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Binding Models and Theory
This page documents the mathematical foundation and thermodynamic models used in BEAST. Each model represents a different chemical equilibrium scenario commonly encountered in NMR titration studies of host-guest systems.
For complete mathematical derivations, see Model Proofs section below the general notes.
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$[H_0]$ : Total host concentration (mol/L) -
$[G_0]$ : Total guest concentration (mol/L) -
$[H]$ : Free (unbound) host concentration -
$[G]$ : Free (unbound) guest concentration -
$[HG]$ ,$[HG_2]$ ,$[H_2G]$ ,$[H_2]$ : Complex concentrations
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$K_{HG}$ : Association constant for H + G ⇌ HG (M⁻¹) -
$K_{HG_2}$ or$K_d$ : Association constant for HG + G ⇌ HG₂ (M⁻¹) -
$K_{H_2G}$ : Association constant for H + HG ⇌ H₂G (M⁻¹) -
$K_{H_2}$ or$K_d$ : Dimerization constant for 2H ⇌ H₂ (M⁻¹)
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$\delta_{obs}$ : Observed chemical shift (Hz or ppm) -
$\delta_{free}$ or$\delta_G$ : Chemical shift of free guest (reference = 0) -
$\Delta\delta$ : Chemical shift change relative to free guest -
$d_{inf}$ ,$d_{HG}$ ,$d_{H_2G}$ : Limiting chemical shifts of bound species
- All models assume guest-observed NMR (monitoring guest signal)
- Chemical shifts of free guest serve as the reference point (Δδ = 0)
- Only species containing guest molecules contribute to the observed shift
- Fast exchange on the NMR timescale is assumed (population-weighted average)
The simplest binding scenario where one host molecule binds to one guest molecule to form a 1:1 complex. This is the most common binding mode in supramolecular chemistry.
Mass balance equations:
Quadratic equation solution:
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$K_{HG}$ : Binding constant (M⁻¹) -
$d_{inf}$ : Limiting chemical shift of bound guest (Hz or ppm)
- Strong binding:
$K_{HG} > 10^5$ M⁻¹ - Moderate binding:
$10^3 < K_{HG} < 10^5$ M⁻¹ - Weak binding:
$K_{HG} < 10^3$ M⁻¹
Sequential binding of two guests to one host molecule. The first guest binds with constant
Cubic equation for free guest concentration [G]:
Where:
$a = K_{HG} \cdot K_{HG_2}$ $b = K_{HG}(2K_{HG_2}[H_0] - K_{HG_2}[G_0] + 1)$ $c = K_{HG}([H_0] - [G_0]) + 1$ $d = -[G_0]$
Solution method: Select the smallest positive real root.
Complex concentrations:
Note: Factor of 2 for
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$K_{HG}$ : First binding constant (M⁻¹) -
$K_{HG_2}$ : Second binding constant (M⁻¹) -
$d_{HG}$ : Chemical shift of guest in HG -
$d_{HG_2}$ : Chemical shift of guest in HG₂
- Positive cooperativity:
$K_{HG_2} > K_{HG}$ (second guest binds stronger) - Negative cooperativity:
$K_{HG_2} < K_{HG}$ (second guest binds weaker) - Non-cooperative:
$K_{HG_2} \approx K_{HG}$
Sequential binding of two hosts to one guest molecule. First, HG forms with constant
Cubic equation for free host concentration [H]:
Where:
$a = K_{HG} \cdot K_{H_2G}$ $b = K_{HG}(2K_{H_2G}[G_0] - K_{H_2G}[H_0] + 1)$ $c = K_{HG}([G_0] - [H_0]) + 1$ $d = -[H_0]$
Solution method: Select the smallest positive real root.
Complex concentrations:
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$K_{HG}$ : First binding constant (M⁻¹) -
$K_{H_2G}$ : Second binding constant (M⁻¹) -
$d_{HG}$ : Chemical shift of guest in HG -
$d_{H_2G}$ : Chemical shift of guest in H₂G
This model is common when:
- Host molecules can aggregate around the guest
- Guest acts as a template for host assembly
- Sandwich complexes form in supramolecular systems
Competitive equilibria where host-guest binding occurs simultaneously with host self-association (dimerization). The host dimer (H₂) does not bind guest in this model.
Cubic equation for free host concentration [H]:
Where:
$a = 2K_{HG}K_{H_2}$ $b = K_{HG} + 2K_{H_2}$ $c = K_{HG}([G_0] - [H_0]) + 1$ $d = -[H_0]$
Solution method: Select the smallest positive real root.
Free guest concentration:
Complex concentrations:
Note: H₂ does not contain guest, so it doesn't contribute to guest-observed shift.
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$K_{HG}$ : Host-guest binding constant (M⁻¹) -
$K_{H_2}$ : Host dimerization constant (M⁻¹) -
$d_{HG}$ : Chemical shift of guest in HG complex
This model applies when:
- Host molecules tend to self-associate in solution
- Dimerization competes with guest binding
- Higher host concentrations reduce apparent binding affinity
- Common in systems with π-π stacking or hydrogen bonding hosts
Complex system with three competing equilibria: 1:1 binding, host dimerization, and 2:1 complex formation. This is the most general model and can describe systems where multiple processes occur simultaneously.
Iterative numerical solution required due to quartic polynomial complexity.
Mass balance equations:
Iterative scheme:
- Initial guess:
$[H]_0$ ,$[G]_0$ - Calculate complexes:
$[HG] = K_{HG}[H][G]$ $[H_2] = K_{H_2}[H]^2$ $[H_2G] = K_{H_2G}K_{HG}[H]^2[G]$
- Update free concentrations from mass balance
- Repeat until convergence (typically
$|\Delta[H]| < 10^{-6}$ )
For guest-observed NMR, only HG and H₂G contribute:
Note: H₂ doesn't contribute (no guest molecule).
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$K_{HG}$ : Host-guest binding constant (M⁻¹) -
$K_{H_2}$ : Host dimerization constant (M⁻¹) -
$K_{H_2G}$ : Second host binding constant (M⁻¹) -
$d_{HG}$ : Chemical shift of guest in HG -
$d_{H_2G}$ : Chemical shift of guest in H₂G
The overall formation of H₂G can occur via:
Sequential pathway:
This model is particularly relevant when:
- Host aggregation is significant
- Multiple binding modes coexist
- Concentration-dependent behavior is observed
- Simple models fail to fit the data adequately
| Model | Use When |
|---|---|
| 1:1 (HG) | Simple binding curve, clear saturation, no evidence of higher-order complexes |
| 1:2 (HG + HG₂) | Two distinct binding regions, biphasic curve, guest in excess |
| 2:1 (HG + H₂G) | Binding increases with host concentration, evidence of sandwich complexes |
| Dimer (HG + H₂) | Non-linear behavior at high host concentration, aggregation-prone hosts |
| Multi (HG + H₂ + H₂G) | Complex behavior, multiple equilibria evident, simple models fail |
BEAST automatically fits all models and ranks them using:
- AIC (Akaike Information Criterion) - Penalizes complexity
- BIC (Bayesian Information Criterion) - More stringent penalty
- R² - Goodness of fit
- Residual diagnostics - Systematic patterns indicate wrong model
Lower AIC/BIC = Better model
When solving cubic equations for [G] or [H]:
- Calculate all roots (may be complex)
- Filter for real roots
- Filter for positive roots
- Select smallest positive root within physical bounds:
$0 < [G] \leq [G_0]$ $0 < [H] \leq [H_0]$
If no valid root is found (rare, usually during initial guess optimization):
- Set
$[H] \approx 10^{-10}$ M or$[G] \approx 10^{-10}$ M - Prevents division by zero
- Allows optimization to continue toward valid parameter space
Iterative solution converges when:
Maximum iterations: 100,000 (typically converges in < 100)
The observed chemical shift is the mole-fraction-weighted average of all species:
Chemical shift changes are reported relative to free guest:
Since
This explains why:
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$\Delta\delta = 0$ when no binding occurs -
$\Delta\delta$ increases toward$d_{inf}$ at saturation - Units are Hz or ppm depending on input data
- Concentrations: mol/L (M)
- Chemical shifts: Hz or ppm (consistent with input)
- Association constants: M⁻¹
- Fast exchange on NMR timescale (averaged signal)
- Guest-observed NMR (monitoring guest nucleus)
- No guest aggregation (only host-guest complexes)
- Closed system (no precipitation or phase separation)
- Temperature-independent K during measurement
- Cannot distinguish between different binding geometries with same stoichiometry
- Requires sufficient data points (≥8-10) for reliable fitting
- Multiple models may fit data equally well (use AIC/BIC for selection)
- Extreme affinity (
$K > 10^8$ M⁻¹) may be difficult to measure accurately
This section provides detailed mathematical derivations for the 1:1, 1:2, and 2:1 host-guest binding models used in BEAST (Binding Evaluation and Analysis Software Tool).
The simplest binding model describes the formation of a 1:1 complex between host (H) and guest (G):
Equilibrium constant:
Total host concentration:
Total guest concentration:
From the mass balance equations:
Substituting into the equilibrium expression:
Rearranging:
Expanding:
Rearranging into standard quadratic form:
Dividing by
Quadratic formula solution:
This model describes sequential binding of two guests to one host:
Equilibrium constants:
Total host concentration:
Total guest concentration:
From the equilibrium expressions:
From host mass balance:
Therefore:
Substituting back:
From guest mass balance:
Multiplying through by the denominator:
Rearranging:
Collecting terms:
Cubic equation in [G] (free guest concentration):
Where:
$a = K_{HG} K_{HG_2}$ $b = K_{HG}(2K_{HG_2}[H_0] - K_{HG_2}[G_0] + 1)$ $c = K_{HG}([H_0] - [G_0]) + 1$ $d = -[G_0]$
This model describes sequential binding of two hosts to one guest:
Equilibrium constants:
Total host concentration:
Total guest concentration:
From the equilibrium expressions:
From guest mass balance:
Therefore:
Substituting back:
From host mass balance:
Multiplying through by the denominator:
Rearranging:
Collecting terms:
Cubic equation in [H] (free host concentration):
Where:
$a = K_{HG} K_{H_2G}$ $b = K_{HG}(2K_{H_2G}[G_0] - K_{H_2G}[H_0] + 1)$ $c = K_{HG}([G_0] - [H_0]) + 1$ $d = -[H_0]$
This model describes host-guest binding in competition with host dimerization:
Equilibrium constants:
Total host concentration:
Total guest concentration:
From the equilibrium expressions:
From guest mass balance:
Therefore:
Substituting back:
From host mass balance:
Multiplying through by
Expanding:
Rearranging:
Cubic equation in [H] (free host concentration):
Where:
$a = 2K_{HG}K_{H_2}$ $b = K_{HG} + 2K_{H_2}$ $c = K_{HG}([G_0] - [H_0]) + 1$ $d = -[H_0]$
This model describes a complex system with three competing equilibria:
Equilibrium constants:
Total host concentration:
Total guest concentration:
From the equilibrium expressions:
From guest mass balance:
Therefore:
Substituting back:
From host mass balance:
Multiplying through by the denominator:
Expanding and collecting terms:
Rearranging into polynomial form:
This leads to a quartic polynomial equation in [H] that typically requires iterative numerical solution.
Due to the complexity of the algebraic solution, this system is typically solved iteratively:
- Initial guess: Start with reasonable estimates for [H] and [G]
- Calculate complexes: Use equilibrium expressions to find [HG], [H₂], [H₂G]
- Update free concentrations: Use mass balance equations
- Check convergence: Repeat until changes are below tolerance
- Calculate chemical shift: Use final concentrations
For guest-observed NMR, only species containing guest molecules contribute:
The chemical shift change relative to free guest:
This model describes systems where:
- Host-guest binding competes with host dimerization
- Host dimers can also bind guest molecules
- The system exhibits complex behavior depending on relative binding constants
- Higher host concentrations can favor either HG or H₂G formation depending on the equilibrium constants
- This model is particularly relevant for systems where host aggregation is significant
The overall equilibrium for H₂G formation can be viewed as:
This can occur through two pathways:
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Sequential pathway:
$H + G \rightleftharpoons HG$ , then$H + HG \rightleftharpoons H_2G$ -
Alternative pathway:
$2H \rightleftharpoons H_2$ , then$H_2 + G \rightleftharpoons H_2G$ (not present in this model)
The overall constant through the sequential pathway:
This relationship helps in understanding the relative importance of the different pathways to complex formation. The formation of H₂G is directly dependent on the availability of both free host and the HG complex.
The observed chemical shift in NMR is the population-weighted average of all species:
For guest-observed experiments:
Chemical shift changes are calculated relative to the free state:
Since
- Guest molecules:
$[G_{\text{free}}]$ and$[G_{\text{in HG}}] = [HG]$ $\Delta\delta = \frac{\Delta\delta_{\text{inf}} \cdot [HG]}{[G_0]}$
- Guest molecules:
$[G_{\text{free}}]$ ,$[G_{\text{in HG}}] = [HG]$ , and$[G_{\text{in HG}_2}] = 2[HG_2]$ $\Delta\delta = \frac{\Delta\delta_{\text{inf1}} \cdot [HG] + \Delta\delta_{\text{inf2}} \cdot 2[HG_2]}{[G_0]}$
- Guest molecules:
$[G_{\text{free}}]$ ,$[G_{\text{in HG}}] = [HG]$ , and$[G_{\text{in H}_2\text{G}}] = [H_2G]$ $\Delta\delta = \frac{\Delta\delta_{\text{inf1}} \cdot [HG] + \Delta\delta_{\text{inf2}} \cdot [H_2G]}{[G_0]}$
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Fielding, L. Determination of association constants (Ka) from solution NMR data. Tetrahedron 2000, 56, 6151-6170. DOI
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Hirose, K. A practical guide for the determination of binding constants. J. Incl. Phenom. Macrocycl. Chem. 2001, 39, 193-209. DOI
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Connors, K.A. Binding Constants: The Measurement of Molecular Complex Stability. John Wiley & Sons: New York, 1987. DOI
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Thordarson, P. Determining association constants from titration experiments in supramolecular chemistry. Chem. Soc. 2011, 40. DOI
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Ulatowski, F.; et. al. Recognizing the Limited Applicability of Job Plots in Studying Host–Guest Interactions in Supramolecular Chemistry. J. Org. Chem. 2016, 81, 1746-1756. DOI
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