Expanding the stoichiometric window for metal cross-linked gel assembly using competition

Significance

This paper reports how metal-coordination cross-linking can bypass traditional stoichiometric limits for macromolecular material assembly. Specifically, we use metal-coordinated hydrogels to demonstrate how to prevent excess cross-linker from dissolving macromolecular networks by using hydroxide ion competition to buffer excess metal ions and thereby protect the network. Additionally we use simulations to develop a thermodynamic framework that predicts the coupled dynamic equilibria conditions that result in this competition-induced network protection effect. This is a critical innovation, not only because it demonstrates how to overcome classical limits on macromolecular material cross-link stoichiometry, but also because it reveals an intrinsic robustness in the self-assembly of metal-coordinate networks, shedding additional light on the emerging prevalence of metal coordination in biological materials.

Abstract

Polymer networks with dynamic cross-links have generated widespread interest as tunable and responsive viscoelastic materials. However, narrow stoichiometric limits in cross-link compositions are typically imposed in the assembly of these materials to prevent excess free cross-linker from dissolving the resulting polymer networks. Here we demonstrate how the presence of molecular competition allows for vast expansion of the previously limited range of cross-linker concentrations that result in robust network assembly. Specifically, we use metal-coordinate cross-linked gels to verify that stoichiometric excessive metal ion cross-linker concentrations can still result in robust gelation when in the presence of free ion competing ligands, and we offer a theoretical framework to describe the coupled dynamic equilibria that result in this effect. We believe the insights presented here can be generally applied to advance engineering of the broadening class of polymer materials with dynamic cross-links.

Recently, the engineering of macromolecular materials has increasingly focused on the incorporation of nonpermanent (transient) cross-links, due to their demonstrated ability to improve both material toughness and stimuli responsiveness (1). A subclass of transient cross-links, metal coordination, has generated particular interest partially due to a growing understanding of its functional role in the extraordinary mechanical properties of mussel holdfast threads (2–7). The protective coating of mussel threads consists of a protein rich in 3,4-dihydroxyphenyl-l-alanine, which strongly binds metal ions such as Fe³⁺ and Al³⁺ in dynamic transient cross-links. Similarly, the histidine-modified collagen-like proteins that make up the core of the threads coordinate with divalent transition metal ions such as Cu²⁺ and Zn²⁺. This metal-coordinate material chemistry has been shown to be critical for the ability of the mussel thread to both dissipate energy from crashing waves and recover mechanical properties after failure (2, 3). Mussel threads contain metal ions in concentrations elevated up to 10⁵ times the amount present in seawater, and mussels are known to be remarkably opportunistic in regard to which metals they incorporate into their threads, plausibly due to the similar effect on mechanical properties of macromolecular materials when cross-linked with certain transition metal-coordinate complexes (6–9). Surprisingly, the mussel also appears to be opportunistic in the amount of metal incorporated in the threads, and here we present evidence that could explain why (7, 10). Grounded in the physical chemistry of metal-coordinate equilibrium dynamics, we demonstrate that robust material cross-linking is thermodynamically favored as long as a minimum amount of metal is available during macromolecular self-assembly. This is a critical finding, not only in the ongoing effort to uncover the mysteries of the molecular assembly of mussel fibers, but also as a possible trick to be utilized in future bio-inspired material manufacturing.

Results and Discussion

Most studies of mussel-inspired metal-coordinated hydrogels have relied on hydrophilic star polymers, typically poly(ethylene glycol) (PEG), end functionalized with a metal-coordinating group. Precise control over the self-assembled state of the network has been demonstrated by using the pH of the hydrogel to tune the number of ligands bound to a metal ion from 1 at low pH (L₁M), 2 at medium pH (L₂M), and 3 at high pH (L₃M) (11). More recently, orthogonal control of the metal–ligand complex cross-linking was demonstrated by using the metal-to-ligand stoichiometry to direct the formation of complexes, ultimately driving the coordination toward L₁M and material disassembly upon increasing metal concentration as shown in Fig. 1 A, i and iii, where a stoichiometrically balanced cross-linked network depercolates when excess cross-linker is added (12). This preference toward forming the L₁M complex is a consequence of the equilibrium constants for coordination following the trend, $K_{{\mathrm{L}}_1\mathrm{M}}>K_{{\mathrm{L}}_2\mathrm{M}}>K_{{\mathrm{L}}_3\mathrm{M}}$, as demonstrated in SI Appendix, Fig. S1 (equilibrium constants are defined as $K_{{\mathrm{L}}_{\mathrm{x}}\mathrm{M}}=\left[L_{x}M\right]/\left(\left[L_{x-1}M\right]\left[L\right]\right)$ and formation constants are defined as ${\beta }_{{\mathrm{L}}_{\mathrm{x}}\mathrm{M}}=\left[L_{x}M\right]/\left({\left[L\right]}^x\left[M\right]\right)$). The reported sensitivity of metal-coordinate network assembly to metal–ligand stoichiometry follows from classical Flory–Stockmayer network theory, which predicts that any stoichiometric imbalance in the direction of either too little or too much cross-linker results in suboptimal network connectivity (13).

Fig. 1.

Simulating the gelation conditions as a function of competitor strength. (A) A classic A₂ + B₃ (A, i) cross-linked network is broken up when (A, iii) excess B is present. The presence of a competitor C does not help increase connectivity when (A, ii) mixed species form, as the effective functionality of B is decreased. The competitor acts to (A, iv) expand the gelation conditions when the species are forced to partition, as a single cross-linker molecule reacts only with either the polymer or the competitor. As a function of competitor B–C strength, (B) the competitor acts to decrease the conditions that result in gel formation when mixed species form, but increases gelation conditions when the species are partitioned. The gelation conditions can be increased as a function of partitioned hydroxide competitor strength for (C) a metal-coordinated hydrogel. The magnitude and duration of the increase in gelation conditions depend on the stoichiometry of the competitor M(OH)_1–3. The lines represent the median simulation result and the shaded region contains 80% of the simulation data.

In general, in a network formed via an A₂ + B₃ cross-linking reaction (where A₂ is a bifunctional polymer and B₃ is a trifunctional cross-linker) a statistical distribution of cross-linked species is expected, as dictated by a reaction probability $p_{\mathrm{A}\mathrm{B}}$ and the stoichiometric ratio of complementary “A” and “B” functional groups (13). Hence, in the limit of excess cross-linker, inhibition of network formation would typically be expected, as shown in Fig. 1 A, iii and SI Appendix, Fig. S2A. However, the addition of a competitor, “C,” that competes for excess B functional groups can alter this outcome, depending on whether mixed cross-linker species will form in the system (14, 15). When mixed species can form, the potential network functionality of cross-linkers will be significantly reduced, as demonstrated in Fig. 1 A, ii. For such systems, great care is therefore taken to remove any contaminant that could behave as a competitor, C, for the cross-linking reaction, A–B, as the competitor would simply further inhibit network percolation (16). However, if a system is able to avoid the formation of mixed cross-linker species, by partitioning the cross-linker that reacts with the competitor from the cross-linker that reacts to form the network, competition can be beneficially utilized to expand the gelation conditions much further into the limiting regime of excess cross-linker concentration, as shown in Fig. 1 A, iv and SI Appendix, Fig. S2B (17).

To illustrate these opposite effects of cross-link competition on network formation, we used gel network percolation theory (18) to compute theoretical gel moduli as a function of competitor strength within a set phase space of cross-linker and competitor concentrations, including the limiting regime of excess cross-linker (see SI Appendix, Figs. S3–S5 and Materials and Methods for more details regarding the computational process). To quantify the likelihood of forming a gel in a given phase space of cross-linker and competitor concentrations, average gel moduli were calculated as a function of competitor strength and then normalized by the corresponding average gel moduli from an equivalent phase space absent competition to provide the metric “relative gelation conditions.” As demonstrated in Fig. 1B, under competitive conditions with mixed species formation (as illustrated in Fig. 1 A, ii), gelation conditions are simply predicted to remain unchanged until the strength of the competitor reaction, ${\beta }_{\mathrm{B}\mathrm{C}}$, begins to overpower the cross-linking reaction ${\beta }_{\mathrm{A}\mathrm{B}}$ (see SI Appendix, Fig. S3 and Movie S1 for more details). In contrast, under competitive conditions with formation of partitioned species (as illustrated in Fig. 1 A, iv), the relative gelation conditions are predicted to increase about 500% up until the strength of ${\beta }_{\mathrm{B}\mathrm{C}}$ competition begins to overcome the cross-linker strength ${\beta }_{\mathrm{A}\mathrm{B}}$ (see SI Appendix, Fig. S4 and Movie S2 for more details).

Next we studied whether this partitioning cross-link competition mechanism could also enhance the relative gelation conditions of mussel-inspired hydrogels by expanding the range of metal–ion concentrations that result in robust network assembly. A free competitor for the cross-linking metal is intrinsically present as hydroxide ions in the solvent of metal-coordinated hydrogels, the concentration of which is given by the gel pH. To represent variable formation constants of hypothetical ligand–metal systems, we simulated a trifunctional metal-coordinating network by randomly generating sets of ${\beta }_{{\mathrm{L}}_{\mathrm{1}}\mathrm{M}}$, ${\beta }_{{\mathrm{L}}_{\mathrm{2}}\mathrm{M}}$, and ${\beta }_{{\mathrm{L}}_{\mathrm{3}}\mathrm{M}}$. We then observed how the gelation conditions for these hypothetical metal-coordinating hydrogels were affected by increasing the strength of the hydroxide competitor, or the competitor reaction formation constant, ${\beta }_{\mathrm{M}{\left(\mathrm{O}\mathrm{H}\right)}_{\mathrm{z}}}$, assuming no formation of mixed cross-linker species (SI Appendix, Fig. S5 and Movie S3). The results of this simulation (given in Fig. 1C) predict an expanded range of competition-enhanced gel network formation for all competitor stoichiometries considered, i.e., M(OH)₁, M(OH)₂, or M(OH)₃ complexes, when the competitor strength ${\beta }_{\mathrm{M}{\left(\mathrm{O}\mathrm{H}\right)}_{\mathrm{z}}}$ is between ${\beta }_{{\mathrm{L}}_{\mathrm{1}}\mathrm{M}}$ and slightly greater than ${\beta }_{{\mathrm{L}}_{\mathrm{3}}\mathrm{M}}$. It is intriguing to observe that an increase in the gelation conditions is predicted even when the competitor formation constant is larger than the metal-coordinate cross-linking reaction ${\beta }_{{\mathrm{L}}_{\mathrm{3}}\mathrm{M}}$, an observation which suggests that metal-coordination cross-linking via pH-coupled competition might allow for more robust material assembly under a broader range of competitive environments. In particular, this prediction suggests that if pH is maintained while metal is added to an aqueous macromolecular metal-coordinate network, excess metal could be buffered from dissolving the network by the formation of metal–hydroxide complexes. In other words, our simulations predict that a self-regulating mechanism of hydroxide complexation will deliver a stoichiometrically balanced amount of metal ions to the network to ensure L₃M coordination, cross-linking, and mechanically robust hydrogels, as long as 2 criteria are met, that the pH is maintained within a given range and that a minimum amount of metal is present.

To test the predictions from our hypothetical simulations, we can use real metal ion to ligand equilibrium formation constants, ${\beta }_{{\mathrm{L}}_{\mathrm{x}}\mathrm{M}}$, and ligand deprotonation constants from established literature, defined and given in SI Appendix, Fig. S6, to calculate the expected concentration of L₁M, L₂M, and L₃M complexes in a metal-coordinate polymer hydrogel at a given pH and metal concentration, given in SI Appendix, Figs. S7–S9 for real, metal–ligand combinations (19–22). This calculation, illustrated schematically in Fig. 2 A and B and described more thoroughly in Materials and Methods on calculating species concentrations, is performed by simultaneously solving the equilibrium equations for each ligand protonation and metal-coordinate complex species, while maintaining the mass balance of the system (23–25). Similar predictive calculations have been done in the past for metal-coordinated hydrogels; however, these predictions neglected the competing role of hydroxide interactions and did not simultaneously explore the variable space of both pH and metal concentration (12, 26, 27).

Fig. 2.

Schematic of the computational process used to predict gel mechanical properties. (A) A system of equations consisting of equilibrium formation constants, $\beta$, from the reactions of (A, i) water self-ionization, (A, ii) ligand protonation, (A, iii) metal and ligands to form L₁M, L₂M and L₃M complexes, and (A, iv) competitive metal–hydroxide complexation are simultaneously solved to calculate (B) the fraction of ligand complexed as an L₁M, L₂M, or L₃M species as a function of pH to (C) predict the phantom network plateau modulus of the hydrogel at a given metal and polymer concentration. The boxed metal–ligand reaction species in A are what is plotted in B to determine the mechanical properties in C.

With the knowledge of the total concentration of polymer chain-terminating L₁M, chain-extending L₂M, and multichain–cross-linking L₃M complexes, the predicted density of cross-links and elastically active chains in a hydrogel can be directly computed, as shown schematically in Fig. 2C and explained in more detail in Materials and Methods. This process thereby provides an experimentally measurable mechanical prediction of the gel plateau modulus, or the elastic modulus at high frequency, across a range of pH values at a given metal and polymer concentration (18, 28). To test these theoretical predictions, we synthesized linear telechelic polymers to ensure that cross-links in the experimental gel networks would arise solely through L₃M coordination. This contrasts the ability of star polymers to form cross-links when either L₂M or L₃M coordination is dominant, due to their multiarm functionality. With a linear polymer, we can therefore identify the presence of L₃M coordination in the gel simply by measuring its plateau modulus, since the plateau modulus is directly proportional to the number of elastically active chains, which arise only from L₃M coordination. Although defects, such as loops, are likely to form, the dynamic nature of our cross-linking, coupled with a constant polymer size and concentration, should result in an approximately constant negative effect on the mechanical properties across the systems examined; therefore they are not included in our model.

Using this method, we generate predictions for the gel plateau modulus across the dual variable space of pH and metal ion concentration, considering both the case where partitioned hydroxide interactions are neglected and where they are included. These predictions correspond to turning on or off the excess metal competition reactions highlighted with the dashed box in Fig. 2A. We first use a linear nitrocatechol-PEG polymer shown in Fig. 3A which has 2 advantages, relative to the catechol ligand found in mussel threads, that we leverage in this study. First, iron- and pH-induced covalent coupling of the ligand is suppressed, resulting in a truly reversible and transient gel (27, 29, 30). Second, the pK_a values of the ligand are lower (6.69 and 10.85 vs. 9.3 and 13.3), facilitating deprotonation of the ligand at a lower pH, allowing coordinate cross-linking in the presence of iron at neutral, even physiological, pH values of above 6 as opposed to above 9 for catechol (11, 22). The predicted plateau moduli for Fe³⁺- and Al³⁺-coordinated gels are given in Fig. 3 C and D and SI Appendix, Fig. S10, respectively. While both Fe³⁺ and Al³⁺ have been reported as transient catechol cross-linkers in mussel threads, they have diverse electronic structures, with iron being a transition metal and aluminum being a posttransition metal (10). Here we study both metal ions to test the generality of the proposed role of competing hydroxide ions in buffering metal-coordinate hydrogel viscoelastic properties by partitioning excess metal cross-linker from the network. For both these systems, neglecting hydroxide interactions generates drastically different predictions than including these interactions, shown for iron in Fig. 3 C and D, respectively. Without hydroxide interactions, at metal concentrations above 1 metal per 3 ligands, or 1/3 equivalents, the ligands are overwhelmed with metal, L₃M formation is suppressed, and a percolated network cannot form. In contrast, with partitioned hydroxide buffering interactions, a stable percolated network can be formed as long as a critical metal concentration around 1/3 equivalents is exceeded, which causes a large expansion of the conditions predicted to result in gelation. Furthermore, an upper pH boundary for gelation is uniquely predicted when hydroxide competition is included. This pH cutoff in gelation occurs at hydroxide concentrations elevated to the point where hydroxide leaches a critical amount of metal from the network, resulting in depercolation of the gel.

Fig. 3.

Theoretical predictions and experimental verification of expanded metal-coordinate gel assembly conditions. (A and B) Chemical structures of linear 5-kDa PEG functionalized with either (A) nitrocatechol or (B) histidine. (C–F) Predicted phantom network plateau moduli for (C) iron- and (E) nickel-coordinated hydrogels neglecting hydroxide interactions and for (D) iron- and (F) nickel-coordinated hydrogels including hydroxide interactions. (G and H) Experimentally measured plateau moduli vs. pH compared with the phantom network predictions neglecting and including hydroxide interactions for (G) iron- and (H) nickel-coordinated hydrogels (where only the nickel ion concentration of 4/3 was investigated). The predictions when neglecting hydroxide interactions for metal ion concentrations of 2/3, 3/3, and 4/3 result in a modulus of zero independent of pH.

The high metal concentration gelation and high pH depercolation are both unique predictions that we can experimentally test to validate our model of hydroxide competition-controlled gelation of metal-coordinating hydrogels. Accordingly, assembly of hydrogels was tested at a range of pH and metal concentrations, and small-angle oscillatory shear rheology was performed to determine any resulting gel plateau moduli as a measure of their equilibrium network structure (representative rheology data are given in SI Appendix, Fig. S11). The measured gel plateau moduli are summarized in Fig. 3G, alongside comparisons to the model predictions from Fig. 3 C and D. The data confirm the predicted gel formation in the regime of excess metal concentration previously thought to be incompatible with gelation. This is a significant result, since it provides experimental support for the proposed role of hydroxide competition in expanding the gelation regime of metal-coordinate hydrogels. In addition, the 1/3 equivalence gels were generally the least stiff. Therefore, this current standard composition in the field of catechol-based metal-coordinate hydrogels might not form the most robust hydrogels.

The window of gelation predicted by our model as a function of pH matches our experiments qualitatively well—at intermediate pH, gels are formed, while at low and high pH, they are not. Furthermore, within the range of pH values where gels are predicted to form, our experimental gel moduli match the predicted gel moduli of the model reasonably well. However, in some instances, especially in the high metal concentration aluminum systems given in SI Appendix, Fig. S10, the experimental gel moduli exceed the model gel moduli predicted from phantom network theory, indicating that our simple model does not yet capture all of the mechanisms of assembly governing network formation in these gels. It is plausible that an affine network model would be more applicable or that supramolecular chain entanglements or higher-functionality metal clusters could be increasing the network cross-link density. However, examination of gel network dynamics obtained from gel frequency sweeps (given in SI Appendix, Fig. S12) revealed that the metal ion concentrations do not significantly impact the relaxation time of the hydrogels, suggesting that the impact of such higher-order supramolecular assembly mechanisms is minor (31). Finally, to support our general assumption that the gels are purely transiently cross-linked and do not contain any covalent cross-links present in similar, catechol-based, gels, we confirmed the reversibility of gel network percolation vs. pH through cyclic pH titrations on single gels formed at different metal ion concentrations (Movies S4–S11 and SI Appendix, Fig. S13). Importantly, these gel titrations also served to verify the existence of the newly predicted gel network depercolation at high pH.

To further demonstrate generality of the proposed effect of hydroxide competition on gelation of metal-coordinate hydrogels, we examined another common mussel-inspired ligand, histidine, which has also been shown to form transient metal-coordinated hydrogels when coordinated with Ni²⁺ (12, 26, 32). Following the same approach as above, we synthesized a histidine functionalized linear polymer as shown in Fig. 3B and used the same algorithm to predict the plateau moduli for histidine hydrogels as a function of pH and metal ion concentration, both neglecting and including the effect of partitioned hydroxide competition, as presented in Fig. 3 E and F. To test our model predictions, we explored the formation of hydrogels at the 4/3 metal ion equivalence concentration, since assembling hydrogels using this metal concentration and a linear polymer architecture is predicted to be impossible when hydroxide competition is neglected, as shown in Fig. 3E. Importantly, as shown in Fig. 3H, we indeed observe strong formation of hydrogels in agreement with our model prediction including hydroxide competition. A small pH offset is observed, which could be caused by a discrepancy between the histamine binding constants used in our computational model and the histidine ligands used in our experimental system. This particular ligand–metal combination highlights the critical importance of including hydroxide interactions when predicting gelation, since the prediction in Fig. 3E shares little overlap with the prediction in Fig. 3F, which potentially explains why no one has previously reported a histidine end-functionalized linear polymer hydrogel network.

Conclusions

Several insights presented in this study should be broadly applicable in the future design of polymeric hydrogels. For example, the discovery that metal coordination can bypass traditional stoichiometric limits could lead to metal coordination becoming a more attractive design choice for use as a sacrificial bond when engineering tough viscoelastic hydrogels. Furthermore, the evidence that higher, stoichiometrically imbalanced, metal ion concentrations allow robust gelation of metal-coordinate systems could lead to more informed design of metal-coordinating gels with a given metal, ligand, and polymer architecture. Additionally, reflecting upon the bio-inspiration for our metal-coordinated hydrogels, it is interesting to speculate that the mussel could be leveraging the robustness in macromolecular material assembly demonstrated here, when secreting its tough, energy-dissipating holdfast fibers. Since the last stage of mussel thread assembly takes place in pH-buffered seawater, perhaps a simple lower bound of metal concentration required to obtain protein network percolation via metal coordination provides the mussel with robust kinetic control during the uniquely fast underwater assembly of its fibers (33). If true, then similar bio-inspired tricks to ensure rapid underwater material assembly should be adaptable to the growing biomedical engineering field of advanced 3D printing of synthetic tissues.

Materials and Methods

For experimental materials and methods, see SI Appendix. The source code used to generate the theoretical mechanical predictions is available for download at Zenodo, https://zenodo.org/record/3461229 (34).

Theoretical Determination of Species Concentrations.

The binding state of the telechelic ligands as a function of pH for a given metal concentration was calculated by modifying the methods presented elsewhere (23–25). For our case, the fundamental components of the system were H, representing a proton; M, representing the metal; and L, representing the ligand. The H component was eliminated by defining a free H concentration to enforce a given input pH on the system. Species were created by combining the individual components using equilibrium binding constants cataloged by Smith and Martell (19–22). Tables cataloging these formation binding constants are provided in SI Appendix, Fig. S6. To convert from the hydroxide components used in Smith and Martell (19–22) to the hydrogen component used in our program, we used the following relation:

$${\beta }_{M{\left(OH\right)}_z,hydrogen}={\beta }_{M{\left(OH\right)}_z,hydroxide}-p{K}_w*z.$$ [1]

The total concentration of each component was fixed within the system. The program used an initial guess of the free component concentrations to calculate the concentration of all of the species using their equilibrium formation equations and binding constants. The total concentration of each component was then calculated and compared with the fixed component concentration restraint. The initial guess was then modified based on the difference between the calculated component concentration and the actual component concentration. This process was repeated until the difference between the calculated component concentration and the actual component concentration was negligible, resulting in the determination of the theoretical equilibrium individual species concentrations at a given pH and total metal and ligand concentration. The pH was then increased and the iterative calculation was performed again to generate species concentrations at different pH values. This entire process, including modifying the pH, was done for a range of discrete total metal concentrations to fully calculate species concentrations with respect to both pH and metal concentration.

Theoretical Prediction of the Plateau Modulus.

Once we knew all of the species concentrations in a system, we then predicted the expected plateau modulus using the Miller–Macosko model (18, 28). To calculate the phantom network plateau modulus, $G_{phantom}$, we needed the concentration of elastically active chains, $c_{chains}$, and the concentration of cross-links, $c_{cross-links}$, as shown in Eq. 2, where $R$ is the ideal gas constant and $T$ is the absolute temperature:

$$G_{phantom}=RT\left(c_{chains}-c_{cross-links}\right).$$ [2]

We could have estimated these parameters directly from the L₃M species concentration; however, to take into account the effect that L₁M species have on terminating a chain and L₂M species have on extending the chain, we considered the probabilistic percolation of the network to get a more accurate prediction by tailoring the methods developed by Miller and Macosko (18). We defined $P\left(F\right)$ as the probability that a chosen path is finite and its complement, $1-P\left(F\right)$, as the probability that a path is infinite. The probability of the path being finite was calculated from the general Eq. 3, where $P_{species}$ is the probability of a given species existing, and $P_{species\hspace{0.17em}leads\hspace{0.17em}to\hspace{0.17em}\mathit{fi}nite\hspace{0.17em}path}$ is the probability that a particular species leads to a finite path:

$$P\left(F\right)=\sum _{species}{P}_{species}*P_{species\hspace{0.17em}leads\hspace{0.17em}to\hspace{0.17em}\mathit{fi}nite\hspace{0.17em}path}.$$ [3]

Eq. 3 simplified for our case into Eq. 4, where $T$, $B$, $M$, and $E$ are the probabilities that a ligand was in the L₃M, L₂M, L₁M, or unbound state, respectively, which were known from the speciation calculations:

$$P\left(F\right)=T*P{\left(F\right)}^2+B*P{\left(F\right)}^1+M*P{\left(F\right)}^0+E*P{\left(F\right)}^0.$$ [4]

Eq. 4 was then solved to determine $P\left(F\right)$. The concentration of 3-fold cross-links was then determined using Eq. 5, where $L$ is the total ligand concentration:

$$c_{cross-links}=T/3*{\left[1-P\left(F\right)\right]}^3*L.$$ [5]

The number of elastically active chains can then be calculated using Eq. 6, giving the necessary variables to calculate the phantom network modulus using Eq. 2:

$$c_{chains}=\left(3/2\right)*c_{cross-links}.$$ [6]

The moduli for the A₂–B₃ covalent gelation simulation were calculated directly using the equations given in Miller and Macosko (18). For the partitioned case, B groups that reacted with C were essentially removed from the system and therefore neglected when calculating the stoichiometric imbalance ratio and the reaction probability $p_{\text{AB}}$ (18).

Simulating Gelation Conditions as a Function of Competitor Strength.

To determine how free competitors affect the gelation conditions, we performed simulations predicting the phantom network plateau moduli as a function of both cross-linker and competitor concentration for a hypothetical set of binding constants representing a hypothetical gel-forming system and then varied the strength of the competitor reaction. Three simulations were performed, 2 classical A₂–B₃ gelations, 1 with mixed competitor species and 1 with partitioned competitor species, and a metal-coordinated hydrogel with partitioned hydroxide as the competitor (SI Appendix, Figs. S3–S5). The ranges of competitor concentrations considered are 0 to 5 n_C/n_A and pH 0 to 14. The cross-linker concentration ranges used are 0 to 5 n_B/n_A and 0 to 5/3 n_M/n_L. The binding constant for the A₂–B₃ gelation, $\text{log}{\beta }_{\text{AB}}$, was a random number between 6 and 16. The binding constants randomly generated for the metal-coordination simulations were a set of $\text{log}{\beta }_{{\text{L}}_1\text{M}}$, $\text{log}{\beta }_{{\text{L}}_2\text{M}}$, and $\text{log}{\beta }_{{\text{L}}_3\text{M}}$ determined with the rules that $\text{log}{\beta }_{{\text{L}}_3\text{M}}<40$, $\text{log}{K}_{{\text{L}}_3\text{M}}>3$, $\text{log}{K}_{{\text{L}}_1\text{M}}>\text{log}{K}_{{\text{L}}_2\text{M}}>\text{log}{K}_{{\text{L}}_3\text{M}}$, and $\text{log}{\beta }_{{\text{L}}_3\text{M}}-\text{log}{\beta }_{{\text{L}}_1\text{M}}<18$ (remember that $\text{log}{\beta }_{{\text{L}}_1\text{M}}=\text{log}{K}_{{\text{L}}_1\text{M}}$, $\text{log}{\beta }_{{\text{L}}_2\text{M}}=\text{log}{K}_{{\text{L}}_1\text{M}}+\text{log}{K}_{{\text{L}}_2\text{M}}$, and $\text{log}{\beta }_{{\text{L}}_3\text{M}}=\text{log}{K}_{{\text{L}}_1\text{M}}+\text{log}{K}_{{\text{L}}_2\text{M}}+\text{log}{K}_{{\text{L}}_3\text{M}}$). A total of 50 sets of hypothetical binding constants were generated for each simulation, with the same set of 50 being used for both A₂–B₃ simulations. The average moduli were calculated by averaging the gelation predictions over the full phase space, first when there was no competitor reaction and then as the competitor reaction strength, $\text{log}{\beta }_{\text{BC}}$, or $\text{log}{\beta }_{{\text{M}\left(\text{OH}\right)}_{\text{z}}}$ was increased by an integer value, where stoichiometry coefficient $z$ was fixed at a value of 1, 2, or 3 to show the effect of competitor stoichiometry. The relative gelation conditions are calculated by dividing the average moduli over the phase space by the average moduli of the phase space obtained when no competitor reaction is present. Please see SI Appendix, Figs. S3–S5 for an illustration of this process.