D. J. Winzor (1996). Interpretation of deviations from pseudo-first-order kinetic behavior in the characterization of ligand binding by biosensor technology. Analytical Biochemistry 236, 275–283. doi:10.1006/abio.1996.0167 unverified
Where non-exponential behaviour comes from, and how to tell the causes apart.
Pol et al., 2016E. Pol, H. Roos, F. Markey, F. Elwinger, A. Shaw, R. Karlsson (2016). Evaluation of calibration-free concentration analysis provided by Biacore systems. Analytical Biochemistry 510, 88–97. unverified
CFCA validation study. NOTE: a candidate DOI of 10.1016/j.ab.2016.08.005 was checked during the audit and resolves to an unrelated paper on RNA lyophilisation, so no DOI is asserted here. Volume and pages should be confirmed before quoting.
Schuck, 1996P. Schuck (1996). Kinetics of ligand binding to receptor immobilized in a polymer matrix, as detected with an evanescent wave biosensor. I. A computer simulation of the influence of mass transport. Biophysical Journal 70, 1230–1249. doi:10.1016/S0006-3495(96)79681-9 verified
Schuck & Minton, 1996P. Schuck, A. P. Minton (1996). Analysis of mass transport-limited binding kinetics in evanescent wave biosensors. Analytical Biochemistry 240, 262–272. doi:10.1006/abio.1996.0356 verified
The two-compartment model in the form most SPR software still implements.
Sigmundsson et al., 2002K. Sigmundsson, G. Másson, R. Rice, N. Beauchemin, B. Öbrink (2002). Determination of active concentrations and association and dissociation rate constants of interacting biomolecules: an analytical solution to the theory for kinetic and mass transport limitations in biosensor technology and its experimental verification. Biochemistry 41, 8263–8276. doi:10.1021/bi020099h unverified
Analytical treatment of combined kinetic and transport limitation, with experimental validation.
Squires et al., 2008T. M. Squires, R. J. Messinger, S. R. Manalis (2008). Making it stick: convection, reaction and diffusion in surface-based biosensors. Nature Biotechnology 26, 417–426. doi:10.1038/nbt1388 unverified
The general transport analysis for any surface-based sensor, in dimensionless form. The clearest statement of when a measured rate is chemistry and when it is delivery.
Vijayendran et al., 1999R. A. Vijayendran, F. S. Ligler, D. E. Leckband (1999). A computational reaction–diffusion model for the analysis of transport-limited kinetics. Analytical Chemistry 71, 5405–5412. doi:10.1021/ac990672b unverified