Article pubs.acs.org/crystal
Computational Modeling and Prediction of the Complex TimeDependent Phase Behavior of Lyotropic Liquid Crystals under in Meso Crystallization Conditions Tu C. Le,† Charlotte E. Conn,† Frank R. Burden,† and David A. Winkler*,†,‡ †
CSIRO Materials Science and Engineering, Bag 10, Clayton South MDC 3169, Australia Monash Institute of Pharmaceutical Sciences, 381 Royal Parade, Parkville 3052, Australia
‡
ABSTRACT: Membrane-bound proteins comprise a very important class of drug targets. Solution of their structures by X-ray crystallography has been hampered by difficulties in crystallizing them in biologically relevant conformations. Novel amphiphilic materials that form bicontinuous cubic phases are being used to support the growth of crystals. However the cubic phase may transit to other lipidic mesophase structures under the influence of the different components within the crystallization screen. Furthermore the mesophases may evolve with time, a process that is poorly understood but potentially critical for controlled crystal growth. Recent advances in high-throughput screening of lipid systems have allowed us to generate a large body of data on the influence of screen components on the cubic phase. However it has been difficult to deconvolute individual effects in the multicomponent system present during a crystallization trial. We have therefore developed robust and predictive computational models that predict how the phase behavior of lyotropic liquid crystals changes over time and under the influence of crystallization additives. Our work demonstrates that the complex phase behavior of amphiphilic nanostructured nanoparticles can be captured with high accuracy using modern, robust machine learning methods. We predicted the existence of individual nanophases with accuracies of 98−99% and the complex coexistence of multiple phases to a similar accuracy using nonlinear models: linear models were not as effective and robust. This approach also allowed us to determine which crystallization screen components were most relevant to the temporal evolution of individual mesophases.
1. INTRODUCTION In meso crystallization uses bicontinuous cubic phases as matrices to support the growth of protein crystals. The technique is of particular importance for membrane proteins (MPs), many of which are important drug targets. In biology, MPs, due to their amphiphilic nature, are normally embedded in the phospholipid bilayers of cell membranes, where they provide signal transduction or ion channel functions. This amphiphilic nature makes them unsuitable for conventional sitting or hanging drop crystallization experiments. Consequently, for structural biology studies, mimicking their native bilayer environment should allow them to fold correctly and facilitate crystallization. Bicontinuous cubic phases, which are based around the fundamental structure of the lipid bilayer, have significant advantages over other methods of MP stabilization and have therefore been employed to incorporate MPs and encourage crystal growth. Experimental work in this area has been extensive and has been conveniently reviewed in recent publications.1−4 However, incorporation of MPs into bicontinuous phases is poorly understood and success rates for MP crystallization remain low. One major issue is the multicomponent nature of crystallization trials. A typical trial consists of adding a complex range of mixtures of poly(ethylene glycol)s (PEG), salts, and buffers (a crystallization screen), to the MP incorporated in a © 2013 American Chemical Society
cubic phase in a multiwell plate. Each individual well contains a single screen consisting of one or more lipids, protein and associated surfactant, and the components of the particular screen. All of these components have been shown to impact the structure of the underlying mesophase.5−11 Individual additives can cause the underlying cubic phase to swell or shrink or, at high enough concentrations, can cause a phase transition to a 1-D lamellar, 2-D hexagonal (HII) or disordered fluid micellar (L2) phase. We have previously suggested5 that, for in meso crystallization, the components of the crystallization screen may impact crystal growth in two ways. First, incorporated PEG and salts may directly influence protein−protein interactions similar to soluble protein crystallization. However the wide range of mesophases observed under crystallogenesis conditions suggests that the underlying lipid nanostructure may play a significant role. An improved understanding of the structural evolution of the lipid mesophase during crystal growth may allow us to elucidate the mechanism of crystallization and improve crystallization success rates significantly. However, the problem of predicting how Received: November 26, 2012 Revised: January 7, 2013 Published: January 25, 2013 1267
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individual components should allow us to determine more accurately the effect of nanostructure on crystal growth. Finally, the ability to predict the effect of individual screen components on the underlying cubic architecture will allow us to assess the compatibility of potential crystallization screens with individual lipid moieties prior to crystallization trials being carried out. This could significantly increase the efficacy of in meso crystallization trials.
individual components will affect the structure of the mesophase, particularly in such complex multicomponent systems, has been widely considered intractable using theoretical modeling approaches. The theoretical models that exist are generally only applicable to much simpler systems containing pure components. Computational models based on assumed phase topologies for single phases have also been reported by Angelov et al. (ref 12 and references cited therein). These provide valuable insight into the growth of cubic membrane structures from small cubosomic structures but cannot make quantitative predictions of multiple phases. The variable space for crystallization trials (the number of factors than can be varied and their concentration range) is very large so it has been difficult to explore using traditional experimental methods. A recently developed high-throughput approach has allowed us, for the first time, to automate sample production and structural analysis using high-throughput synchrotron SAXS experiments.13 Using this protocol, we have studied the complex structural evolution of the monoolein (MO) cubic phase under the influence of the pH-, anion-, and cation-testing (PACT) crystallization screen.6 Typical cubic mesophases are illustrated in Figure 1.
2. EXPERIMENTAL SECTION 2.1. Data. The data used for modeling purposes were obtained from the work of Conn et al.6 Samples consisted of the lipid monoolein (MO, 0.21 mg) hydrated at a ratio 60 wt % lipid to 40 wt % solvent. For all samples, the solvent contained the amphiphilic amyloid-β peptide (1− 42) at a concentration of 0.2 mM. At this concentration, the peptide has no effect on the cubic mesophase structure.6 Samples were set up robotically within a 96-well plate using the method described in Darmanin et al.13 Each of the 96 wells on the plate has two subwells in which duplicates were generated. PACT crystallization screen (0.14 μL) was immediately dispensed directly into each subwell. In addition, 20 μL of PACT crystallization screen was added into the reservoir to maintain environmental hydration levels. The PACT crystallization screen used was made up at the C3 Collaborative Crystallization Centre, Parkville, Australia, with water containing 0.02% sodium azide (NaN3) to act as an antimicrobial agent. Figure 2 summarizes the components of the PACT crystallization screen. The screen contains poly(ethylene glycol) (PEG) with a molecular weight of 1500, 3350, or 6000. In addition a variety of salts were included allowing us to examine the effect of individual cations and anions on mesophase formation. The pH value within individual wells was buffered to a value between pH 4 and 9 using nine different buffer systems. Plates were analyzed 1, 5, 7, and 21 days after addition of screen to track the structural evolution of the lipid mesophases over time under the influence of the components of the screen. Figure 3 is a schematic representation of the complex phase behavior observed under the influence of the PACT crystallization screen. In the absence of screen components, MO forms a QIID bicontinuous cubic phase at 40% amyloid-β peptide (0.2 mM) in PBS. Here the majority of wells retain a cubic phase up to 3 weeks following addition of screen. Both QIIG and QIID phases are observed. In addition an inverse hexagonal (HII) phase was observed in some wells. An additional unassigned peak is observed for both subwells in well D8, 7 days after addition of screen. For some wells, particularly at longer time scales, no diffraction was observed indicating a loss of long-range order in the underlying phase. For a few wells, the reproducibility of the phases as assessed by the duplicate samples was much lower than for the remainder of the plate. In these cases, the mesophase adopted differed between the two subwells. For the most part, the non reproducible wells are in rows A and H where it appears that some wells are subjected to an increased drying effect. This data together with that for row E where the pH was not controlled was excluded from the modeling (white shading in Figure 3). 2.2. Quantitative Structure−Property Relationship (QSPR) Modeling. The availability of large quantities of data from high throughput experiments has allowed us to develop computational models predicting the complex phase behavior of amphiphilic components using modern machine learning methods. These methods are well matched to large complex multidimensional data sets. They allow linear and nonlinear models of optimal parsimony to be developed that relate the components of the screen and time to the existence of various nanophases. The methods are complementary to theory-based methods that make predictions of phase behavior in the absence of experimental data but that are limited to small systems consisting of pure components and few dimensions (experimental variables). Multiple linear regression with expectation maximization (MLREM) and Bayesian regularized artificial neural networks (BRANN) methods14,15,18 have been used to investigate the relationship between the crystallization conditions and mesophase behavior. Model inputs included experimental parameters such as time, pH, molecular weight and concentration of poly(ethylene glycol) (PEG), concentrations and
Figure 1. Structure of the inverse-bicontinuous diamond, QIID (Pn3m), gyroid, QIIG (Ia3d), and primitive, QIIP (Im3m) cubic phases.
We could qualitatively account for many of the observed structural effects with respect to the known effects of the PACT crystallization screen components on the underlying mesophase. However, due to the large number of interlinked components, it was difficult to deconvolute the effects of individual components without the aid of computational modeling methods. Consequently, we used a Bayesian regularized neural network14,15 to generate quantitative structure−property relationships (QSPRs) between the individual PACT screen components and the observed mesophase behavior over time. This versatile machine-learning method can in principle model any complex relationship given sufficient data. The problems of overtraining or overfitting data or the risk of chance correlations of standard neural networks are overcome by the inclusion of a regularization step controlled by Bayesian statistics. This results in the optimum balance between bias (model not sufficiently complex to capture underlying relationships in data) and variance (model overly complex and fits noise). This approach has recently predicted the effect of incorporated drugs on the structure of lipid mesophases in nanoparticulate drug delivery vehicles16 and in materials more generally.17 In this paper, we apply it to understand, model, and predict the influence of the PACT crystallization screen components with time on the phase behavior of MO. Unlike theoretical and topology-based models, our computational modeling method is well matched to situations where large quantities of multidimensional data exist and where complex interactions between the components are likely to occur. A more quantitative understanding of the effect of 1268
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Figure 2. Description of PACT screen conditions with different PEG molecular weights, pH values, salt, and buffer additives.
Figure 3. The mesophase adopted by MO, 40 wt % amyloid-β peptide in PBS (0.2 mM), 1, 5, 7, and 21 days after addition of the PACT screen. Phases observed include inverse bicontinuous cubic QIID and QIIG phases and an inverse hexagonal HII phase. prediction models combined to predict the complex phase diagram where multiple phases may coexist. The BRANNs consisted of one input, one hidden, and one output layer. The number of nodes in the input layer of the network was equal to the number of experimental conditions and the output layer had only
identities of cations and anions in the salts, and concentrations of buffers. As pH value is a descriptor for modeling, data for all samples of row E on the plate were not considered for generating or testing the QSPR models, because pH was not controlled for these samples. Separate models were built for each phase, and all individual phase 1269
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Figure 4. Diamond inverse-bicontinuous QIID cubic phase diagrams predicted by the best (a) MLREM and (b) BRANN models. Incorrect model predictions are circled. one node corresponding to the existence (1) or absence (0) of a given phase. Two or three hidden layer nodes (where the model computation is performed) were found to be sufficient to build good models, and increasing the number is unnecessary because the Bayesian regularization automatically controls the complexity of the models to optimize predictivity.19 The number of effective weights used in Bayesian neural network models asymptotes to a constant value as the number of hidden layer nodes increased. Details of Bayesian regularization applied to backpropagation neural networks can be found in recent publications.14,15,18 MLREM and BRANN with a Laplacian prior (BRANNLP) algorithms14,15 were used to generate sparse models (which have the best predictivity) and to identify experimental conditions that affected
the mesophase behavior most strongly. They use the well-known sparsity-inducing properties of a Laplacian prior to prune out the least informative descriptors (experimental conditions) as the sparsity control is progressively increased. The BFGS18 method was used to train the models until a maximum in the Bayesian evidence was attained (maximum likelihood method). This avoids the need to use a validation set to determine when network training should be stopped to prevent overtraining. Sigmoidal transfer functions were used in the hidden and output layers, and linear transfer functions in the input layer. To generate QSPR models predicting individual liquid crystalline phases, the data set was separated into a training set (80%) and a test set (20%) using the k-means clustering algorithm. The training set was used to 1270
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Figure 5. Gyroid inverse-bicontinuous QIIG cubic phase diagram predicted by the best (a) MLREM and (b) BRANN models. Incorrect model predictions are circled. generate the models relating the experimental conditions to the phase behavior. The test set was never used in the model development process and was held aside so that predictions of these data could be compared with the experimental data for the test set. This provided a means for estimating the ability of computational models to predict the behavior of new experimental conditions. Linear and nonlinear models were produced using MLREM and BRANNs methods, respectively.
behavior for data obtained 1, 5, and 7 days after addition of the crystallization screen, Figure 4a. However this linear method did not predict accurately the existence of the QIID phase 21 days after addition of the screen. We suggest that this reflects the increased instability of this phase over longer time scales that cannot be predicted using a linear model. The best MLREM model gave 24 prediction errors, most of which were for samples with PEG molecular weight of 1500 and no salt additives after 21 days. The nonlinear BRANN method was substantially more successful in predicting the relationship between the exper-
3. RESULTS AND DISCUSSION 3.1. Performance of Phase Prediction Models. For the QIID phase, the MLREM method accurately predicted the phase 1271
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Figure 6. Inverse hexagonal HII phase diagram predicted by the best (a) MLREM and (b) BRANN models. Incorrect model predictions are circled.
imental crystallization screen conditions and the QIID phase behavior. As shown in Figure 4b, the best BRANN model with 17 effective weights generated only 1 prediction error for the training set and no errors for the test set. That is, of 319 data points, 318 (99.7%) were predicted correctly. We found a similar behavior when modeling performance for the QIIG phase as for the QIID phase, with the 21 day data also being poorly predicted by the MLREM method. We hypothesize that this reflects the transition from a QIID to a QIIG phase, which was typically observed over longer time scales, making it difficult for the linear
model to capture the relationship between experimental conditions and QIIG phase formation. The best MLREM model gave 27 prediction errors, 20 of which were for the training set and 7 for the test set (Figure 5a). These errors also mainly occurred in samples with no salt additives and PEG molecular weight of 1500. Better predictions were made by the nonlinear BRANNs method, as shown in Figure 5b. The best BRANN model with 16 effective weights gave high accuracy, with 4 prediction errors for the training set and 2 for the test set. Out of 319 data points, 313 (98.1%) were predicted correctly. 1272
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Figure 7. Truth tables for individual phase prediction from best BRANN models.
Figure 8. Phase coexisting diagram predicted by the best BRANN models. Incorrect predictions for any of the phases at a given experimental point are circled.
8, to an accuracy of 950 out of 957 (99.3%) individual phase existence data points or 312 out of 319 (97.8%) samples with phase coexistence. This is a more stringent test of the phase prediction error because an error in a single phase of a multiple coexistence of phases counts as an error overall. This illustrates the excellent ability of these methods to model the effects of crystallization screen conditions and the time evolution on nanophase formation. 3.2. Identifying the Most Important Crystallization Screen Conditions. The sparse selection of the most relevant descriptors by MLREM and BRANNLP may provide useful insight into the crystallization screen components that affect mesophase formation. This is of particular importance for the complex multicomponent data set modeled here that reflects the structural evolution of a lipid mesophase with time during a typical in meso crystallization trial. Our modeling approach provides a quantitative analysis of the effect of each individual crystallization screen component on the structural evolution of the lipid mesophase with time. We begin by considering the screen components and experimental parameters selected as most relevant by the
For the inverse hexagonal HII phase, the crystallization screen descriptors were insufficient to predict the existence of this phase. The models predicted that this phase would be absent in all experiments. However, we discovered that the prediction performance could be improved substantially by the inclusion of descriptors to the HII model that accounted for the presence or absence of the QIID and QIIG phases. These indicator variables denoting the presence or absence of these other phases could be calculated quite accurately from the cubic phase models. The nonlinear BRANN method again improved the performance of models predicting the HII phase. These refined modeling techniques resulted in only 4 errors for the linear MLREM model (at late time points) and no errors for BRANN model for the HII phase for all 319 data points as shown in Figure 6. The truth tables showing the modeling results for individual phases are illustrated in Figure 7. We could not construct models for those wells where no phase was detected or where an additional unassigned SAXS peak was observed. Combining the best individual phase prediction results shows that, using BRANN method, our approach could predict the complex coexistence of multiple nanophases, illustrated in Figure 1273
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formation of the QIIG, QIID, and HII phases. We note that PEG concentration is highly correlated to PEG molecular weight. Specifically, the lowest molecular weight PEG (1500) has a slightly higher concentration than the two higher molecular weight PEG molecules (3350 and 6000). The PEG concentration is therefore inversely related to the PEG molecular weight for this data set. Previous research by others and us has suggested that PEG molecular weight is the important descriptor; we therefore neglect the effect of PEG concentration in the following discussion. The presence of the citrate ion was also found to be important in the formation of the QIID and QIIG phases. Finally various buffers were found to be important. Many of these screen component observations are in agreement with
BRANNLP models, Table 1. PEG molecular weight, pH, and the presence of the Na+ ion were selected as important in the Table 1. Significant Factors Selected by the Best BRANNLP Phase Prediction Models phase
descriptors
QIID
time, PEG molecular weight, PEG concentration, pH, concentrations of sodium, citrate, MIB, BTP time, PEG molecular weight, PEG concentration, pH, concentrations of sodium, magnesium, zinc, citrate, SPG, MIB, Tris buffer PEG molecular weight, pH, concentrations of sodium, potassium, SPG, HEPES, QIID and QIIG phase indicator variables.
QIIG HII
Figure 9. Significant descriptors selected by the best MLREM models. The coefficients show the importance of the factor to the existence of the phase (positive) or lack of the phase (negative). 1274
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our preliminary analyses.6 We and others have previously observed that PEG molecular weight has a significant effect on the formation of QIID and QIIG phases. Specifically, higher molecular weight PEG molecules will induce a QIID to QIIG transition, while the QIID phase is typically retained in the presence of lower molecular weight PEG molecules.6,10,20 We have also previously observed qualitatively the importance of the citrate ion and suggested that this reflects its position in the Hofmeister series.6 We discuss the effect of individual anions and cations in relation to their position in the Hofmeister series in more detail in the subsequent section. However, the BRANNLP model is unable to provide information on whether a particular descriptor promotes or inhibits formation of individual phases, which limits the information that can be extracted using this method. In contrast, while the MLREM models were slightly less accurate at predicting individual phase formation, this method does provide coefficients for individual descriptors not only allowing us to determine whether each descriptor inhibits or promotes a particular phase but also providing an indication of how strong the effect is. Figure 9 illustrates the contributions individual crystallization screen components make to the formation of the QIID, QIIG, and HII phases as determined by MLREM modeling. For the QIID phase, the most important descriptors suggested by the MLREM models were the PEG molecular weight, time, and concentration of citrate. This is in good agreement with the most significant screen components suggested by the nonlinear BRANNLP method (time, PEG molecular weight, pH, [Mg2+], [citrate ion]). The QIID phase formation is promoted by PEG molecules with lower molecular weights. As described above, this is in full agreement with previous studies suggesting that PEG molecules of higher molecular weight promote a QIID−QIIG transition while the QIID phase is retained in the presence of lower molecular weight PEG.5,10,20 In addition the QIID phase is inhibited at longer time scales, as is immediately apparent in Figure 9. Finally, the QIID phase is strongly promoted by the presence of the citrate ion. Again this is in agreement with our earlier analysis that the effect of individual anions and cations is strongly correlated with their position in the Hofmeister series. The Hofmeister series classifies ions as chaotropes (water-structure breakers) or kosmotropes (water-structure makers).21 Due to their influence on the interfacial area of individual mesophases, kosmotropes tend to promote phase transitions in the sequence Lα → QIIG → QIID → HII while chaotropes promote the reverse phase sequence. A typical Hofmeister series for anions (from kosmotropic to chaotropic) is citrate ≈ tartrate (≈ malonate) > sulfate > phosphate > acetate (≈ formate) > F− > Cl− > Br− > I− > nitrate > SCN−. A corresponding series for cations (from chaotropic to kosmotropic) is NH4+ > K+ > Na+ > Li+ > Mg2+ > Ca2+. Promotion of the QIID phase by the citrate anion reflects its position in the Hofmeister series as a strong kosmotrope. The citrate anion is the only anion capable of retention of the QIID phase under the competing influence of high molecular weight PEG, which is promoting the QIIG phase. Both of the other anions selected, thiocyanate and fluoride, inhibit QIID phase formation, consistent with their chaotropic nature. The behavior for cations is also consistent with their position in the Hofmeister series: Mg2+, which is a kosmotrope, promotes the QIID phase, while the chaotropic cations, Na+, NH4+, and K+ inhibit QIID phase formation. The effect of cations in general is smaller than that of anions, consistent with previous observations.
We now consider individual crystallization screen buffers selected as important parameters affecting phase formation. Our earlier experimental work had suggested that the impact of individual buffers on mesophase structure was negligible.6 Here the buffers sodium acetate, MES, HEPES, and BTP were shown to significantly inhibit QIID phase formation. In contrast succinate phosphate glycine (SPG), malonate imidazole borate (MIB), propionate cacodylate-bis tris propane, malate MES-tris, and tris have a minor promoting effect on QIID phase formation. We note that the effect of the buffer correlates strongly with the effect of the PEG present in the same well. With the exception of tris, those buffers that promote the QIID phase (SPG, MIB, PCBT, MMEST) coexist in the same wells as lower molecular weight PEG, which is known to support QIID phase formation. Similarly those buffers that inhibit the QIID phase (sodium acetate, MES, HEPES, and BTP) all coexist with higher molecular weight PEG, which promotes a QIID−QIIG transition. This strongly suggests that the model may have incorrectly identified the buffer as being a significant descriptor, when the effect is due to the PEG molecules with which it coexists. For the QIIG phase, the most relevant screen components and experimental factors suggested by MLREM were the citrate concentration, time, and concentration of SPG buffer. These descriptors were also selected as significant by BRANNLP modeling, as well as the molecular weight of PEG and the presence of sodium acetate, MIB, and bis-tris propane buffers. As expected the significant screen components that promote the QIID phase inhibit the QIIG phase and vice versa. Higher molecular weight PEG molecules and longer time scales both promote the QIIG phase as shown in Figure 9. We again find that the position of individual cations and anions in the Hofmeister series is strongly correlated with their effect on formation of the QIIG phase. The kosmotropic cations (magnesium, calcium, zinc, and lithium) and anions (citrate) inhibit QIIG phase formation. In contrast the chaotropic thiocyanate anion has a small promotion effect. We note that the model has selected factors that inhibit the QIIG phase in preference to those that promote it. Again the effect of buffers is strongly correlated with the effect of the coexisting PEG molecule. SPG, PCBT, and MMEST are shown to inhibit QIIG phase formation and all coexist with lower molecular weight PEG, which also inhibits QIIG phase formation. In contrast those buffers that coexist with higher molecular weight PEG, known to promote QIIG phase, are shown to promote QIIG phase. For the HII phase, the most relevant experimental factors suggested by MLREM were the citrate and tartrate concentrations, time, and concentration of sodium. These crystallization screen components were also selected as significant by BRANNLP modeling, as well as the molecular weight of PEG, the presence of sodium, and the concentrations of SPG and HEPES. The strong promoting effect of the citrate and tartrate anions is in good agreement with their highly kosmotropic nature, which promotes formation of the highly curved HII phase. In contrast virtually all chaotropic anions were shown to inhibit this phase as expected. The model identified all cations, whether kosmotropic or chaotropic, as promoting the HII phase. This may reflect the somewhat weaker effect of cations in the Hofmeister series compared with anions. Both longer time scales and high pH were determined to have a promoting effect. BTP buffer had a small promoting effect that is almost certainly correlated with the coexistence of the strong kosmotropic anions citrate and tartrate with this buffer. In summary, the model has allowed us to successfully identify the most relevant crystallization screen components and 1275
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(2) Angelova, A.; Angelov, B.; Garamus, V. M.; Couvreur, P.; Lesieur, S. J. Phys. Chem. Lett. 2012, 3 (3), 445−457. (3) Caffrey, M. Ann. Rev. Biophys. 2009, 38, 29−51. (4) Caffrey, M.; Cherezov, V. Nat. Protoc. 2009, 4 (5), 706−731. (5) Conn, C. E.; Darmanin, C.; Mulet, X.; Hawley, A.; Drummond, C. J. Soft Matter 2012, 8 (26), 6884−6896. (6) Conn, C. E.; Darmanin, C.; Mulet, X.; Le Cann, S.; Kirby, N.; Drummond, C. J. Soft Matter 2012, 8 (7), 2310−2321. (7) Conn, C. E.; Darmanin, C.; Sagnella, S. M.; Mulet, X.; Greaves, T. L.; Varghese, J. N.; Drummond, C. J. Soft Matter 2010, 6 (19), 4838− 4846. (8) Conn, C. E.; Darmanin, C.; Sagnella, S. M.; Mulet, X.; Greaves, T. L.; Varghese, J. N.; Drummond, C. J. Soft Matter 2010, 6, 4828−4837. (9) Conn, C. E.; Mulet, X.; Moghaddam, M. J.; Darmanin, C.; Waddington, L. J.; Sagnella, S. M.; Kirby, N.; Varghese, J. N.; Drummond, C. J. Soft Matter 2011, 7, 567−578. (10) Cherezov, V.; Fersi, H.; Caffrey, M. Biophys. J. 2001, 81 (1), 225− 242. (11) Joseph, J. S.; Liu, W.; Kunken, J.; Weiss, T. M.; Tsuruta, H.; Cherezov, V. Methods 2011, 55 (4), 342−349. (12) Angelov, B.; Angelova, A.; Garamus, V. M.; Drechsler, M.; Willumeit, R.; Mutafchieva, R.; Stepanek, P.; Lesieur, S. Langmuir 2012, 28 (48), 16647−16655. (13) Darmanin, C.; Conn, C. E.; Newman, J.; Mulet, X.; Seabrook, S. A.; Liang, Y.-L.; Hawley, A.; Kirby, N.; Varghese, J. N.; Drummond, C. J. ACS Comb. Sci. 2012, 14 (4), 247−252. (14) Burden, F. R.; Winkler, D. A. QSAR Comb. Sci. 2009, 28 (6−7), 645−653. (15) Burden, F. R.; Winkler, D. A. QSAR Comb. Sci. 2009, 28 (10), 1092−1097. (16) Le, T. C.; Mulet, X.; Burden, F. R.; Winkler, D. A. Mol. Pharmaceutics 2012, accepted. (17) Le, T.; Epa, V. C.; Burden, F. R.; Winkler, D. A. Chem. Rev. 2012, 112 (5), 2889−2919. (18) Burden, F. R.; Winkler, D. A. J. Med. Chem. 1999, 42 (16), 3183− 3187. (19) Tarasova, A.; Burden, F.; Gasteiger, J.; Winkler, D. A. J. Mol. Graph. Model. 2010, 28 (7), 593−597. (20) Vargas, R.; Mateu, L.; Romero, A. Chem. Phys. Lipids 2004, 127 (1), 103−111. (21) Lo Nostro, P.; Ninham, B. W. Chem. Rev. 2012, 112 (4), 2286− 2322.
experimental factors inhibiting and promoting a particular phase. In general, the factors selected by the model were consistent with those inferred from our earlier experimental analysis. Specifically, the QIIG phase is promoted at longer time scales, by higher molecular weight PEG, and by chaotropic anions and cations. In contrast, the QIID phase is promoted by lower molecular weight PEG and kosmotropic cations and anions. The model is particularly useful in the wide range of individual cations and anions that it selected as important components. Our qualitative analysis was only able to determine a single anion as significant, the citrate anion, which is the strongest kosmotrope. While the model also identified the citrate anion as one of the most important descriptors, it additionally correctly selected many more cations and anions as either promoting or inhibiting individual mesophases. The model allows successful determination of the effect of individual components in a multicomponent system, but care must be taken to avoid inferences based on highly correlated factors in the experiments. Experimental design can now be used to probe a wider range of experimental conditions and avoid highly correlated conditions now that we have shown that machine and statistical learning methods are well suited to modeling and predicting complex phase behavior in the presence of crystallographic screens. The ability of the models to predict the impact of the crystallization screen components on the resulting nanophases formed will significantly accelerate the development of future in meso crystallization systems for structural studies of MPs.
4. CONCLUSION Our work demonstrates that the complex time-dependent phase behavior of amphiphilic nanostructured mesophases under the influence of diverse crystallization screens can be modeled with high accuracy. A Bayesian regularized nonlinear neural network can model the phase behavior of the system and predict the behavior of systems not used in the training model. This technique is relatively simple to apply and is the only predictive modeling method currently available. The concentrations of PEG and citrate as well as the time scale were suggested by our models to be the most important descriptors driving the existence of the diamond and gyroid inverse-bicontinuous phases. The inclusion of other phase indicators in the inputs greatly improves the predicting accuracy of the inverse-hexagonal HII phase.
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AUTHOR INFORMATION
Notes
The authors declare no competing financial interest.
ACKNOWLEDGMENTS The authors gratefully acknowledge the financial support from the CSIRO Advanced Materials Transformational Platform. T.C.L. acknowledges support from the Chemical Structure Association Trust Jacques-Émile Dubois Award. In addition, the authors acknowledge the use of the C3 Collaborative Crystallisation Centre, CSIRO, Parkville, Australia, and the SAXS beamline at the Australian Synchrotron for collection of data used in this study. Dr. Xavier Mulet is acknowledged for producing Figure 1 in this manuscript.
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REFERENCES
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