Predicting and optimizing asymmetric catalyst performance using the principles of experimental design and steric parameters

Article abstract of DOI:10.1073/pnas.1013331108

Using a modular amino acid based chiral ligand motif, a library of ligands was synthesized systematically varying the substituents at two positions. The effects of these changes on ligand structure were probed in the enantioselective allylation of benzald

Full text of DOI:10.1073/pnas.1013331108

Predicting and optimizing asymmetric catalyst
performance using the principles of experimental
design and steric parameters
Kaid C. Harper and Matthew S. Sigman1
Department of Chemistry, University of Utah, 315 South 1400 East, Salt Lake City, UT 84112
Edited by Eric N. Jacobsen, Harvard University, Cambridge, MA, and approved December 10, 2010 (received for review September 6, 2010)
Using a modular amino acid based chiral ligand motif, a library of
ligands was synthesized systematically varying the substituents at
two positions. The effects of these changes on ligand structure
were probed in the enantioselective allylation of benzaldehyde,
acetophenone, and methylethyl ketone under Nozaki-Hiyama-
Kishi conditions. The resulting three-dimensional datasets allowed
for the construction of mathematical surface models which de-
scribe the interplay of substituent effects on enantioselectivity for
a given reaction. The surface models were both extrapolated and
manipulated to predict the enantioselective outcomes of several
previously untested ligands. Analyses were also used to predict
optimal ligand structure of a minimal dataset. Within the dataset,
a linear free energy relationship was also discovered and a direct
comparison of both the linear prediction as well as the three-
dimensional prediction illustrates the potential predictive power
of using a three-dimensional model approach to asymmetric cata-
lyst development.
catalysis ∣ enantioselective ∣ modeling ∣ carbonyl allylation
centerpiece of modern organic chemistry is the development
Aof new catalytic enantioselective methods to obtain valuable
enantiomerically enriched synthetic building blocks (1, 2). Asym-
metric catalysis, in practice, initiates from the discovery of a new
catalytic reaction or identification of a reaction of interest. Sub-
Fig. 1. (A) Library of modular chiral oxazoline ligands for the enantio-
selective allylation of benzaldehyde under Nozaki-Hiyama-Kishi conditions.
(B) Three-dimensional scatterplot for the allylation of benzaldehyde where
the X and Y axes are the adjusted Charton steric parameter for positions X
sequently, a number of chiral ligand classes are experimentally
explored in hope of finding a “lead” which generates a promising
enantiomeric ratio (er). Further optimization of the reaction
conditions and ligand structure can ultimately yield a mature
catalytic asymmetric method. This process is highly empirical and
the results can be unsatisfactory for a given reaction. The empiri-
cism inherent in reaction development has been addressed by
computational chemistry via stereocartography and various other
methods (3–10). Even with the impressive impact computational
chemistry has had on the field, the small energy differences in the
diastereomeric transition states (∼2–3 kcal∕mol or 8.2–12.3 kJ∕
mol) leading to enantiomerically enriched products are not easily
rationalized. Additionally, these methods generally depend on a
detailed understanding of the parent chiral catalyst structure.
Furthermore, kinetic analyses of asymmetric catalytic reactions
often reveal the general complexities of catalysis and highlight
the importance of the Curtin-Hammett principle (11, 12), but
do not often expose the key catalyst structural features respon-
sible for enantioselection. These issues highlight an underlying
challenge in the field of asymmetric catalysis: how does one de-
sign a ligand for a given reaction type without engaging in a long-
term, empirical investigation of multiple ligand classes?
and Y in ligand 1 and the enantioselectivity described in terms of free energy
(ΔΔG‡) (kcal∕mol) is the Z axis.
distribution of enantiomers (R vs. S) is directly related to the dif-
ferences in free energy arising in the diastereomeric transition
states, we were able to correlate er with the corresponding Char-
ton steric parameter. We had hoped that the resulting LFERs
would be a useful tool in optimizing ligands as well as de novo
catalyst design. However, at the conclusion of these studies, we
were puzzled at the observed breaks in the linear correlations
mitigating the ability to predict the outcome of entirely new cat-
alyst structures by extrapolation (20, 21). Herein, we report a
more sophisticated approach to the problem in which a library
of catalysts is used to generate a three-dimensional relationship
of free energy to substituent size leading to successful predictions
of catalyst performance.
A goal of our program has been to utilize classic linear free
energy relationships (LFER) to predict the enantioselective
outcomes of new catalytic systems (13). We have recently dis-
closed the use of steric parameters originally developed by Taft
and modified by Charton (14–17) to quantitatively evaluate
ligand effects on enantioselectivity in several Nozaki-Hiyama-
Kishi (NHK) carbonyl allylations (18, 19). Because the product
Author contributions: M.S.S. and K.C.H. designed research; K.C.H. performed research;
M.S.S. and K.C.H. analyzed data; and M.S.S. and K.C.H. wrote the paper.
The authors declare no conflict of interest.
This article is a PNAS Direct Submission.
1To whom correspondence should be addressed. E-mail: sigman@chem.utah.edu.
This article contains supporting information online at www.pnas.org/lookup/suppl/
doi:10.1073/pnas.1013331108/-/DCSupplemental.
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PNAS ∣ February 8, 2011 ∣ vol. 108 ∣ no. 6 ∣ 2179–2183

Fig. 4. Comparison of predictions made by the LFER and three-dimensional
model to the experimentally determined values (er = enantiomeric ratio).
failed to substantiate any relationships between X and Y. The
dimensionality of the dataset led to the abandonment of this type
of linear regression analysis.
To approach this issue in a more rigorous manner, the three-
dimensional dataset required a three-dimensional function (a
surface) to accurately model the data. Inspection of the dataset
indicated that adequate surface models could be achieved
through simple polynomial functions. Polynomial models were
attractive due to their simplicity wherein the functions would con-
tain steric parameters for X and Y as the independent variables
and enantioselectivity (expressed in terms of the free energy,
ΔΔG‡) as the dependent variable. It quickly became evident that
the development of three-dimensional surface models would
be a highly iterative and subjective process, particularly when
using different algorithms to create models. In order to limit
the subjectivity involved, principles in experimental design were
employed (25–28).
The first key principle in fitting a polynomial model to the data
was modifying the Charton parameters. A problem with higher
order polynomials is that the greater the distance from the origin
the more difficult it becomes to distinguish second order from
third order character. To better determine the true polynomial
character of the dataset, the steric parameters were translated
from their reported values to new values which centered about
zero (Fig. 2A). This translation was accomplished by determining
the center point for the range of parameters used in the library
and setting it equal to zero. No information is lost in the transla-
tion as these parameters were originally based on relative rates
(14–17). New values for each substituent were calculated based
on the new center point with no change in the relative values.
Simply, the parameter values coincide with the center point in
our ligand libraries’ range along both the X and the Y axes.
The next key principle in developing a model to fit the data was
assembling a surface function. Polynomials were initially selected
through inspection of the raw two-dimensional data slices, obser-
ving linear and quadratic character. Full third order polynomials
of the form
Fig. 2. (A) Graphical representation of the translation of the original Char-
ton values to those used in this study. (B) Array of all 25 ligand substituent
values in the XY plane.
Results and Discussion
The modular ligand scaffold developed for the NHK allylation
presented us with the unique ability to vary the substituents at
both the X and Y positions of ligand 1 (Fig. 1A). It was hypothe-
sized that both the X and Y positions may contribute synergis-
tically to the selectivity of the system. Taking advantage of the
ligand’s modularity and the commercial availability of amino
acids, we initially synthesized a 25 member library (22–24).
The library was tested initially in the NHK allylation of benz-
aldehyde. Benzaldehyde was selected as a model substrate be-
cause we had not extensively examined this reaction with ligand
scaffold 1 (21). The library of ligands was evaluated at random
under the conditions shown in Fig. 1 and each data point
presented was reproduced. The raw data shown in Fig. 1B was
initially analyzed as data slices in two dimensions by holding
one variable constant and evaluating the variation in the other
variable. Two-dimensional analysis was simpler but ultimately
Fig. 3. (A) The raw data overlaid with the surface model given by ΔΔG‡ ¼ 0.931 þ 0.576 Y − 0.905X2 − 1.005Y2 − 0.502XY − 0.407X3 − 0.475YX2. R2 ¼ 0.78.
(B) Contour plot of the surface model. (C) Linear free-energy correlation for ligands with X ¼ Me.
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Fig. 5. (A) Pseudo 3 × 3 data array used in analyzing the optimal ligand structure for the allylation of benzaldehyde. The X ¼ tBu and Y ¼ CEt3 ligand has been
replaced with the X ¼ i Pr and Y ¼ CEt3 ligand. (B) The raw data for the allylation of benzaldehyde overlaid with the three-dimensional surface described by
the equation ΔΔG‡ ¼ 0.857 − 0.489X þ 0.813X2 − 0.361Y2 − 0.422XY − 0.571YX2. R2 ¼ 0.97: (C) Contour plot of the surface model.
shown in Fig. 2B. Experimental design dictates that to maximize
ΔΔG‡ ¼ z0 þ aX þ bY þ cX2 þ dY2 þ fXY þ gX3 þ hY3
statistical significance the data array should be symmetrical about
þ iXY2 þ jYX2
the center point of the graph (25–28). This example shows the
data slightly weighted toward the third quadrant. Nonetheless,
a surface model given by the following equation was constructed:
were the initiation point of each process. The coefficient values
(z0,a,b,c,d,f,g,h,i,j) were solved using multivariable linear least
squares regression analysis.* In order to perform the regression,
two matrices were mapped: (i) the design matrix consisting of the
independent variable values and (ii) the corresponding response
matrix of measured enantioselectivity described in terms of free
energy. The coefficient values could then be calculated according
to least squares regression
ΔΔG‡ ¼ 0.931 þ 0.576Y − 0.905X2 − 1.005Y2 − 0.502XY
− 0.407X3 − 0.475YX2:
Fig. 3A is a graphical representation of this equation and
Fig. 3B is a contour plot of the same model. Close inspection of
the data revealed a LFER for the ligand series in which X ¼ Me
(Fig. 3C). The LFER predicts that installing a very large group at
the Y position would result in a highly enantioselective catalytic
system (Fig. 4). The three-dimensional surface prediction differs
from the linear estimate considerably, predicting a more modest
enantiomeric ratio (er). Synthesis and evaluation of the ligands
bearing a much larger group on Y results in a breakdown of
the linear model but to our delight the three-dimensional analysis
accurately predicts the performance of three new catalysts.
By using the entire dataset, the outcomes of new catalysts can
be predicted accurately through extrapolation. The magnitude of
the extrapolation along the Y axis is almost double the data range
indicating the robustness of this model. However, as one removes
data from this model, the number of degrees of freedom in
the analysis is reduced and the error of the prediction becomes
greater. Additionally, the model relies on crossterms of X and Y
providing evidence of the hypothesized synergistic effect.
While extrapolation can yield important results, the power
of using experimental design lies in the ability to simply define
maxima and minima within a given domain with statistical signif-
icance. In the realm of asymmetric catalysis, the most powerful
application would be to quickly identify the optimal ligand struc-
ture from a set of systematically varied ligands. However, synth-
esis of 25 or even 16 ligands to determine optimal structure may
be impractical. Therefore, analysis of a much simpler 3 × 3 design
matrix was performed including the newly synthesized ligands
where Y ¼ CEt3. The design of this matrix encompasses all of
the synthetically practical values for X and Y and the full domain
of available Charton parameters. It should be noted that the
adjusted Charton values have changed to include this expanded
domain as compared to the analysis of the 5 × 5 matrix (Fig. 5A).
The entry for X ¼ tBu and Y ¼ CEt3 is missing because that
ligand proved impractical to synthesize and has been substituted
with the X ¼ i Pr and Y ¼ CEt3. A ramification of this substitu-
tion is that areas of the model beyond this value would carry less
significance. A surface model given by the equation was deter-
mined:
C ¼ ðXTXÞ ðXTYÞ;
−1
where C is the matrix of calculated coefficients, X is the design
matrix, XT is the transpose of the design matrix, and Y is the
response matrix. The preliminary model was then simplified by
eliminating terms with significant covariance and simultaneously
maximizing the goodness of fit. The goodness of fit was deter-
mined primarily through the errors associated with the coefficient
values but other relevant statistics including R2 and ANOVA
analysis were taken into account.
Some considerations in our method of analysis warrant discus-
sion. The first consideration is that Charton steric parameters
are not continuous. As an example, there is no value between
H and Methyl (Me). Additionally, purely carbon frameworks
at the X and Y position were solely evaluated because the inclu-
sion of heteroatoms might perturb the system in ways not related
to their steric influence. These issues limit the ability to use the
principles of experimental design in their purest form. The sec-
ond major consideration is the synthetic availability of ligands. It
was observed in the synthesis of the X ¼ tBu series of ligands, that
reactions proceed prohibitively slow for groups larger than
Y ¼ tBu. The final consideration is that the complexity of the
polynomial models might not arise purely from steric interactions
in the transition state. Some of the complexity contained in the
models might arise from deficiencies is the parameters used
†
(29) .
Initially, a model was developed based on all 25 ligands eval-
uated. The alignment of data points collected in the XY plane is
*For more details regarding how equations were solved and for step-by-step walkthrough
of how models were developed, the reader is referred to the SI Appendix.
†In the following paragraphs, we will describe the methods used for different subsets of
data to make meaningful predictions. All of the models described below were derived
using the process described above and in the SI Appendix. Statistical analyses were also
made on each individual model and we refer the reader to the SI Appendix for a step by
step example of how the analyses were performed.
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evaluated (Fig. 6 A and B). The resultant linear correlation gives
a slope of 0.93 where 1.00 would be a perfect fit. This excellent
correlation confirms the reliability of the model. For similar ana-
lysis of all of the models, see the SI Appendix.
Next, we moved to the allylation of acetophenone under
conditions shown in Fig. 7A. Development of a model for the
3 × 3 dataset described above led to the following equation:
ΔΔG‡ ¼ 1.282 þ 0.034X − 0.724X2 − 0.432Y2 − 0.073XY
þ 0.240Y3:
The surface and contour graphs in Fig. 7 present a maximum
which is nearest the modified Charton values for ligand 1d.
The model correctly predicts the catalyst which we had previously
optimized and published for the allylation of a variety of aromatic
ketones including acetophenone (21). Again, this ligand was not
utilized in the analysis. An interesting feature of this surface is
that it crosses zero on the Z axis representing a change in product
facial selectivity. In the initial studies, this ligand was found
primarily through empirical ligand evaluation. Even though this
ligand performed well, it was never known to be the optimal
ligand structure. A power of this three-dimensional modeling
method is the level of confidence in identifying the optimal cat-
alyst is considerably higher.
Finally, we applied this system to probe an interesting and
synthetically promising reaction class. The NHK allylation of
aliphatic ketones in Fig. 8 does not result in high selectivity using
our previously published ligands (and many analogs). Our efforts
to develop a useful, enantioselective variant have thus far been
unsuccessful. Therefore, we sought to evaluate the simplest ali-
phatic ketone, methylethyl ketone, with the ligand library and use
the results to guide the development of a highly enantioselective
catalyst. Using the approach described for the determination of
the optimal catalyst for both benzaldehyde and acetophenone,
the reaction using the simple 3 × 3 ligand set was evaluated. The
resulting data was modeled by the following equation:
Fig. 6. (A) Prediction vs. measured enantioselectivity for the evaluated cat-
alysts in benzaldehyde allylation. (B) Plot of measured vs. predicted ΔΔG‡.
ΔΔG‡ ¼ 0.857 − 0.489X þ 0.813X2 − 0.361Y2 − 0.422XY
− 0.571YX2:
ΔΔG‡ ¼ −0.200 − 1.036X − 0.177Y þ 0.294X2 þ 0.126Y2
þ 0.129XY þ 2.6120X3 þ 0.341YX2:
The local maximum of the graph and contour plot in Fig. 5 shows
that the optimal ligand for the reaction would be either of the
ligands in Fig. 6A (X ¼ Et or i Pr). Neither of these ligands
was used in the experimental data processed to create the surface
model. The predictions are lower than their experimentally
determined values; however, of all 29 ligands evaluated for this
reaction, these two ligands gave the highest er. Close comparison
of this new model with a smaller dataset as compared to the
model in Fig. 3 shows a modest change in the maximum of the
plot but a similar general shape. Both do predict that the ligand
with Y ¼ tBu and X ¼ i Pr as the optimal ligand structure.
To further validate the simpler 3 × 3 dataset model, the mea-
sured ΔΔG‡ was compared to the predicted ΔΔG‡ for the ligands
The model confirmed our empirical observation that the best
selectivity we could hope for using this ligand scaffold was a mere
40% enantiomeric excess. Interestingly, the best result obtained
in the screen was using the ligand where X ¼ Me and Y ¼ Me,
indicating that smaller features were potentially desirable. The
ultimate conclusion from this analysis is that the current ligand
scaffold will most likely not result in the desired outcome for this
difficult reaction. Therefore, we have since diverted our attention
to developing new scaffolds that may hold more promise. This
example illustrates that the three-dimensional model analysis
Fig. 7. (A) Predicted optimal catalyst for acetophenone allylation. (B) Surface described by the equation ΔΔG‡ ¼ 1.282 þ 0.034X − 0.724X2 − 0.432Y2−
0.073XY þ 0.240Y3R2 ¼ 0.87 (C) Contour plot of the surface model.
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the performance of previously unknown catalysts. Of potential
practical importance, a 3 × 3 matrix of ligands was evaluated
using this approach to successfully predict the best catalyst for
a given reaction even though this catalyst was not used in the
analysis. Additionally, evaluation of methylethyl ketone using the
three-dimensional modeling approach revealed the use of the
current ligand template would most likely not result in an effec-
tive asymmetric catalyst (29)‡.
This approach does not rely on precise knowledge of a cata-
lyst’s solution structure which is elusive in many reactions. To
employ a three-dimensional free energy relationship analysis re-
quires the ability to readily synthesize catalysts wherein at least
two structural variables can be independently and systematically
varied. One can envision using any thermodyanamic or kinetic
parameter of specific substituents [e.g., Hammett parameters
(30), cone angle, or pKa values] to construct three-dimensional
free energy relationships to facilitate catalyst design in broad
areas of catalysis. Additionally, one is not limited to three dimen-
sions as other reaction parameters (solvent, temperature, and
concentration) often used in experimental design could be eval-
uated simultaneously with ligand substituents and modeled to
find both the optimal ligand and conditions. These approaches
are currently under investigation in our laboratory in the context
of developing new and improved catalytic processes.
Fig. 8. Surface describing the allylation of methylethyl ketone ΔΔG‡
− 0.200 − 1.036X − 0.177Y þ 0.294X2 þ 0.126Y2 þ 0.129XY þ 2.6120X3
0.341YX2. R2 ¼ 0.97.
¼
þ
can be useful in determining the optimal ligand structure, but also
can show the limitations of ligand structure on the enantioselec-
tivity of a system.
ACKNOWLEDGMENTS. We would like to thank Professor Joel Harris for critical
discussions on data analysis. This work was supported by the National Science
Foundation (CHE-0749506).
In conclusion, we have developed a unique means to analyze
ligand steric effects on enantioselective reactions. A ligand library
with two independently tunable substituents was evaluated for
the NHK allylation of benzaldehyde, acetophenone, and methyl-
ethyl ketone. The resultant ΔΔG‡ derived from the measured
enantiomeric ratios were plotted vs. Charton steric parameters
for the two groups. Using the principles of experimental design
to manipulate and fit the data to a modified third order polyno-
mial provided a surface model that could successfully predict
‡It should be noted the cases presented here have utilized the manipulation of Charton
steric parameters. In these studies, meaningful data and results have been observed;
however, situations have been and may be encountered where these steric parameters
do not appear adequate (29). Charton’s correlation to Van Der Waals radii might be
insufficient in cases where conformational constraints limit free rotation about bonds.
One must take care in choosing steric parameters and substituents to incorporate into
the experimental design based on the problem at hand.
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