Thermodynamics, kinetics, density, and design

Advanced Multivalent Binding Model Calculator

Explore independent, sequential, cooperative, avidity, surface, and kinetic models. Calculate state probabilities, mass balance, apparent affinity, residence time, uncertainty, sweeps, fits, and optimized constructs.

1Model and preset

Choose the binding framework and starting scenario.

Identical, noninteracting sites with binomial state probabilities.
A different Kd controls every sequential step.
An empirical Hill slope controls response steepness.
Cumulative Adair constants define the binding polynomial.
One additional proximity-assisted engagement is modeled.
Valency, effective concentration, geometry, and density combine.
Surface density and mobility receive emphasis.
Association, dissociation, washout, and loss pathways are shown.

2Ligand and receptor

g/mol
Required for mass units

3Affinity, kinetics, and cooperativity

M⁻¹s⁻¹
s⁻¹
°C
kJ/mol

4Effective concentration and geometry

nm
nm
nm
nm
nm
nm

5Surface density and mobility

µm⁻²
µm⁻²
µm²
µm⁻²

6Competition and nonspecific binding

7Kinetic simulation

s
s
s⁻¹
s⁻¹
s⁻¹

8Curves, sweeps, and uncertainty

%
%
%

9Experimental fitting

One concentration and response pair per line.

10Design optimization

µm⁻²
µm⁻²
nm
nm
Results appear above the form after submission.

Formula used

Independent sites

θ = [L] / (Kd + [L])
P(i) = C(n,i) θi(1 − θ)n−i

Independent sites share one affinity and do not communicate. State probabilities follow a binomial distribution. The mean bond count is probability weighted.

Binding polynomial

Q = 1 + β₁[L] + β₂[L]² + ··· + βₙ[L]ⁿ
P(i) = βᵢ[L]ⁱ / Q

Sequential Kd values produce cumulative association constants. Adair mode accepts those cumulative constants directly. Every state weight is normalized by the binding polynomial.

Hill model

θ = [L]h / (Kdh + [L]h)

The Hill coefficient controls empirical steepness. It does not necessarily equal the physical site count. Mechanistic interpretation requires independent evidence.

Avidity approximation

B = 1 + (Ceff/Kd)·G·ω·S·I
Avidity factor = Bv−1
Kdapp = Kdcompetition / Avidity factor

G combines distance, linker, orientation, steric, activity, and accessibility factors. S represents density, contact area, and mobility. I is an intramolecular correction. This transparent expression is phenomenological, not universal.

Competition

Kdcompetition = Kd(1 + [C]/KdC)

This equation models simple competition at the same site. Multivalent competitors may require a larger state model. Competitor valency remains documented for extension.

Kinetics

kobs = kon,eff[L] + koff,eff
Association = θeq(1 − e−kobst)
Dissociation = Signal₀e−koff,efft

Rebinding reduces the effective dissociation rate. Internalization and degradation add irreversible losses. Full transport and state-transition kinetics need specialized models.

Free energy and mass balance

ΔG° = RT ln(Kd / 1 M)
[L]total = [L]free + [R]total⟨bonds⟩

Kd is converted to molar units before free-energy calculation. The mass-balance equation is solved by bisection. This matters near stoichiometric conditions.

How to use this calculator

  1. Select the simplest model compatible with available data.
  2. Enter total ligand and receptor concentrations with correct units.
  3. Set ligand and receptor valencies.
  4. Use a monovalent intrinsic Kd whenever possible.
  5. Configure effective concentration, linker reach, spacing, orientation, and sterics.
  6. Add receptor density, contact area, and mobility for surface systems.
  7. Enable a competitor only when it shares the relevant site.
  8. Use exact mass balance when receptor depletion can matter.
  9. Run uncertainty analysis with realistic coefficients of variation.
  10. Compare models and inspect warnings before interpretation.
  11. Fit experimental data as an empirical cross-check.
  12. Export CSV, JSON, or a print-ready PDF report.

Interpretation and limitations

Affinity and avidity

Affinity describes one interaction. Avidity reflects multiple engagements, rebinding, geometry, and presentation. Apparent Kd therefore depends on the selected model and assay context.

State probabilities

State zero is unbound. Higher states represent more simultaneous engagements. The full state is the maximum engagement allowed by the smaller valency.

Superselectivity

The density selectivity alpha is a local logarithmic slope. Values above one indicate sharper-than-proportional density response. The value changes across receptor-density ranges.

Uncertainty

Monte Carlo intervals include only selected parameter variation. They exclude unmodeled structural and measurement uncertainty. Use experimentally justified distributions.

Optimization

The optimizer searches bounded valency and linker grids. It rewards target binding and selectivity while mildly penalizing complexity. The result is exploratory rather than manufacturing guidance.

Frequently asked questions

Which Kd should be entered?

Enter a monovalent intrinsic Kd when available. Entering an avidity-enhanced Kd may double count multivalent effects.

Why is apparent Kd extremely small?

High effective concentration, favorable geometry, density, cooperativity, and valency multiply. Review each assumption and compare against experimental data.

Does Hill coefficient equal valency?

No. It is mainly an empirical slope parameter.

When is exact mass balance important?

Use it when receptor concentration is not negligible relative to ligand. It is especially relevant near stoichiometric conditions.

Can this fit SPR or BLI sensorgrams?

It provides a compact kinetic approximation and an equilibrium Hill fit. Full sensorgram analysis may require transport, heterogeneity, and rebinding models.

Can heteromultivalent systems be modeled?

The current state engine uses one effective receptor class. Add receptor-specific state weights and affinity matrices for a fully heterogeneous system.

Is the optimizer definitive?

No. It is a bounded exploratory search based on the selected phenomenological equations.

Can results be saved as PDF?

Use Print or PDF, then select the browser save-as-PDF destination.

Is this suitable for clinical decisions?

No. Validate equations, parameters, and numerical behavior before regulated, diagnostic, or clinical use.

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Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.