Periodic Nucleation of Calcium Phosphate in a Stirred Biocatalytic Reaction

Abstract Highly ordered superstructures composed of inorganic nanoparticles appear in natural and synthetic systems, however the mechanisms of non‐equilibrium self‐organization that may be involved are still poorly understood. Herein, we performed a kinetic investigation of the precipitation of calcium phosphate using a process widely found in microorganisms: the hydrolysis of urea by enzyme urease. With high initial ratio of calcium ion to phosphate, periodic precipitation was obtained accompanied by pH oscillations in a well‐stirred, closed reactor. We propose that an internal pH‐regulated change in the concentration of phosphate ion is the driving force for periodicity. A simple model involving the biocatalytic reaction network coupled with burst nucleation of nanoparticles above a critical supersaturation reproduced key features of the experiments. These findings may provide insight to the self‐organization of nanoparticles in biomineralization and improve design strategies of biomaterials for medical applications.

UV-vis spectrophotometer (VWR, UV3100-PC) in experiments with lower amounts of calcium ion where the absorbance is expected to be proportional to the amount of precipitate ( Figure S1).
The precipitate was filtered and rinsed with doubly deionized water. After air drying, the powder was dispersed on carbon conductive adhesive tape, and coated with platinum and gold alloy. Raman spectroscopic studies were performed on a Thermo Scientific TM DXR TM Raman microscope using a green laser (λ = 532 nm), operating at 5 mW laser power. For each sample, 30 spectra were averaged with an exposure time of two seconds. Scanning electron microscopic (SEM) images were recorded by a Hitachi S-4700 field emission scanning electron microscope, operating at 10 kV acceleration voltage. Brightfield optical microscopy images were obtained using a Leica TCS SP8 Confocal Microscope.
The SEM of samples collected with low urea and low [Ca 2+ ] indicate formation of roughly spherical particles or aggregates less than 100 nm in size ( Figure S2). The precipitate formed did not react with HCl and the Raman spectra indicated that it contained entirely calcium phosphate; there was no evidence of calcite under these conditions. There were typical peaks associated with PO4 vibrational modes at 420 cm -1 and 580 cm -1 ; the broad peak at 952 cm -1 ( Figure S3) corresponds to the symmetric stretching mode (ν1) of the tetrahedral PO4 group and is indicative of amorphous calcium phosphate (ACP) or poorly crystalline hydroxyapatite (Ca10(PO4)6(OH)2, HAP). [1] The absence of signals at 1103 cm −1 and 1073 cm −1 suggests there is no incorporation of carbonate ions in the samples. [2] With higher [Ca 2+ ] and [urea], the precipitate reacted with HCl and the Raman spectra ( Figure S4) showed that it contained a mixture of calcium phosphate with PO4 peaks at 420, 580 and 952 cm -1 and calcium carbonate with CO3 peaks at 712, 1086, 1437 and 1747 cm -1 . The characteristic strong peak at 1086 cm -1 is indicative of calcite. The SEM contained a mixture of submicron particles and micron scale crystals ( Figure S5).

Model
The system involves enzyme urease in phosphate with substrate (urea), calcium chloride and acid under well-stirred, closed batch conditions. The main processes are (3.1) the enzyme catalysed reaction, (3.2) the equilibria that govern the pH and (3.3) the precipitation.

Enzyme-catalysed reaction
The enzyme catalysed hydrolysis of urea, U, yields ammonia and carbon dioxide: where the rate of the enzyme catalysed reaction is given by:

Equilibria
The pH is determined by the following reversible reactions (for simplicity acid is included as H + rather than H3O + ): The desorption of gaseous CO2 or NH3 from the surrounding solution are not included here as the behaviour is qualitatively reproduced in sealed containers, nor H2CO3 as this species rapidly forms CO2.

Coupling to precipitation
The calcium precipitates with both the inorganic carbon and phosphorous. The precipitation of calcium phosphate is complicated involving many crystalline species such as hydroxyapatite (Ca10(PO4)6(OH)2) and tricalcium phosphate (Ca3(PO4)2 where {} denotes activities: {} = γici where ci is the concentration of the i th species and γ is the activity coefficient that depends on ionic strength: I = 0.5∑cizi 2 . Precipitation involves nucleation, growth, aggregation/breakage and Ostwald ripening. We only consider nucleation here as the experiments suggest that nucleation occurs periodically. The steady state nucleation rate, J, in number of nuclei per unit volume per unit time, from classical nucleation theory is given by: Where A is the pre-exponential factor and B = Cmσ 3 ω 2 /(kT) 3 where Cm is a shape factor (for spheres Cm = 16π/3), ω is the molecular volume, σ is the surface tension of the nucleating surface in water, k is the Boltzmann constant and T is the temperature. To imitate burst nucleation of spheres of precipitate of a critical radius, rc then the rate of production of solid, in moles per unit volume of solution per unit time, was approximated by: where KJ = AVc/Vm with Vm is the molar volume; 1 is for ATCP and 2 is for CaCO3. The critical volume VC depends on the radius: rc = 2ωσ/(kTlnS) and hence KJ = KJ'/(lnS) 3 where KJ' = 4π/3*(2ωσ/kT) 3 . Inspired by the theory of LaMer, to delay the onset of nucleation the parameter p(S) was introduced as a probability factor which depends on supersaturation: The value of Scrit1 was given by Scrit1 = 1 + R1[Ca 2+ ]/x where x is the inorganic counterion and R1 is a constant. The probability of nucleation depends on the ratio of metal ion, A, to counterion, B. A step function or a Gaussian probability function were used in earlier work with maximum probability at [A] = [B] and an decreasing probability of nucleation when A/B increased. [3] These simulations also took incorporated the stochastic nature of precipitation. Here, the dependence of Scrit on the ratio [Ca 2+ ]/[PO4 3-] was intended to take into account the fact that precipitation was less likely as the concentration of phosphate was decreased for experiments where [Ca 2+ ] > [PO4 3-]. With [Ca 2+ ] < [PO4 3-], all available calcium was consumed as soon as precipitation commenced; this scenario was not explored in simulations.
The nucleation terminated when S ≤ Scrit2 = 1 + R2[Ca 2+ ]/x and then p = 0. This is different from the LaMer model, in which precipitation initiated and terminated above a single threshold. We found that oscillations occurred for 1 < Scrit2 < Scrit1; a better fit to experimental data was obtained using the same dependence of Scrit2 on the ratio [Ca 2+ ]/[PO4 3-] as Scrit1 but with R2 << R1.

Equations and parameters
There are 15 chemical species in reactions 1 -6 (not including water). This results in 15 coupled odes. The values of all the rate constants and parameters taken in this work are shown in Table 1.
The equilibria rate constants are well established [4] , but the enzyme constants vary depending on the source. The enzyme constants are in line with earlier work [5] . The enzyme in the ode file is given in units/ml to more easily compare to experiments and the maximum rate for ET = 19 unit/ml is vmax = ET*k1e = 19 units/ml*2 x 10 -5 M/units/ml/s = 3.8 x 10 -4 M/s (and k1 in s -1 = k1E /5.3 x 10 -8 M/units/ml).
The ionic strength here ranged from I = 0.05 -2 M and can be assumed to be constant as it was dominated by large initial concentrations of ions from CaCl2. The activity coefficients can be calculated from the Pitzer equation for high ionic strength, however the latter includes empirical parameters that were not evaluated in the complex mixtures used here. [6] It is expected that the inclusion of activity coefficients will not impact the qualitative results and so we used concentrations in the simulations, rather than activities. The aim of the simulations was to help explain the dynamic behaviour, rather than reproduce the results quantitatively.