The Double Nucleation Model for Sickle Cell Haemoglobin Polymerization: Full Integration and Comparison with Experimental Data

Abstract

Sickle cell haemoglobin (HbS) polymerization reduces erythrocyte deformability, causing deleterous vaso-occlusions. The double-nucleation model states that polymers grow from HbS aggregates, the nuclei, (i) in solution (homogeneous nucleation), (ii) onto existing polymers (heterogeneous nucleation). When linearized at initial HbS concentration, this model predicts early polymerization and its characteristic delay-time (Ferrone et al. J Mol Biol 183(4):591–610, 611–631, 1985). Addressing its relevance for describing complete polymerization, we constructed the full, non-linearized model (Simulink^®, The MathWorks). Here, we compare the simulated outputs to experimental progress curves (n = 6–8 different [HbS], 3–6 mM range, from Ferrone’s group). Within 10% from start, average root mean square (rms) deviation between simulated and experimental curves is 0.04 ± 0.01 (25°C, n = 8; mean ± standard error). Conversely, for complete progress curves, averaged rms is 0.48 ± 0.04. This figure is improved to 0.13 ± 0.01 by adjusting heterogeneous pathway parameters (p < 0.01): the nucleus stability (σ₂ μ _cc : + 40%), and the fraction of polymer surface available for nucleation (ϕ), from 5e⁻⁷, (3 mM) to 13 (6 mM). Similar results are obtained at 37°C. We conclude that the physico-chemical description of heterogeneous nucleation warrants refinements in order to capture the whole HbS polymerization process.

Appendix

1.1 The Double Nucleation Model

As mentioned in Sect. 2, the core equations of the double nucleation model of HbS polymerization relate to the three main kinetic equations involving monomers and polymers:

• removal of free HbS monomers from the solution by the elongation of polymers (or “fibers endings”; see Eaton and Hofrichter 1990); this corresponds to core equation 1,

• production of polymers by the two nucleation pathways; this corresponds to core equation 2, which is actually sub-described into (2a), for homogeneous pathway, and (2b) or (2c) for heterogeneous pathway (depending on j*),

• conservation of total concentration of HbS monomer; this corresponds to core equation 3.

1. Removal of HbS monomers from the solution by the elongation (core equation 1):

$$ \frac{-dC}{dt}=k_+ \cdot(\gamma\cdot C -\gamma_S \cdot C_S)\cdot C_p $$

(1)

Along time-wise integration, this equation yields C; the process stops at solubility, C _s .

2. Generation of polymer fibers (elongating fibers endings; core Eq. 2) by homogeneous (2a) and heterogeneous (2b) pathways:

$$ \frac{dC_{p}}{dt}=\nu_{hon}+\nu_{hen} $$

(2)

with

$$ \nu_{hon}=\left(\frac{k_+\cdot K_{i^\ast}}{\gamma_{i^\ast +1}} \right) \cdot \left(\gamma\cdot C \right)^{i^\ast + 1} $$

(2a)

(v _hon , homogeneous pathway, i.e. addition of a monomer to the homogeneous nucleus, in solution) where,

• i ^* represents the size of the critical nucleus (on a monomer basis)

  $$ i^\ast=\frac{-(4\cdot R\cdot T+\delta_1\cdot\mu_{pc})}{R\cdot T\cdot \hbox{ln} (S)} $$

  For brevity, equations for activity coefficients (“gamma’s”) are omitted here; they are detailed in Ferrone et al. 1985b.

• S, supersaturation  = γ·C/ γ _S ·C _S

• δ₁ and δ₂ are parameters describing the fraction of contacts, as normally present in the “infinite” polymer, that are actually formed in an homogeneous aggregate of size “i*” δ₁ = 1.29, δ₂ = 0.84 (see Ferrone et al. 1985b)

• $K_{i^\ast}$, apparent equilibrium constant for the aggregation of “i*” monomers into the critical homogeneous nucleus (F. Ferrone, unpublished: modified form of Eq. A3.13 in Ferrone et al. 1985b):

  $$ \hbox{ln}\left(K_{i^{\ast}}\right)=\xi\cdot\hbox{ln} \left(\hbox{ln}\left(S\right)\right) - \frac{\xi}{\hbox{ln}\left(S\right)} \cdot \hbox{ln}\left(\gamma\cdot C\right) +\xi\cdot\left(1-\hbox{ln}\left(\xi\right)\right)+\hbox{ln}\left(\gamma_s\cdot C_s\right)+\hbox{ln}\left(\frac{\sqrt{8}}{\rho}\right)+\left(\delta_2-1\right) \frac{\mu_{pc}}{R\cdot T} $$

1.1.1 Calculation of the Heterogeneous Pathway Rate

For the calculation of v _hen , rate of addition of a monomer to the heterogeneous nucleus (onto existing polymers), two cases are distinguished, depending on j ^*, the actual size of the heterogeneous nucleus, as compared to the j _max limit (j _max = −σ₂μ_cc/σ₁μ_cc), corresponding to the size beyond which its attachment area to the polymer (thus the chemical interaction potential) does not increase any more (Ferrone et al. 1985b):

- case 1 : $j^\ast < j_{max}$

$$ j^\ast=\frac{\xi_2}{\hbox{ln}(S)+\xi_1} $$

with

$$ \xi_1= \frac {-\sigma_1\cdot \mu_{cc}}{R\cdot T},\;\xi=-(4+\frac{\delta_1 \mu_{pc}}{R\cdot T}),\;\xi_2=\xi+4+\frac{\sigma_2 \mu_{cc}}{R\cdot T} $$

in which case,

$$ \nu_{hen}=k_{+}\cdot\left[\frac{\ln(S+\xi_{1})}{\xi_{2}}\right]^{\xi_2}\cdot e^{\xi_2}\cdot\gamma\cdot C\cdot\Gamma\cdot C_{poly} $$

(2b)

- case 2: $j^{\ast}\ge j_{max}$

$$ j^{\ast}=i^{\ast}+(1/ln\,S) $$

in that case,

$$ \nu_{hen}=k_{+}\cdot C_{j^{\ast}}\cdot(\gamma\cdot C) \cdot\Gamma $$

(2c)

with

$$ \ln(C_{j^\ast})= \ln (\Phi\cdot(C_{0}-C))+j^{\ast}\cdot\ln(S)+\frac{\delta_1\cdot\mu_{pc}+3\cdot R\cdot T }{R\cdot T}\cdot \ln(j^\ast)+\frac{\delta_2\cdot\mu_{pc}}{R\cdot T}-\frac{\mu_{jCmax}}{R\cdot T}-3\cdot\ln(j_{max}) $$

μ _jCmax , the chemical potential of the critical heterogeneous nucleus (F. Ferrone, unpublished correction of a typographical error of Eq. A3.22, in Ferrone et al. 1985b), is

$$ \mu_{jCmax}=\frac{-\sigma_2 \mu_{cc}}{\ln (\frac{1}{j_{max}})+1} $$

3. Concentration of polymerized monomers, calculated from total HbS conservation (core equation 3):

$$ C_{poly}= C_{o}-C $$

(3)

Model variables and units are presented in Sect. 2. Main HbS-related constants used:

• V, monomer specific volume = 0.0497 l/mM (0.767 cm³/g)

• C _pp , monomer concentration in polymer phase = 10.8 mM/l (0.69 g/cm³)

• ρ, monomer density, relative to polymer = 0.537 (C _PP  × V)