Biacore application guides
Principles
of kinetics
and affinity
analysis
2
Introduction
This Application guide gives an overview of the theoretical
principles of kinetics and affinity determination with Biacore
systems, insofar as the theory is relevant to interpretation
of the experimental results.
Practical aspect of kinetics and affinity determination are
covered in a separate Application guide.
3
Kinetic determination
Interaction kinetics are determined from the change in response as a function of time, as represented in the sensorgram.
Sensorgrams are recorded for a series of analyte concentrations and evaluated together as one data set, and a
mathematical model of the interaction is fitted to the experimental data.
The analyte concentrations may be injected in separate cycles with surface regeneration between the cycles (multi-cycle
analysis) or sequentially in a single cycle with no regeneration between injections (single-cycle analysis), as illustrated
below. In either case, a separate fitted curve is obtained for each analyte concentration.
In multi-cycle kinetics and affinity determinations, each sample is injected in a separate cycle.
The concentration series is presented as an overlay plot aligned at the start of the injection in
the evaluation software. Arrows in the illustration mark the start of sample injections.
In single-cycle determinations, the samples are injected sequentially in the same cycle.
Arrows in the illustration mark the start of sample injections.
-100 -50 0 50 100 150 200 250 300
25
20
15
10
5
0
-5
RU
S
Time
Response
Multi-cycle kinetics and affinity
RU
Response
Single-cycle kinetics and affinity
-200 0 200 400 600 800 1000 1200
S
Time
30
25
20
15
10
5
0
-5
-10
4
Affinity determination
For affinity determination, sensorgrams are recorded for a series of analyte concentrations, and the steady state affinity is
determined from a plot of the steady state response against concentration (usually referred to as R
eq
versus C). One single
curve is fitted to the whole data set. Like kinetics, affinity may be measured using single-cycle or multicycle format.
0 5e-8 1e-7 1.5e-7 2e-7 2.5e-7
Concentration
Response
100
90
80
70
60
50
40
30
20
5
Curve fitting principles
Introduction
Both kinetics and affinity are evaluated by fitting a mathematical model of the interaction to the experimental data. While
a close fit between the model and the data provides some confidence in the numerical results, obtaining a good fit is not
in itself evidence that the model describes the physical reality of the interaction. The fitting procedure does not have any
“knowledge” of the biological significance of parameters in the model equations, and it is wise always to examine the results
obtained for reasonableness of the values obtained. In addition, any mechanistic conclusions drawn for the interaction
from fitting results (e.g., concerning multiple interaction sites or conformational changes) should ideally be tested using
independent techniques.
Fitting procedure
Kinetic and affinity parameters are extracted from experimental data by an iterative process that finds the best fit for a set
of equations describing the interaction. The fitting process begins with initial values for the parameters in the equations,
and optimizes the parameter values according to an algorithm that minimizes the sum of the squared residuals.
The interaction equations may be created from a description of the interaction model or entered as mathematical
expressions. Equations for kinetic analysis are essentially differential equations (describing the rate of change of response
with time) that cannot be mathematically integrated.
6
Local and global parameters
Parameters in the fitting equations are treated as local variables, global variables, or constants as described in
the table to the right.
When the data set contains multiple curves, fitting can be performed with local or global parameter
settings. This applies primarily to kinetic analysis. Steady state affinity determination results in a single
curve for the data set, so that the local/global distinction is not relevant.
•
Local parameters are assigned independent values for each curve in the data set
•
Global parameters have the same value for all curves in the data set
Evaluating kinetics with global rate constants gives a more robust value for the rate constants, although
the curves may fit the experimental data more closely if all parameters are fitted locally. This is because
local fitting allows variation between the constants obtained from different curves: when the constants
are fitted globally, this variation appears in the closeness of fit rather than the reported values. Rate
constants are always global in predefined kinetic models.
In general, rate constants should be fitted as global parameters and bulk refractive index contribution
as a local parameter. The analyte binding capacity of the surface R
max
is normally set to global, but may
be evaluated as a local parameter if there is reason to believe that the surface capacity may vary
between cycles in serial mode or channels in parallel mode.
Parameter type Description
Local variables Assigned an independent value for each curve in the data series
(or sample injection in single-cycle kinetics).
Global variables Have one single value that applies to the whole data series.
Constants Have a fixed value that is not changed in the fitting procedure.
7
Statistical parameters
The closeness of fit between the experimental data and the fitted curve is formally
described by a set of statistical parameters, described in the table to the right.
Parameter Description
Chi-square A measure of the closeness of fit, calculated as the average squared residual (the difference between the
experimental data and the fitted curve).
where r
f
is the fitted value at a given point
r
x
is the experimental value at the same point
n is the number of data points
p is the number of fitted parameters
chi-square =
n – p
( r
f
– r
x
)
2
n
∑
1
Standard error (SE) A measure of the parameter significance, reported separately for each parameter.
The parameter significance indicates the extent to which a change in the parameter value affects the closeness
of fit. A parameter with low significance can have a wide range of values without affecting the fit.
T-value The parameter value divided by the standard error. This can make it easier to compare the significance of
parameters with widely differing absolute values.
Uniqueness (U-value) An estimate of the uniqueness of the calculated values for rate constants and R
max
. For correlated parameters,
the fitting procedure can determine their relative magnitudes but not absolute values. For example, knowing
the affinity gives the ratio but not the values for rate constants. The U-value is determined by testing the
dependence of the fit on correlated variations in pairs of parameters, and is reported as a single value for the
whole fitting. U-values above about 25 indicate that two or more of the parameters (rate constants and R
max
) are
correlated and the absolute values cannot be determined. If the U-value is below about 15 the parameter values
are not significantly correlated.
Note: Some Biacore systems do not report a U-value.
8
The mass transfer coefficient can be normalized for molecular weight and adjusted
approximately for the conversion of surface concentration to RU, to give a parameter
referred to as the mass transfer constant k
t
(units RU·M
-1
· s
-1
):
where G is the conversion factor from surface concentration to RU. The value of G is
approximately 10
9
for proteins on Sensor Chip CM5.
The mass transfer constant can be further modified to give the flow rate-independent
component (units RU · M
-1
s
-2/3
m
-1
), referred to as t
c
.
Different software versions may report different combinations of k
m
, k
t
and t
c
.
Fitting models for kinetics
Mass transfer in kinetic models
Analyte must reach the sensor surface in order to interact with the surface-bound ligand. Free analyte is depleted at the
surface by interaction with ligand, and is replenished by diffusion-driven transfer from the bulk solution. If transfer is slow
compared with binding of analyte to the ligand, the transport process will limit the observed binding rate, at least partially.
All kinetic models except 1:1 dissociation include a term for mass transfer of analyte, allowing rate constants to be
extracted from partially mass transfer limited data.
Mass transfer parameters
Mass transfer is described in terms of transfer of analyte (A) between bulk solution and the surface, with the same
rate constant in both directions. Only analyte at the surface can interact with ligand (B). As an example, the simple
1:1 interaction scheme may be represented as
The rate of mass transfer under the conditions of non-turbulent laminar flow that prevail in the flow cell is characterized
by the mass transfer coefficient k
m
(units m · s
-1
):
Parameter Description
D Diffusion coefficient of the analyte (m
2
· s-
1
)
f Volume flow rate of solution through the flow cell (m
3
· s-
1
)
h, w, l Flow cell dimensions (height, width, length in m)
A
bulk
B AB+A
surface
k
m
k
m
k
a
k
d
k
t
k
m
× MW × G
=
=
t
c
k
t
3
√
f
K
m
( )
0.98
1/3
=
D
2
•
f
0.3
•
h
2
•
w
•
l
9
1:1 binding
This is the simplest model for kinetic evaluation, and is recommended as default unless there is good experimental reason
to choose a different model. The model describes a 1:1 interaction at the surface:
A + B = AB
Default initial values for the 1:1 binding model are listed below.
Parameter Description Fit Value
ka Association rate constant (M
-1
s-1) Global 1e5
kd Dissociation rate constant (s
-1
) Global 1e-3
Rmax Analyte binding capacity of the surface (RU) Global Ymax
tc Flow rate-independent component of the mass transfer constant Global 1e8
RI Bulk refractive index contribution Constant 0
10
1:1 dissociation
This model fits the dissociation phase of the sensorgrams to an equation for exponential decay, representing dissociation
of a homogeneous 1:1 complex. The fitting is independent of analyte concentration. The equation includes an offset term
to allow for a nonzero residual response after completion of the dissociation.
Default initial values for the 1:1 dissociation model are listed below.
Note: This model cannot handle initial response changes resulting from bulk refractive index contributions. If the
sensorgrams show bulk contributions, remove data ranges so that the fitting starts after the bulk change is
complete.
Note: This model does not take account of the effect of mass transport limitations.
Note: Application of this model can be sensitive to initial parameter settings. If a good fit cannot be obtained with
apparently reasonable dissociation data, try adjusting the initial values to correspond more closely with
expected results.
Parameter Description Fit Value
kd Dissociation rate constant (s
-1
) Global 1e-3
offset Residual response above baseline after complete dissociation (RU) Local 0
R offsetR
0
e
–k
d
(t–t
0
)
= +
11
Bivalent analyte
This model describes the binding of a bivalent analyte to immobilized ligand, where one analyte molecule can bind to
one or two ligand molecules. The two analyte sites are assumed to be equivalent. The model may be relevant to studies
among others with signaling molecules binding to immobilized cell surface receptors (where dimerization of the receptor
is common) and to studies using intact antibodies binding to immobilized antigen. As a result of binding of one analyte
molecule to two ligand sites, the overall binding is strengthened compared with 1:1 binding. This effect is often referred to
as avidity.
A + B = AB
AB + B = ABB
Note: Once analyte is attached to the ligand through binding at the first site, interaction at the second site does not
contribute to the SPR response. For this reason, the association rate constant for the second interaction is reported
in units of RU
-1
s
-1
, and can only be obtained in M
-1
s
-1
if a reliable conversion factor between RU and M is available.
For the same reason, a value for the overall affinity or avidity constant is not reported.
Default initial values for the Bivalent analyte model are listed below.
Parameter Description Fit Value
ka1 Association rate constant for the first site (M
-1
s
-1
) Global 1e5
kd1 Dissociation rate constant for the first site (s
-1
) Global 1e-3
ka2 Association rate constant for the second site (RU
-1
s
-1
) Global 1e-3
kd2 Dissociation rate constant for the second site (s
-1
) Global 1e-3
Rmax Analyte binding capacity of the surface (RU) Global Ymax
tc Flow rate-independent component of the mass transfer constant Global 1e8
RI Bulk refractive index contribution Constant 0
12
Heterogeneous ligand
This model describes an interaction between one analyte and two independent ligands. The binding curve obtained is simply
the sum of the two independent reactions. The relative amounts of the two ligands does not have to be known in advance.
A + B1 = AB1
A + B2 = AB2
Note: The model is limited to two ligands because the fitting algorithm tends to become unstable with more components,
and three or more ligand species cannot be reliably resolved.
Default initial values for the Heterogeneous ligand model are listed below.
Parameter Description Fit Value
ka1 Association rate constant for the first ligand (M
-1
s
-1
) Global 1e5
kd1 Dissociation rate constant for the first ligand (s
-1
) Global 1e-3
ka2 Association rate constant for the second ligand (M
-1
s
-1
) Global 1e5
kd2 Dissociation rate constant for the second ligand (s
-1
) Global 1e-3
Rmax1 Analyte binding capacity of the first ligand (RU) Global Ymax
Rmax2 Analyte binding capacity of the second ligand (RU) Global Ymax
tc Flow rate-independent component of the mass transfer
constant
Global 1e8
RI Bulk refractive index contribution Constant 0
13
Two state reaction
This model describes a 1:1 binding of analyte to immobilized ligand followed by a conformational or other change that
stabilizes the complex. To keep the model simple, it is assumed that the changed complex can only dissociate through
reversing the conformational change:
A + B = AB = AB*
Note: Conformational changes in ligand or complex do not normally give a response in Biacore systems. A good fit
of experimental data to the two-state model should be taken as an indication that conformational properties
should be investigated using other techniques (e.g., spectroscopy or NMR), rather than direct evidence that a
conformational change is taking place.
Default initial values for the Two state reaction model are listed below.
Parameter Description Fit Value
ka1 Association rate constant for analyte binding (M
-1
s
-1
) Global 1e5
kd1 Dissociation rate constant for the complex (s
-1
) Global 1e-2
ka2 Forward rate constant for the stabilizing change (s
-1
) Global 1e-3
kd2 Reverse rate constant for the stabilizing change (s
-1
) Global 1e-3
Rmax Analyte binding capacity of the surface (RU) Global Ymax
tc Flow rate-independent component of the mass transfer
constant
Global 1e8
RI Bulk refractive index contribution Constant 0
14
Fitting models for affinity
Steady state affinity
This model calculates the equilibrium dissociation constant K
D
for a 1:1 interaction from a plot of steady state binding levels
(R
eq
) against analyte concentration (C). The equation includes an offset term which represents the response at zero analyte
concentration.
Note: Reported K
D
values that are higher than half the highest analyte concentration used should be treated with caution.
If the response against concentration plot does not flatten out sufficiently because the concentrations are not high
enough in relation to the K
D
value, the reported value may be unreliable.
Default initial values for the Steady state affinity model are listed below.
Parameter Description Fit Value
Rmax Analyte binding capacity of the surface (RU) Global Ymax
offset Response at zero analyte concentration Global Ymax/5
R
eq
=
+ offset
CR
max
K
D
+ C
15
Steady state affinity (constant Rmax)
This model uses the same equation as the simple steady state affinity model, but sets the R
max
parameter
to a constant. The value for R
max
is obtained for each analyte from a value entered for a control analyte
and the relative molecular weights of control and sample (see the Application guide Fragment and small
molecule screening with Biacore™ systems for more details. The value may be adjusted for assay drift
using repeated control samples.
Steady state affinity (constant Rmax and multi-site)
This model fits data from interactions that exhibit binding to multiple sites. Two sites are accommodated
in the model.
The model uses a constant R
max
value for one site, defining the expected stoichiometry, and a fitted value
for the other site, which can give an apparent value with undefined stoichiometry.
1
The default initial value is set to Constant = Ymax if no input value for R
max
has been provided.
Default initial values for the Steady state affinity (constant Rmax and multi-site) model are listed below.
Parameter Description Fit Value
KD Equilibrium dissociation constant for the main (strong) binding (M) Global Xmax
KD2 Equilibrium dissociation constant for the secondary (weak) binding (M) Global 100*Xmax
Rmax Analyte binding capacity for the main binding (RU) Constant (Input)
1
Rmax2 Analyte binding capacity for the secondary binding (RU) Global Ymax
offset Response at zero analyte concentration Global Ymax/5
R
max
R
max
analyte control
= ×
MW
analyte
MW
control
R
eq
=
+ offset
+
K
D1
+ C
CR
max1
K
D2
+ C
CR
max2
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