Optimal sequencing for drug discovery in Ewing’s sarcomaDiana Negoescu, Peter Frazier, Warren B. Powell

Department of Operations Research and Financial Engineering, Princeton UniversityJeffrey A. Toretsky, Sivanesan Dakshanamurthy

Georgetown University

Approach

AcknowledgementsThe research was performed under the supervision of Peter Frazier and Prof. Warren Powell at Princeton University, and Professors Jeff Toretsky and Sivanesan Dakshanamurthy at Georgetown University. We also thank Dr. Andrew Mulberg for providing the introduction.

For further information…Please contact me at negoescu@princeton.edu. I would be happy to share with you more on the current state of my thesis research.

Conclusions and Future Work

IntroductionEwing’s sarcoma is a small round-cell tumor typically arising in the bones, and rarely in soft tissues, of children and adolescents.

In the US, 650-700 children and adolescents younger than 20 years of age are diagnosed with bone tumors each year, of which approximately 200 are Ewing's sarcomas (Ries et al. (1999)).

The 5 year survival rate is of about 58%, but children with the metastatic disease at diagnosis have a much lower prognosis: 18 – 30% (Shankar et al. (2003)).

It has been discovered recently that, genetically, Ewing's sarcoma is the result of a translocation between chromosomes 11 and 22, which fuses the EWS gene of chromosome 22 to the FLI1 gene of chromosome 11 (Owen et al. (2008)).

A medical research group at the Lombardi Cancer Center at Georgetown University has selected a chemical compound as a candidate for treating Ewing's sarcoma. This chemical operates by preventing two proteins, RNA Helicase and EWS-FLI, from binding with each other, thus stopping the spread of the disease. The research group is now searching for derivatives of this compound that could block binding with even greater efficiency. However, synthesizing each compound takes a few days, and there is a very large number of molecules that could be tested.

Our problem: given the data we have available thus far, and taking into account that molecules with similar structures might have similar properties, can we systematically tell which compound to test next?

Assessing the value of a molecule

The Correlated Knowledge Gradient (CKG) (Frazier et al. (2007))

Results for Free Wilson Model

How CKG works

Results Modified Free Wilson ModelNon-Informative Prior

Informative Prior

Non-informative Prior

Informative Prior

Fig. 1 Child with Ewing’s sarcoma

Fig. 2 Five year survival rates

Fig. 4 Ewing’s sarcoma cells

Fig. 5 Sarcoma of the femur

Fig. 6 Lab equipment at the Lombardi Comprehensive Cancer Center

Substituents -> H F Cl Br I Me H F Cl Br I Me Compound # 1 1 0 0 0 0 0 1 0 0 0 0 0 2 1 0 0 0 0 0 0 1 0 0 0 0 3 1 0 0 0 0 0 0 0 1 0 0 0 4 1 0 0 0 0 0 0 0 0 1 0 0 5 1 0 0 0 0 0 0 0 0 0 1 0 6 1 0 0 0 0 0 0 0 0 0 0 1 7 0 1 0 0 0 0 1 0 0 0 0 0 8 0 1 0 0 0 0 0 1 0 0 0 0 9 0 1 0 0 0 0 0 0 1 0 0 0 10 0 1 0 0 0 0 0 0 0 1 0 0 11 0 1 0 0 0 0 0 0 0 0 1 0 12 0 1 0 0 0 0 0 0 0 0 0 1 13 0 0 1 0 0 0 1 0 0 0 0 0 14 0 0 1 0 0 0 0 1 0 0 0 0 15 0 0 1 0 0 0 0 0 1 0 0 0 16 0 0 1 0 0 0 0 0 0 1 0 0 17 0 0 1 0 0 0 0 0 0 0 1 0 18 0 0 1 0 0 0 0 0 0 0 0 1 19 0 0 0 1 0 0 1 0 0 0 0 0 20 0 0 0 1 0 0 0 1 0 0 0 0 21 0 0 0 1 0 0 0 0 1 0 0 0 22 0 0 0 1 0 0 0 0 0 1 0 0 23 0 0 0 1 0 0 0 0 0 0 1 0 24 0 0 0 1 0 0 0 0 0 0 0 1 25 0 0 0 0 1 0 1 0 0 0 0 0 26 0 0 0 0 1 0 0 1 0 0 0 0 27 0 0 0 0 1 0 0 0 1 0 0 0 28 0 0 0 0 1 0 0 0 0 1 0 0 29 0 0 0 0 1 0 0 0 0 0 1 0 30 0 0 0 0 1 0 0 0 0 0 0 1 31 0 0 0 0 0 1 1 0 0 0 0 0 32 0 0 0 0 0 1 0 1 0 0 0 0 33 0 0 0 0 0 1 0 0 1 0 0 0 34 0 0 0 0 0 1 0 0 0 1 0 0 35 0 0 0 0 0 1 0 0 0 0 1 0 36 0 0 0 0 0 1 0 0 0 0 0 1

Position X Position Y

Two methods can be used:

• The BIAcore method: detect optically if the target protein binds with the compound (Raghavan & Bjorkman (1995)). This technique is accurate, but is difficult to perform because the compounds tend to aggregate when in the fluid.

• Protein displacement: combine the target protein with the chemical compound to be tested, and then mix with the secondary protein. Move the second protein into a second container, and see if any target protein has moved along with it (Angelakou et al. (1999)). This technique is less accurate than the BIAcore method.

Fig. 7 BIAcore machine

Modeling the relationship between the structure and the value of a molecule

Define

• a substituent to be an atom or group of atoms substituted in place of a hydrogen atom on the parent chain of a hydrocarbon. The molecule in Fig. 8 has two positions, X and Y, at which substituents can be attached;

• ai as the contribution of substituent i; si is an indicator variable whose value is 1 if substituent i is present and 0 otherwise;

• μ as the biological activity value of the unsubstituted parent structure.

Fig. 8 molecule of

disubstituted

N,N- Dimethyl-α-

Bromophenethylamines

Free Wilson Model Modified Free Wilson ModelAssumptions:

• each substituent has a strictly additive contribution

• contributions of any two different substituents are independent.

Model the value of a compound as

V = Σaisi + μ

Model the covariance between compounds i and j as

Cov(i,j) = ΣlVar(ai), where l is a counter over all common substituents to compounds i and j.

Assumptions:

• substituents do not have additive contributions

• contributions of any two different substituents are independent.

Model the value of a compound as:

V = Σaisi + μ + b

Model the covariance between compounds i and j as

Cov(i,j) = ΣlVar(ai) + σb21{i=j}, where l is a counter

over all common substituents to compounds i and j.

Method

Bayesian approach:

• Assume we have a budget of N measurements;

• Assume measurements come from a multivariate normal distribution;

• Start with a belief on the values of the compounds, given by a mean vector μ and a covariance matrix Σ;

1. Decide what to measure and make the measurement;

2. Update the mean vector μ and the covariance matrix Σ;

• Repeat steps 1 and 2 until all N measurements have been made.

The measurement decisionMake each decision so as to maximize the increase in knowledge (the gradient) from measuring a specific compound. Mathematically, this is

where Sn is the belief state after measurement n, and x is a compound.

Fig. 9 Illustration of KG for independent measurements

The CKG policy chooses the molecule x that maximizes νCKG,n, which is the amount by which the solution is expected to improve, and is illustrated for the case of independent measurement as an example in Fig. 9. In the example, choice 4 has the current highest mean, but choosing alternative 5 could improve what we believe to be the best value. The shaded area under the Gaussian curve is the probability that choice 5 is better than the current best value, and the knowledge gradient is the expected amount by which the new best value will increase if we choose compound 5.

When updating our belief, we keep in mind that measuring a compound teaches us something about other compounds that share its substituents.

Fig. 10 Compounds’ Representation

Fig. 11 The molecule that generates the compounds of Fig. 10

Fig. 12 First 4 measurements in a sequence of 19 measurements made by the CKG algorithm under the Pure Free - Wilson model for the 36 compounds data set shown in Fig. 10. After each measurement, not only does the variance of the measured compound decrease, but also the variances of the compounds that share a substituent with it.

Fig. 13 Measurements 7 -10 of the sequence started in Fig. 12

Fig. 14 Measurements 11-14 of the sequence started in Fig. 12

Fig. 15 A sample path using a data set of 1000 compounds, when our initial belief has a high uncertainty (non-informative prior). We plot the opportunity cost after each measurement, defined as the difference between the true best value and the true value of the current best compound.

If the chemists have an idea about the mean and variance of the substituent contributions, use these values as an informative prior.

Let m be the mean of the substituent contributions, mainMol the value of the unsubstituted molecule, v the variance of the substituent contributions. The initial belief is:

where i is the number of substituents present in compound x, j is the number of substituents common to compounds x and x’, and R is a noise term simulating the error in the prior belief about the value of the unsubstituted molecule.

Fig. 16 Sample path using the informative prior.

The best compound is found after about 60 measurements.

Best compound found after 15 measurements.

Fig. 17 The true values of the compounds chosen in the sample path at each step.

Fig. 18 Four sample paths using the Non-informative prior. The best compound is found after about 55 measurements.

Fig. 19 Four sample paths using the informative prior. Best compound is usually found after about 50 measurements.

• Results so far indicate that the CKG algorithm could be used to improve efficiency in drug discovery for Ewing’s sarcoma. This conclusion is made assuming that the additive Free-Wilson model is accurate.

• The current procedure requires enumerating all possible compounds, limiting its application to small molecules (< 1000 combinations).

• We are working on methods which can handle on the order of 1000 parameters, making it possible to handle molecules with millions of combinations.

• Further research needs to consider more realistic models than Free-Wilson.

ReferencesAngelakou, A., Valsami, G., Macheras, P. & Koupparis, M. (1999), ‘A displacement approach for competitive drug – protein binding studies’, European Journal of Pharmaceutical Sciences 9(2), 123-130.

Frazier, P., Powell, W.B. & Dayanik, S. (2009), ‘The knowledge-gradient policy for correlated normal rewards’, INFORMS Journal on Computing.

Frazier, P., Powell, W.B., & Dayanik, S. (2008), ‘A knowledge-gradient policy for sequential information collection’, SIAM Journal of Control and Optimization.

Free, S. & Wilson, W. (1964), ‘Contribution to structure-activities studies’, J Med Chem 7, 395-399.

Owen, L., Kowalewski, A. & Lessnick, S. (2008), ‘EWS/FLI Mediates Transcriptional Repression via NKX2. 2 during Oncogenic Transformation in Ewing’s Sarcoma’, PLoS ONE.

Raghavan, M. & Bjorkman, P. (1995), ‘BIAcore: a microchip-based system for analyzing the formation of macromolecular complexes’, Structure 3(4), 331-333.

Shankar, A., Ashley, S., Craft, A. & Pinkerton, C. (2003), ‘Outcome after relapse in an unselected cohort of children and adolescents with Ewing’s Sarcoma’, Medical and Pediatric Oncology 40(3), 141-147.