Universal characteristics of chemical synthesis and property optimization

Abstract

A common goal in chemistry is to optimize a synthesis yield or the properties of a synthesis product by searching over a suitable set of variables (e.g., reagents, solvents, reaction temperature, etc.). Synthesis and property optimizations are regularly performed, yet simple reasoning implies that meeting these goals should be exceedingly difficult due to the large numbers of possible variable combinations that may be tested. This paper resolves this conundrum by showing that the explanation lies in the inherent attractive topology of the fitness landscape specifying the synthesis yield or property value as a function of the variables. Under simple physical assumptions, the landscape is shown to contain no suboptimal local extrema that could act as traps on the way to the optimal outcome. The literature contains broad evidence supporting this “OptiChem” theory. OptiChem theory implies that increasing the number of variables employed should result in more efficient and effective optimization, contrary to intuition.

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1. Introduction

Two general goals in chemistry are to optimize synthesis yields and the properties of the products. With several hundred years of chemical experience to draw on, these optimization goals are consistently achieved for many different objectives. Synthesis and property optimization is widely successful even though there is no a priori reason to expect that optimization in chemistry should be easy, or even practically feasible. On the contrary, simple reasoning suggests that optimization in chemistry should be exceedingly difficult. Importantly, the notion of the observed “ease” of performing optimization in chemistry refers to considering the apparent mathematical complexity of the tasks, as explained below, and not to the often difficult and time consuming overhead of setting up and performing individual experiments.

In a typical reaction optimization, the goal is to obtain the maximal product yield by changing a small number of suitable variables, such as the concentration of reagents and catalysts, the choice of solvent(s), reaction temperature, etc. In addition, when kinetics plays a role, the reaction time may also be a variable. Even when using chemical intuition to choose which values of the variables to sample, it may be expected that finding the absolute best values that maximize reaction yield should require testing the majority of their possible combinations. Typically, no more than ∼5–10 variables are chosen in order to avoid the “curse of dimensionality”,¹ which states that the number of possible experiments (i.e., unique combinations of variable values) grows exponentially with the number of variables. The large number of possible independent experiments could render reaction optimization practically infeasible as the size of the pool of variables rises.

Contrary to the reasoning above, experience shows that synthesis and property optimizations in chemistry are far easier to achieve than the curse of dimensionality suggests. Furthermore, quantitative evidence for optimization being extremely efficient in chemistry comes from many areas. Typically, “smart” optimization methods such as factorial design,² directed evolution,^3,4 or genetic algorithms⁵ are used to find the best values of the variables. Efficient optimization has been observed in organic synthesis,^6–8 discovery of functional proteins,^9–26 optimization of catalytic activity,^27–39 and the properties^40–50 of materials. The typical numbers of variables and resulting possible experiments, as well as the number of experiments actually performed for these objectives are shown in Table 1.

Table 1 Evidence from diverse areas of synthesis and property optimization shows that a very small fraction of the number of possible experiments is typically performed to achieve the desired outcome

Efficient Optimizations in Chemistry
Goal	Method	Number of variables	Number of possible experiments	Number of required experiments	References
Synthesis yield	Factorial design	2–10	∼10^2–10^4	∼10–100	6–8,66–89
Protein function	Directed evolution	2–100	∼20^2–20^100	∼10–10^5	9–26
Catalytic activity	Genetic algorithm	4–20	∼10–10^11	60–1000	27–39
Material properties	Genetic algorithm	3–10	∼10–10^23	100–1000	40–50

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The success of chemistry is often attributed to effectively employing practical “rules” governing chemical properties. However, aside from empirical evidence arising from the use of these rules, the underlying reason for why chemistry beats the curse of dimensionality has remained a puzzle. Since this behavior is evident across diverse disciplines in chemistry, as shown in Table 1, the basis for beating the curse of dimensionality must have a generic foundation rooted in some fundamental principles. In order to address this issue, we express optimization of a chemical reaction or property in terms of a fitness function J (e.g., percent synthesis yield), which is dependent on the values of the variables (e.g., concentration of reagent, choice of solvent, etc.). The functional relationship between J and the variables employed defines the fitness landscape.

A synthesis or property optimization may be expressed as an excursion over the fitness landscape, with the goal of finding the absolute best value of J. The topology of the fitness landscape (i.e., number and location of maxima, minima, and saddle points) plays an important role in determining whether optimization will be easy, or even feasible. In many areas of chemistry, fitness landscapes are believed to be rugged (i.e., exhibit many local minima and maxima), for example in the context of optimization of synthesis conditions,⁵¹ material properties,³³ and protein function.⁵² The assumption of ruggedness is intuitively reasonable, considering the complex relationship between a synthesis yield or material property and the variables involved.

The concept of a “landscape” in chemistry is commonly associated with free energy landscapes. If minimization of free energy is the goal, for example in the case of protein folding,⁵³ then optimization takes place on the free energy landscape where the variables are the atomic coordinates (e.g., torsional angles).^54–56 In the latter case, such landscapes inherently have constraints on the variables because the atoms are of fixed type and bonding arrangement. Constrained free energy landscapes are known to have a rugged topology, with the number of local minima increasing exponentially with the number of atoms.⁵⁷ In the context of this work, free energy landscapes are a particular class of fitness landscapes. For a general chemical synthesis or property optimization landscape, the constraints can be mild or lifted because the composition may be varied, as well as processing conditions such as temperature and reaction time. This paper considers the topological analysis of these general chemical fitness landscapes. The effects of significant imposed constraints will be considered in this context.

The effect of the topology on the ease of optimization is shown in a graphical illustration of a simple two-dimensional landscape J(x,T) in Fig. 1, where x represents the concentration of a reagent and T is temperature. The landscape in Fig. 1 (a) contains one global maximum and two “traps” at locations M, X₁, and X₂, respectively. Since searching on this landscape through intuition or other means can easily lead to a suboptimal solution (e.g., beginning from the values of x and T corresponding to point B in Fig. 1 (a), leading to the trap X₁), a significant portion of the possible variable values would need to be sampled in order to assure finding the global maximum M. In contrast, a trap-free landscape is often easy to climb, as shown in Fig. 1 (b). The absolute best fitness can be reached when starting from almost any point on the landscape, as indicated by the two paths C and D leading to two equivalent optima at locations M₁ and M₂, and an exhaustive sampling of all combinations of x and T is unnecessary. Thus, ascertaining the true nature of the topology for the fitness landscape governing chemical optimization is important for determining the ease of finding an optimal solution.

Fig. 1 Schematic of two possible generic classes of fitness landscapes illustrated with two variables x (concentration of a reagent) and T (temperature), where J(x,T) represents the percent synthesis yield. The landscape in (a) contains three maxima, one of which is globally optimal (M) and two that are suboptimal (X₁ and X₂). The landscape in (b) contains two homologous globally optimal maxima M₁ and M₂ linked by a saddle point S. Constraining the variables, for example fixing the temperature to 40 °C, introduces false traps T₁ and T₂.

In this work, we show that the inherent topology of fitness landscapes in chemistry explains why the curse of dimensionality can be beaten and identifies practical procedures that can further accelerate chemical optimization. Surprisingly, the topology of fitness landscapes in chemistry may be generically determined, without reference to any specific reaction or property for optimization. This paper will refer to the analysis of fitness landscape topology and its consequences as “OptiChem” theory. In addition to the basic considerations that the maximum synthesis yield is 100% and that physical properties are generally bounded, the conclusions of OptiChem theory rest on two basic assumptions. (i) The optimization goal must be well-posed, meaning that the desired value of J is attainable by some combination of the chosen variables. An example of an ill-posed objective would be a synthesis target containing an atom in an unattainable valence state. (ii) The number and range of the variables must be large enough so as to ensure that free movement over the landscape is permitted, i.e., there are no practical constraints on the variables. For example, imposing the constraint of fixing the temperature at 40 °C while varying the concentration x on the landscape shown in Fig. 1 (b) (the dashed vertical line) would result in encountering two apparent maxima T₁ and T₂, which are referred to as false traps. Satisfaction of the assumptions (i) and (ii) above provides the basis for the mathematical analysis establishing the fitness landscape topology in chemistry.

The remainder of the work is organized as follows: Section 2 presents the mathematical foundations of OptiChem theory. The latter material draws upon aspects of quantum mechanics and the reader may skip to Section 3, which states the predictions of OptiChem theory and provides an overview of evidence of chemical landscapes in the literature supporting these predictions. Section 4 provides an outlook towards the practical uses of OptiChem theory and offers concluding remarks.

2. Theory

The mathematical foundations of OptiChem theory rely on the general quantum mechanical description of an open system (i.e., a quantum system, such as a molecule, interacting with an environment) and draw from the properties of density matrices⁵⁸ and convex analysis.⁵⁹ The reader may choose to skip over this section along with the detailed Supplemental Material† and go directly to Section 3, where the consequences of OptiChem theory may be understood without the mathematical analysis.

We seek to determine the topology of the chemical fitness landscape J, which is the mapping of a suitable set of variables p = {p₁,p₂,…} to the fitness value (e.g., in the illustration of Fig. 1, the temperature is p₁ = T and the reagent concentration is p₂ = x, with the fitness value being J). The behavior of the fitness landscape J(p) with respect to the variables p is determined by its fundamental quantum-mechanical form⁶⁰

J(p) = Tr[ρ(p)O], (1)

where ρ(p) is the quantum state of the system (described by a density matrix ρ dependent on the variables p) and the operator O is associated with a laboratory observable (e.g., synthesis yield). The optimization goal is to identify a set of optimal variables p^* that produces the maximum value J^* = J(p^*).

The state ρ(p) may be determined by the action of a Kraus map (see Electronic Supplemental Material† for details) upon the initial density matrix ρ₀,⁶¹ for example where ρ₀ is the state of the system before the reaction occurs. The Kraus map is written in terms of a matrix K^p, where the superscript denotes the dependence on the variables p. For a system in state ρ₀ with m accessible energy levels, the matrix K^p can be expressed as

where each entry K^p_ij is an m × m dimensional matrix. The matrix K^p captures how the variables p determine the value of J. In this fashion, the matrices K^p_ij act collectively on the state ρ₀ to generate the state ρ(p). This state ρ(p) = ΦK^p(ρ₀) is an m × m matrix with the elements μ,ν = (1,2,...,m) given by , where 〈μν〉 = μ + m(ν − 1) and analogously for 〈στ〉, and K^p_{〈μν〉,〈στ〉} is the matrix element of K^p. (see Electronic Supplemental Material† for details). The dependence of K^p upon p could be very complex in many chemical applications; however, the mere existence of the matrix K^p and the ability to fully manipulate it are sufficient to establish the fitness landscape topology.

Using the above formulation, the fitness J in eqn (1) becomes

The set of all Kraus matrices generated by sampling the variables p is denoted as {K}. Upon satisfaction of the two assumptions (i) and (ii) in Section 1, it follows that {K} forms a convex set, meaning that for any Kraus matrices K₀ and K₁, along with any λ ∈ [0,1], their sum K_λ = λK₀ + (1 − λ)K₁ is also an acceptable Kraus matrix, as shown in the Electronic Supplemental Material†. Since J is a linear function of the Kraus matrices {K}, it also follows that J is a convex function over the set {K}.⁵⁹ Under these circumstances, convex analysis leads to the conclusion that the resulting fitness landscape contains no local suboptimal maxima to act as traps.⁶² Thus, there is always a steadily climbing path to the top of the landscape starting from any initial location (i.e., specified by the choice of reagents, processing conditions, etc.), as long as the assumptions (i) and (ii) in Section 1 are satisfied.^63,64 A similar analysis (shown in the Supplemental Material†) demonstrates that there are multiple (possibly connected) global optima of J.

3. OptiChem theory predictions and their assessment

In Section 2 and the Electronic Supplemental Material†, we proved the existence of a trap-free topology of the fitness landscape for chemical optimization, subject to the assumptions that the fitness objective is well posed and no practical constraints are imposed on the variables. We also demonstrated that when the above assumptions are satisfied, there exist multiple optimal solutions, i.e., different combinations of the variables can produce the same optimal synthesis yield or property value. These findings lead to three predictions of OptiChem theory:

(a) Using more variables should make optimization more effective and efficient: Since a trap-free landscape only exists with sufficient flexibility in the choice of variables, choosing a greater number and range of variables should accelerate the optimization process in terms of reaching the globally optimal fitness value. This conclusion is counterintuitive with respect to common belief and practice in chemistry. The wide success of chemical optimization with as few as two or three variables demonstrates that the number of variables required to form a trap-free landscape may be small in some cases.

(b) Observed suboptimal trapping indicates operation with significantly constrained variables: As a corollary to prediction (a), observation of traps on the landscape implies a violation of assumption (ii), such that some significant limitations are present on the variables, assuming that the objective is well-posed under assumption (i). For example, if the temperature is fixed to 40 °C in Fig. 1 (b), two apparent sub-optimal maxima T₁ and T₂ may be encountered, both of which are false traps.

(c) Homologous multiple optimal solutions may exist: The allowed presence of multiple global optima on the fitness landscape is consistent with the existence of “homologous” solutions to chemical objectives. For example, for the objective of finding a solvent that produces the fastest alkylation rate of sodium diethyl n-butylmalonate,⁶⁵ any polar aprotic solvent should produce similar high alkylation rates; these homologous solvents constitute multiple optimal solutions to the objective.

The degree to which the predictions of OptiChem theory hold in the laboratory may be assessed through examination of the extensive literature reporting chemical fitness landscapes. Several issues arise when considering OptiChem theory in practical laboratory optimizations. First, all synthesis or property optimization efforts necessarily constrain the number of variables to a modest set, which could introduce false traps on the landscape because the actual required number of variables can be much larger. Second, various intrinsic features of the variables can limit their dynamic range, e.g. solubility limits may constrain the allowed concentration of a reagent. Finally, the optimal value J^* may be unknown a priori for property optimization, and sometimes less than 100% molar yield in a synthesis is the maximum achievable value, thereby making it difficult to determine if traps are present. Despite these caveats, the overwhelming finding is that reported fitness landscapes are almost all trap-free. Although we cannot claim to have performed an exhaustive search of the literature, overall 142 separate fitness landscapes were identified,^66–130 with 123 appearing trap-free and with 19 containing traps.¹³¹ In some of the latter cases, the traps can be attributed to variable constraints explicitly discussed by the authors; the remaining works make no mention of the presence or absence of traps. Trap-free landscapes have been reported for the chemical synthesis and property goals listed in Table 2. From the list in Table 2, four illustrative studies producing fitness landscapes are summarized in this work; similar landscapes are observed in the references above.

Table 2 Evidence for trap-free fitness landscapes from the chemical literature, grouped by optimization goal. In the synthesis objectives indicated by a ^*, the goal is an organic compound, and either enzymatic or biological catalysts (e.g., yeast or bacteria) are employed; the concentration of the catalyst is one of the variables

Trap-free Landscapes in the Chemical Literature
	Optimization Goal	References
Synthesis	Organic compounds	66–74
	Polymers	75–77
	Enzyme-catalyzed^*	78–83
	Biologically catalyzed^*	84–89
	Material catalytic activity	90–104
Material Properties	Luminescence	105–115
	Color	116,117
	X-ray spectral structure	118,119
	Mechanical constants	120–123
	Dielectric constants	124–126
	Electrical resistivity	127–130

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One of the most common objectives in chemistry is to find reaction conditions that produce the highest yield of a desired product. Many studies use methods such as Design of Experiment^6,7 to optimize reaction conditions by sampling the available search space and fitting the data points to a polynomial, generating a fitness landscape, while others plot experimental yields directly as a function of the variables. The reported studies almost always produced trap-free landscapes of reaction yield with respect to the variables. Most reported fitness landscapes, obtained by both piecewise linear interpolation and polynomial fitting, contain a single maximum and no saddle points. A typical example of a landscape obtained by direct interpolation of experimental points is shown in Fig. 2 (a), where the lipase-catalyzed production of isopropyl esters of sunflower oil was optimized using enzyme concentration and the molar ratio of oil to alcohol substrates as variables.⁸¹ Some fitness landscapes also contain saddle points. For example, the optimization of palladium-catalyzed cyanation of aryl bromides using the ratio of the ligand [(t-Bu)₃PH]BF₄ to Pd metal and added volume equivalents of water as variables¹³² revealed a landscape with two disconnected maxima and a saddle point, as shown in Fig. 2 (b).

Fig. 2 (a) Yield of isopropyl ester as a function of catalyst concentration C(%) and substrate molar ratio (MR). The landscape is obtained by linear regression of experimental data points.⁸¹ (b) Yield of cyanation product of aryl bromides as a function of ligand : Pd ratio and volume equivalents of water added.¹³² The landscape is obtained from a polynomial fit of the data points.

Optimization of solid-state catalytic activity is another common goal. Catalytic activity landscapes are often constructed using the mole fraction of the individual catalyst components as variables. For example, the oxidation of isobutane to methacrolein, isobutene, and CO₂ was examined using ternary metal catalysts, where the most effective catalysts contained Mo, V, and Sb as variables.⁹⁰

Each of the three resulting activity landscapes for the respective products was trap-free, and a large optimal domain of functionally homologous catalysts is present on the landscape for methacrolein, as shown in Fig. 3 (a), consistent with prediction (c) of OptiChem theory. The landscape for isobutene formation with Mo, V, and Sb as variables is shown in Fig. 3 (b) to contain a single optimal point. When Sb was replaced by Bi, the isobutene landscape was found to contain traps (Fig. 3 (c)), but the maximal catalytic activity from the Mo–V–Bi landscape is reduced to 160% of the reference catalyst from over 1200% of the reference for the Mo–V–Sb library. The suboptimal fitness and presence of false traps on the landscape shows that the choice of Bi instead of Sb produces a significant constraint on the variables, in accordance with prediction (b) of OptiChem theory.

Fig. 3 Relative catalytic activity (compared to a fixed literature catalyst) for oxidation of isobutane to form (a) methacrolein with a Mo–V–Sb oxide library, (b) isobutene with a Mo–V–Sb library, and (c) isobutene with a Mo–V–Bi library.⁹⁰ The landscape for methacrolein formation in (a) contains a large optimal set of functionally homologous solutions with the same activity. The landscape for isobutene formation using Sb is trap free, but that using Bi contains false traps.

Various properties of molecules and materials are also common targets for optimization. For solid-state materials, the variables are often the mole fractions of the components, producing landscapes similar to the catalytic activity landscapes discussed above. Some property optimizations producing trap-free fitness landscapes include luminescence following optical excitation, X-ray spectral structure, mechanical properties, and electrical properties. For optimization of molecular properties, the variables are frequently the functional groups at two or more substitution sites on a molecular scaffold. For example, electrochemical properties were optimized for tetramers with functional group substitution at one site on each monomer unit, for a total of four variables.¹³³ Plotting the variables (i.e., substituents) in order of electron donating ability revealed a fitness landscape that is trap-free to within experimental noise (Fig. 4) and contains a large domain of homologous molecules exhibiting low oxidation potential (red, lower right-hand corner), in accordance with prediction (c).

Fig. 4 The first oxidation potential of oligomers with respect to the electron donating character of substituents.¹³³ The substituents are grouped as R₁/R₂ (vertical axis) and R₃/R₄ (horizontal axis) on the same scaffold structure, which resulted in the trap-free landscape shown in the figure. Functionally homologous sets of substituents are found at low oxidation potential (red). The grey squares [a] denote unsynthesized compounds.

4. Discussion

Prior to addressing practical implications of OptiChem theory, it is informative to examine apparent exceptions to the proof of trap-free landscape topology, with a notable example being constrained free energy landscapes, e.g. for protein folding. In the latter case, the fixed nature of the atoms and their bonding arrangement constitutes a severe constraint on the variables. Thus, the documented presence of many traps on these landscapes^53,57,134 is not an exception to OptiChem theory, but rather a situation in accordance with prediction (b) of OptiChem theory in Section 3. The efficiency of finding global free energy minima for these landscapes has been attributed to a “funnel” structure containing thermally surmountable small energy barriers separating local traps from the global minimum.^54,55,135 Thus, under the right circumstances, general synthesis and property landscapes with mild traps arising from constraints may be successfully traversed, especially with guidance by stochastic algorithms operating in an analogous fashion to thermal fluctuations on free energy landscapes.

The surprising observation is that OptiChem theory's prediction of finding trap-free landscapes widely holds for most chemical applications in the presence of seemingly severe constraints. One possible explanation is the opportunity to draw on hundreds of years of collective experience to effectively choose a good set of variables and then proceed to systematically find their optimal values. The evident broad scale success of the latter optimizations, however, can only readily occur because the underlying fitness landscape topology for chemical optimization is so favorable, as set out by OptiChem theory.

OptiChem theory has important practical implications that may improve upon current operations for chemical optimization procedures. The requirement of employing sufficiently flexible variables to take full advantage of the attractive landscape topology leads to the conclusion that optimization should become more effective and efficient as the number and range of the variables increases, contradicting intuitive expectations and the curse of dimensionality. OptiChem theory thus implies, in accordance with prediction (a) above, that the most efficient method for optimizing nominally complex chemical objectives is to simultaneously change all important variables, which is likely best performed with automated high-throughput synthesis and assaying machines guided by advanced pattern-recognition algorithms.^136,137 Finally, the success of chemistry is often attributed to the existence of a modest number of “rules” having wide applicability, where traditional discovery of such rules typically follows from lengthy empirical observations. OptiChem theory opens up a systematic means to identify new rules using the metric of landscape topology as a guiding principle. For example, previously unknown structure–property relationships might be deduced based on seeking a small set of variables that produce a trap-free property landscape, as was shown in Ref. 133.

The landscape concepts leading to OptiChem theory were originally developed in the context of laser control of quantum systems.^138,139 In the latter case, the goal is to achieve some specified behavior of the target quantum system, such as selective bond breaking,¹⁴⁰ maximal fluorescence from a chromophore,¹⁴¹etc.. The variables controlling the quantum system specify the features of a laser field that steers the quantum system to the desired behavior. In contrast to the situation for OptiChem theory, the chemical constituents (e.g., atoms, molecules, or materials) are normally not viewed as variables in laser control. The landscape for laser control with the field structure specifying the variables was demonstrated to have a trap-free topology, and now OptiChem theory shows that the concepts extend to more general optimization in chemistry even without laser fields. Importantly, the diverse domains of laser control and OptiChem share a common landscape topology readily amenable to optimization. Furthermore, chemical optimization and laser control could be combined in order to achieve demanding objectives such as the discovery of materials with specific optical properties.

The chemical sciences are arguably one of the most important endeavors garnering enormous practical benefit. While success can be achieved by employing rules and intuitive simple optimization procedures, left wanting has been a basic explanation for why chemistry is much easier to optimally perform than intuition would suggest. The inherent trap-free chemical landscape topology revealed by OptiChem theory provides the foundation to finally answer this fundamental question.

Acknowledgements

The authors acknowledge support from NSF grant CHE-0718610 and DOE grant DE-FG02-02ER15344. K.W.M. acknowledges the support of an NSF graduate student fellowship. J.D. was supported, in part, by U.S. Department of Energy Contract No. DE-AC02-76-CHO-3073 through the Program in Plasma Science and Technology at Princeton. A.P. acknowledges partial support from RFFI 08-01-00727a, NS-7675.2010.1, and EMALI European Union Marie Curie Teaching-Research Network Contract No. MRTN-CT-2006-035369

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Footnote

† Electronic supplementary information (ESI) available: Proof of fitness landscape topology. See DOI: 10.1039/c0sc00425a

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