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# Pricing financial options using Monte Carlo simulations

In this blogpost we will dive into the world of finance, and specifically option pricing. We are going to illustrate how we can use Monte Carlo simulations to price financial options, by looking at two examples. In both of these examples no explicit solution formula exists, making the Monte Carlo approach useful in pricing these options.

If you are not familiar with Monte Carlo simulations or financial options, or could use a reminder, I suggest that you take a look at the post Introduction to Monte Carlo simulations and option pricing where we introduced all these concepts. In that post we:

• Introduced derivative assets and financial options, especially the European call option

• Looked at our first model for stocks, namely the standard Black&Scholes model

• Introduced the concept of Monte Carlo simulations

• Looked at our first pricing example

The examples in this post are definetely more challenging mathematically, so make sure that you have a grasp on the basics from the introductory post.

Let's get started.

## Arithmetic basket call in the standard Black&Scholes model

For our first example, we will use the standard Black&Scholes model, i.e. the same one used in the introduction post about Monte Carlo simulations and option pricing. Here the stock dynamics follow a Geometric Brownian Motion:

dS(t) = rS(t)dt + \sigma S(t)dW(t)

with the solution:

S(T) = S(0)e^{(r-\frac{\sigma^2}{2})T + \sigma W(T)} = e^{\log(S(0)) + (r-\frac{\sigma^2}{2})T + \sigma W(T)}

where W(T)\sim N(0,T), so S(T) has a log-normal distribution. The contract we will look at now, is very similar to a European call option, but instead of having just one underlying stock, we take the arithmetic average of several stocks (n of them). This contract is called an arithmetic basket call option, and has the following payoff at time T:

\Phi(S_1(T),...,S_n(T)) = \max\left((\frac{1}{n}\sum_{i=1}^n S_i(T)) - K, 0\right)

Even if we know the distribution of every stock S_i, we do not know the distribution of the arithmetic average of n of them (the log-normality is not preserved under summation). We can therefore not find an explicit formula like the Black&Scholes formula for European call options in this case for the option price:

\Pi(0) = e^{-rT}\mathbb{E}\left[\max\left((\frac{1}{n}\sum_{i=1}^n S_i(T)) - K, 0\right)\right]

Our solution will be to use Monte Carlo simulations. For simulation purposes, we will use that the log stock prices \log(S_1(T)),...,\log(S_n(T)) follow a multivariate normal distribution. For simplicity, we assume that all stocks have the same initial value S(0)=s, same volatility \sigma and that the correlation \rho between all the stocks are the same. The log stock prices will then follow a multivariate normal distribution with the following expected value vector \mathbf{\mu} and covariance matrix \Sigma:

\boldsymbol{\mu} = \begin{bmatrix} \log(s) + (r-\frac{\sigma^2}{2})T \\ \vdots \\ \log(s) + (r-\frac{\sigma^2}{2})T \end{bmatrix} \quad \text{and} \quad \boldsymbol{\Sigma} = \begin{bmatrix} \sigma^2 & \sigma^2\rho & \dots & \sigma^2\rho \\ \sigma^2\rho & \sigma^2 & \dots & \sigma^2\rho \\ \vdots & \vdots & \ddots & \vdots \\ \sigma^2\rho & \sigma^2\rho & \dots & \sigma^2 \end{bmatrix}T

Note that \boldsymbol{\mu} is an n\times 1-vector and \boldsymbol{\Sigma} is an n\times n-matrix. This is useful because we can now simulate from this multivariate normal distribution and then transform the values using the exponential function to get simulations of the stocks. We thus have the following Monte Carlo sampler for estimating the price of the arithmetic basket call option:

\begin{aligned} &\textbf{for } i=1 \textbf{ to } N \\ &\quad\quad \text{draw } [\log(S_1(T)),...,\log(S_n(T))]_i \sim MVN(\boldsymbol{\mu}, \boldsymbol{\Sigma}) \\ &\quad\quad \text{transform to } [S_1(T),...,S_n(T)]_i \text{ with the exponential function} \\ &\quad\quad \text{compute arithmeric average } (\frac{1}{n}\sum_{j=1}^n S_j(T))_i \\ &\textbf{end} \\ &\Pi(0)_N^{MC} = \frac{1}{N}\sum_{i=1}^N e^{-rT}\cdot\max\left((\frac{1}{n}\sum_{j=1}^n S_j(T))_i - K, 0\right) \end{aligned}

So, assuming that we have n=10 underlying stocks, that s=100, the strike price K=90, the risk free interest rate r=1\%, the stock volatility \sigma=0.2, the correlation rho=0.4 and that the time to maturity T=1 year, we can implement the following Monte Carlo sampler in Python:

Running a simulation (with N=100000) gives us the following price estimate and 95% confidence interval for the Monte Carlo estimated price:

\begin{aligned} \Pi&(0)_N^{MC} = 12.308 \\ I_{\Pi(0)} &= (12.236, 12.381) \end{aligned}

As mentioned in the first post, we will get different values if we were to repeat the simulations. So if you try and implement this by yourself you would not get exactly the same values, but they should however be very similar.

## European call option in the Heston model

When comparing actual options prices on various financial markets with the prices given by the standard Black&Scholes model, it becomes clear that they generally do not match very well. The standard Black&Scholes model is just not complex enough to explain the stock's price process. There are several adjustments one could make to get a better model. One major drawback of the standard Black&Scholes model is that the stock volatility is constant and deterministic, while the actual stock volatility is a stochastic process just like the stock's price process. We will therefore now look at a model with stochastic stock volatility, namely the Heston model.

In the Heston model, we have dynamics given as stochastic differential equations, for both the stock and the volatility. These dynamics looks as follows:

\begin{aligned} &dS(t) = rS(t)dt + \sqrt{V(t)}S(t)(\sqrt{1-\rho^2}dW^{(1)}(t) + \rho dW^{(2)}(t)) \\ &dV(t) = \kappa(\theta - V(t))dt + \sigma\sqrt{V(t)}dW^{(2)}(t) \end{aligned}

where S(t) is the stock's price process as before, and V(t) is the stock's volatility process. W^{(1)}(t) and W^{(2)}(t) are here two independent standard Brownian Motions. Also, r is the risk free interest rate, and \rho, \kappa, \theta and \sigma are model parameters.

We return to the European call option that we saw in the introductory post. As you might remember, this means that we want to find the option price:

\Pi(0) = e^{-rT}\mathbb{E}[\max(S(T)-K,0)]

In the standard Black&Scholes model we could find an explicit formula for this price. In the Heston model however, this is not the case. The reason is that we cannot find a closed form solution to the system of stochastic differential equations for S(t) and V(t) above. We will therefore need numerical methods and Monte Carlo simulations to generate a sample of stock prices S(T) that we can use to estimate the option price.

So we cannot solve these stochastic differential equations analytically, and will instead rely on numerical methods. As always when dealing with continuous functions in numerical analysis, we are going to use some kind of discretization scheme to solve these equations. This means that instead of dealing with this as a continuous problem over the interval [0,T], we are reducing this continuous interval to discrete time points \{t_0,t_1,...,t_{n-1},t_n\} where we have that t_0=0 and t_n=T. We will then, using some discretization scheme, calculating the values of S(t) and V(t) at these time points only.

We start with the differential equation for S(t). We will here use something called the Euler Scheme. For a stochastic differential equation on the form:

dS(t) = g(S(t))dt + h(S(t))dW(t)

where g and h are functions, the Euler scheme is:

S_i = S_{i-1} + g(S_{i-1})\Delta + h(S_{i-1})\Delta W

for i=1,...,n, where \Delta=t_i-t_{i-1} is the size of each time step and \Delta W=W_{t_i}-W_{t_{i-1}} \sim N(0,\Delta) is the increment of the standard Brownian Motion. Comparing with the equation for S(t), the Euler scheme in this case becomes:

S_i = S_{i-1} + rS_{i-1}\Delta + \sqrt{V_{i-1}}S_{i-1}(\sqrt{1-\rho^2}\Delta W^{(1)} + \rho\Delta W^{(2)})

for i=1,...,n. Note that the stochastic part here are the \Delta Ws, so the Monte Carlo aspect will be to simulate these normal random variables. Doing this scheme N times, we will have a Monte Carlo sample of size N of the stock prices S(T)=S_n. We must of course simulate the process for V(t) as well. To ensure that the V_is take on only positive values, we add another term to the discretization scheme. We will not go into details here but this is called the Milstein scheme (this extra term can be seen in the code in the end). If you are interested you can check out the Wikipedia page.

So, we have the following Monte Carlo sampler for estimating the price of the European call option:

We assume that the initial stock price is S(0)=100, the initial volatility is V(0)=0.16, the strike price K=90, the risk free interest rate r=1\% and the time to maturity is T=1 year. Also, we assume that our model parameters have the values \kappa=10, \theta=0.16, \sigma=0.1 and \rho=-0.8. Then, using n=252 time steps in the discretization (typically 252 trading days on a year, so daily time steps as T=1 year), we can implement the following Monte Carlo sampler in Python:

Running a simulation (with N=100000) gives us the following price estimate and 95% confidence interval for the Monte Carlo estimated price:

\begin{aligned} \Pi&(0)_N^{MC} = 21.071 \\ I_{\Pi(0)} &= (20.866, 21.276) \end{aligned}

## Conclusion

In this plogpost, we made use of the concepts introduced in

Introduction to Monte Carlo simulations and option pricing

to price two quite tricky financial options using Monte Carlo simulations: the arithmeric basket call option in the standard Black&Scholes model, and the European call option in the Heston model. To summarize the highlights, we have:

• Introduced the arithmetic basket call option

• Looked at an example of how we can simulate using multivariate distributions

• Introduced the Heston model for stocks

• Introduced discretization and the Euler scheme for solving stochastic differential equations

• Seen two examples of Monte Carlo samplers for estimating expected values

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# Browse Dissertations and Theses - Mathematics by Contributor "Bergvelt, Maarten J"

• (2017-10-30)
In this thesis we prove the existence of Jordan Decomposition in $D_{G/k}$, the ring of invariant differential operators on a semisimple algebraic group over a field of positive characteristic, and its corollaries. In ...

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Here's the question you clicked on:

## GOODMAN Group Title How many 7 digit phone numbers can be formed if the first digit cannot be 0 or 1? 2 years ago 2 years ago

• This Question is Closed
1. imranmeah91

well, let's write out each posibility for each digit and then multiply

2. GOODMAN

I was told to use the Counting Principle

3. imranmeah91

first one can't be 0 or 1 , so we got only 8 possibilities

4. RyanE

8*10*10*10*10*10*10

5. imranmeah91

the other digits have 10 possibilties so we multiplies them all

6. GOODMAN

Okay, so we delete 2 outta the 10 possibliity.

7. imranmeah91

like Ryan did above

8. lgbasallote

you're not planning on stalking someone are you @GOODMAN /:) he

9. GOODMAN

Wait, how do we know that the others have 10 possibilities, doesnt it apply to the others as well?

10. lgbasallote

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11. imranmeah91

well, only first digit can't be 0 or 1 ; other digits can be any number from 0 to 9

12. GOODMAN

Ohhh, okay, got it!

13. GOODMAN

I need help with one more if you dont mind..

14. GOODMAN

A photographer is taking a picture o f a bride and groom together with 6 attendants. How many ways can he arrange the 8 people in a line if hte bride and groom stand in the middle?

15. imranmeah91

@satellite73

16. satellite73

yes?

17. imranmeah91

little help with photographer problem

18. satellite73

ok ____, ___, ____, B G ____, ____, _____

19. satellite73

$6!$

20. GOODMAN

Thats it?

21. imranmeah91

yep

22. GOODMAN

How?

23. satellite73

that looks right to me

24. agreene

what all of a sudden this has to be in base 10?

25. satellite73

this might surprise you and this might amaze you

26. satellite73

but i am going to knock out joe frazier !

27. GOODMAN

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28. satellite73

btw it is $6!=6\times 5\times 4\times 3\times 2=720$

29. GOODMAN

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30. satellite73

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31. satellite73

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32. lgbasallote

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33. GOODMAN

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Article | Open | Published:

Predicting conformational ensembles and genome-wide transcription factor binding sites from DNA sequences

Abstract

DNA shape is emerging as an important determinant of transcription factor binding beyond just the DNA sequence. The only tool for large scale DNA shape estimates, DNAshape was derived from Monte-Carlo simulations and predicts four broad and static DNA shape features, Propeller twist, Helical twist, Minor groove width and Roll. The contributions of other shape features e.g. Shift, Slide and Opening cannot be evaluated using DNAshape. Here, we report a novel method DynaSeq, which predicts molecular dynamics-derived ensembles of a more exhaustive set of DNA shape features. We compared the DNAshape and DynaSeq predictions for the common features and applied both to predict the genome-wide binding sites of 1312 TFs available from protein interaction quantification (PIQ) data. The results indicate a good agreement between the two methods for the common shape features and point to advantages in using DynaSeq. Predictive models employing ensembles from individual conformational parameters revealed that base-pair opening - known to be important in strand separation - was the best predictor of transcription factor-binding sites (TFBS) followed by features employed by DNAshape. Of note, TFBS could be predicted not only from the features at the target motif sites, but also from those as far as 200 nucleotides away from the motif.

Introduction

The physical basis of protein-DNA interactions has been explained from the perspective of direct recognition of nucleic acid bases by complementary TF residues or through an indirect recognition of sequence-dependent DNA structure, more appropriately termed as base and shape readout respectively in the recent literature1,2,3,4. While the former ignores the differential accessibilities of DNA bases in the double helix, the latter assumes the existence of a unique and exclusive structure of the DNA. Base readout having been the primary focus of investigations, numerous methods to model sequence features of TF binding sites as a consensus motif or position weight matrices (PWMs) have been successfully developed5,6,7,8,9. However, the number of studies focusing on DNA shape or conformational dynamics has been limited2, 3, 10,11,12,13. Part of the problem is the lack of tools to rapidly estimate the DNA conformation or its dynamics directly from the sequence. The DNA shape prediction tool, “DNAshape” provided a major innovation in the computational determination of DNA shape and allowed genome-scale study of the contribution of DNA shape to TF binding site recognition2. Trained on Monte Carlo simulation data, DNAshape takes a nucleic acid sequence as its input and predicts four sequence-dependent shape features. It uses a five-nucleotide window (for the base-pair features) and a six-nucleotide window for the base-step features and can be thought of as a dictionary to translate an exhaustive set of small DNA fragments into their corresponding shape features. Use of the four features, predicted by DNAshape has resulted in highly accurate classifiers to distinguish between TF binding and non-binding DNA sequences e.g. in explaining the data from the DREAM5 competition for predicting TF binding specificities14. Despite these successful implementations and substantial evidence that DNA shape encoded in the sequences is critical for TF recognition, the body of work available on the subject is limited compared to sequence-only analyses. For example, most of the TF target search methods such as Transfac or Uniprobe have not yet implemented a shape based analysis15, 16. Even in the studies employing DNA shape, the focus has been limited to a relatively small number of shape features. It may be intuitive that a simpler model, utilizing fewer features is easier to interpret, but we believe it may not necessarily reflect the entire picture of TF-target recognition. Consequently, it is helpful to explore more shape features and hence verify both the completeness and competitiveness of DNAshape. For this purpose, predictive models for much larger set of DNA shape features need to be developed.

We have previously developed techniques to thread DNA sequences onto the structure of a known protein-DNA complex and to determine the energy of cognate and designed sets of sequences17,18,19,20,21,22,23. This approach was based on developing a statistical force field from the observed co-variances in 12 DNA shape features. Trained force field features could estimate the intrinsic sequence energies for a given DNA shape and thereby return the specificities of sequence-dependent structures. Using a combination of base readout energies and those predicted from our statistical force fields, we successfully explained the specificities of the observed DNA sequences in the known protein-DNA complexes. Realizing the inadequacies of the data taken from the crystal structures of protein-DNA complexes, we have also employed molecular dynamics (MD) derived shape data to make more accurate estimates of force field parameters20. Separately, we have tried to explore and predict the sequence-dependence of DNA solvent accessibility by analyzing their structures available from the Protein Data Bank (PDB)24. However, in most of these works, we focused on shape specificity in terms of sequence-structure relationships and the shape dynamics itself was not incorporated into the predictive models. Moreover, our evaluation of sequence specificities was focused on high-resolution structures of protein-DNA complexes and not on the genome-wide TF-target associations.

Here, we present a novel approach, called DynaSeq to model DNA sequence specificities using MD-derived sequence-dependent conformational ensembles, instead of their static values. We define and predict an ensemble each for an exhaustive set of 13 shape features or conformational parameters, of which 12 (Shift, Slide, Roll, Buckle, Helical Twist, Propeller Twist, Stagger, Shear, Tilt, Rise, Stretch and Opening) completely describe its atomic structure, and the 13th (Minor Groove Width; MGW) is used to draw comparisons with DNAshape features25. DynaSeq is a set of support vector regression models (SVRs), trained to predict conformational ensemble occupancies at different nucleotide positions in a given sequence environment. Ensemble definitions are obtained from a data pool of MD simulation snapshots for 136 unique tetrameric DNA sequences and ensemble occupancies are obtained at each base position in each sequence independently. Subsequently trained models are capable of predicting ensemble occupancies for any nucleotide-sequence in much the same way as DNAshape but for a much larger feature set. Upon comparison between DNAshape features and the corresponding values derived from DynaSeq, we observe a good agreement in the predicted values for the common feature sets. Yet the latter allowed us to investigate the predictability of TFBS from other shape features both as ensembles of single individual parameters as well as a single superset of them all.

For a large scale assessment of the power of the different feature sets of DynaSeq (and for comparison, DNAshape) to predict TFBS, we utilized the available data set of TFBS for a large number (1312) of TFs identified by the Protein Interaction Quantification (PIQ) algorithm under similar conditions for the same cell type (mouse embryonic stem cells; mESCs)26. Developing predictors employing ensemble occupancies of one (out of 13) shape feature at a time we assessed the ability of individual conformational parameters to model TFBS. Results indicated that “base pair opening” was the most powerful predictor in majority of the cases followed by some of the features currently employed in DNAshape. For models employing a large feature set of all ensemble occupancies of all parameters, D ynaSeq successfully explained the observed genome-wide binding data with high accuracy. Thus, our results suggest that the performance of DynaSeq is comparable to and arguably better than the similar models built on DNAshape features. Both DNAshape and DynaSeq based models revealed that in most cases, TFBS could be predicted not only from the shape of the target motifs but also from the flanking regions as far as 200 bases away in a 5′ or a 3′ direction.

Taken together, this study provides a novel approach to study DNA structural dynamics at a genomic scale and indicates that information about TF-DNA binding is contained not only in the exact site of TF-binding but also extends to much larger flanking region of DNA. The dictionary of the current implementation of DynaSeq can be accessed from http://dynaseq.sciwhylab.org.

Results

The overall design of the study is shown in Fig. 1. As shown, the study consists of three components viz. (1) Defining and generating a conformational ensemble for an exhaustive set of tetrameric DNA sequences flanked by a GCGC tetramer on both 3′ and 5′ terminals. (2) Developing and benchmarking DynaSeq as a sequence-based tool for predicting shape ensemble occupancies of nucleotides in different sequence environments and finally (3) Applying DynaSeq to predict genome-wide binding sites for 1312 TFs and identifying the most predictive shape feature in each of them. Results obtained from these analyses are presented below.

Global conformational ensemble

DynaSeq consists of 65 SVR models trained on Molecular dynamic (MD) simulation data of 136 tetranucleotides, represented by five ensemble bins for each of the 13 helical/step conformational features for each base of the DNA sequence. The observed values of conformational features from the pooled poses of MD trajectories (544 × 105 values) have been used to define five equal probability bins. The exact ranges or bin boundaries, are shown in Supplementary Figure SF1(a). By definition each bin is expected to have an equal (20%) occupancy, in the global ensembles. However, individual bases and their sequence neighbors, specifically alter these globally deduced values, quantifying the sequence-dependent DNA shape dynamics. General variations in the occupancies across individual ensembles in the MD data (corresponding to various bases under different environments) are plotted in the form of their standard deviations in Supplementary Figure SF1(b). At the first level of specificity, each of the four bases (pooling together all occurrences of the corresponding base irrespective of their flanking bases) has a unique ensemble profile as shown in Supplementary Figure SF1(c). Explicit interpretation of this profile is not possible, but the existence of specific variations, which may play a role in recognizing unknown complementary TF structures, can be noted. For example, the first bin of shear is depleted in the case of Adenine and Cytosine and enriched for Guanine and Thymine. On the other hand both of the triple-hydrogen bond forming bases, Guanine and Cytosine, have an enriched first bin for stretch, which is opposite to what we observe for Adenine and Thymine. Subtle variations are also observed for other conformational features. In addition to the base-wise variations, sequence neighbors also specifically alter these conformational ensembles, leading to the sequence-dependent DNA conformational dynamics (for reference, specificities caused by a single base neighbor for all 64 possible combinations are shown in Supplementary Figure SF1(d)). Modulation of ensemble bin occupancies by neighboring nucleic acid bases forms the basis of DynaSeq. Conceptually, this is similar to the sequence-dependent shape features in DNAshape. However, the origin of reference data, scope and the content of the two tools are different as outlined above.

DynaSeq: Sequence to ensemble predictor

Cross-validation and prediction performance

To develop DynaSeq and evaluate its power to predict shape ensembles defined from our MD trajectory data, we created independent training and test data sets in a leave-one-out fashion; trained the ensemble occupancies for all base positions in 135 DNA sequences and tested the predictive power for the left-out 136th. Results from an exhaustive set of 136 combinations were pooled and evaluated.

Each of the 65 SVR models in DynaSeq, takes identities (A, C, G or T) of a DNA base and its sequence neighbors within a defined window as inputs and is trained to return the corresponding ensemble bin occupancies as the output. The whole set of 65 occupancy values returned from the models is also referred to as predicted ensemble in the manuscript. Figure 2 summarizes the prediction performance of cross-validated SVR models. We trained and tested various window spans and found that a 5-nucleotide window is optimum for the prediction models on the current data. Based on this optimized model, most of the ensemble bin occupancies are predicted well (~80% with an absolute error of 5 percentage points (Fig. 2b)). A high correlation (R = 0.94) between the predicted and observed values in the entire population range further indicates the stability of the prediction model (Fig. 2c). Furthermore Fig. 2(c) and (d) indicate that the populations, highly skewed from their global 20% average value, are also estimated well, albeit with a slightly higher error rate for the first and last bins than for the values in the middle. However, the worst-case Mean Absolute Error (MAE) is still less than 6% (observed in the first bin of slide).

Cross-validation results provided above indicate the robustness of the predictions of sequence-dependent ensembles across 136 independently generated MD trajectories on different unique tetramers. To further evaluate the power of DynaSeq models critically, we performed additional benchmarks as described below.

Comparison with DNAshape on the common descriptors

We have used 3DNA25 a widely popular DNA structural analysis tool to describe DNA shape features. 3DNA generates two types of conformational features from the atomic coordinates of a DNA structure: the deformations in complementary bases, also called base-pair features and the deformation with respect to the stacking of base pairs along the helical axis, also called the base-step features. DNAshape assigns base-step and base-pair parameters to a pair of bases and an individual base respectively, which is technically appropriate but causes an offset between the two sets of values. To describe a DNA sequence and corresponding shape more uniformly, in DynaSeq, we assigned both the base-pair and base-step parameter to a single base position as described in supplementary methods. Thus DynaSeq global dictionary for the finally selected 5-base window consists of 45 values, whereas the number is 46 for DNAshape.

Further, DynaSeq models DNA dynamics instead of its static structures and therefore a comparison with DNAshape was made after converting the DynaSeq-predicted ensembles to their averaged shape feature values (see Methods).

Having addressed the issues described above, we computed DNAshape and DynaSeq features for all the possible 6-mers (46 = combination; with third base position in DynaSeq corresponding to 4th in DNAshape for the base-step parameters) and evaluated the agreement between DynaSeq-derived equilibrium values and the corresponding DNAshape predictions (Fig. 3). We observed a good correlation between the predicted values by the two methods with Pearson’s correlation ranging from 0.61 to 0.73 (Fig. 3(a)). We also observed that several conformational features in DynaSeq are not well correlated with any of the four DNAshape features, thereby suggesting that DynaSeq provides more clues about the DNA conformations than DNAshape (Fig. 3(b)). However, at the outset it is unclear whether this information, even though different from DNAshape is actually useful for modeling transcription factor specificities or otherwise. In the following sections, we present results that highlight what DynaSeq could achieve that DNAshape could not.

Evaluating sequence-specificity of known three-dimensional DNA-structures

DNAshape predicts only four shape features, which are not adequate to model complete DNA structures in atomic details. On the other hand DynaSeq is capable of predicting all the parameters required for rebuilding complete three-dimensional structures for DNA and could therefore be useful in docking and other problems of protein-DNA complex design. We evaluated whether the DynaSeq-predicted structures of DNA sequences in the protein data banks (PDB) are any closer to the reported structures compared to those predicted for a set of randomly generated sequences of the same length. To avoid, confounding factors from other proteins, we used only the free DNA structures for this purpose. The results indicate that the known DNA structures could be favorably recalled from their predicted ensembles using only the DNA sequence with an RMSD (root-mean square deviation) of 4.2 Å compared to 7.5 Å for the randomly generated sequences (Z-score = −1.48) (summary in Table 1; Detailed results in Supplementary Table ST1). These results provide promise for DynaSeq’s ability to help in more accurate design and modeling of protein-DNA complexes27.

DynaSeq’s ability to predict TF specificities for DREAM5 data sets

Dialogue for Reverse Engineering Assessments and Methods (DREAM) is a series of crowdsourcing challenges to solve biological and medical problems28. The fifth of this series of challenges, DREAM5 consisted of a competition to predict TF binding specificities for an exhaustive list of fixed length DNA sequences in a hold-out blind prediction manner14. Results of one binding assay were made available to train models and the predictions on a similar assay were tested in a blind experiment. Even though the training and test data sets are somewhat redundant in the two cases, it would be interesting to note how well DynaSeq could have performed by doing a retrospective analysis. DynaSeq provided highly accurate recall of TF specificities on DREAM5 data sets trained in the manner similar to the conditions of the competition (Supplementary Methods and Supplementary Figure SF2). Prediction was evaluated by AUC (area under the curve) of a receiver operating characteristic curve (i.e. true positive rate plotted against false positive rate), which is used in optimal model selection.

When using the common features in DNAshape and DynaSeq, AUC results agreed with each other (Figure SF2(a)). When a consensus was taken between results obtained by DNAshape and DynaSeq models, the AUC shows significant improvement over DNAshape-only results with a mean AUC rising from 75.68% to 77.60% (p-value from a paired Student’s t-test = 0.0015) (Figure SF2(b)). Finally, when all the 65 features were used in the prediction model, DynaSeq outperformed DNAshape by raising the average AUC to 93% (a gain of 18% AUC) (Figure SF2(c)). In particular, DynaSeq with the full set of features improved lower AUCs yielded by DNAshape (Figure SF2(d)). Some of this improvement could be caused because of high redundancy in DREAM5 experiments and larger number of features in DynaSeq compared to DNAshape. Even so, in terms of the conditions set out in DREAM5, the proposed DynaSeq approach had a clear advantage over DNAshape and at the very least provides an alternative route to perform DNA shape analysis at a large scale.

It should be noted that, while this work was being prepared for submission, another study using DREAM5 data for benchmarking DNAshape performance in identifying TFBS was reported29. However, that work corresponds to a different perspective, demonstrating how k-mer information could be combined with shape features and the detailed results about AUC, we employed here are not available. Our focus in the above has been how DNAshape and DynaSeq predicted features produce similar or different levels of TFBS specificity predictions.

Predictability of TFBS from DynaSeq features in a sliding window

We next set out to evaluate DynaSeq’s performance in predicting TFBS using a sliding window on and around the sequence motifs. For this purpose we used the recently reported binding site information for a comprehensive list of TFs identified by a computational algorithm (Protein Interaction Quantitation or PIQ) using DNase-Seq data30. A number of attributes of this database make it a good choice to evaluate and compare TF binding site prediction methods. First, the database reports the binding sites of a large number of TFs (1,312). Secondly, binding sites are assigned based on the experimental chromatin accessibility data (DNase-Seq) followed by PWM-based motif assignment, thereby ensuring that the annotated regions are indeed the binding sites of a given TF (and not its co-factor, as could be the case in a ChIP-Seq experiment). Thirdly, genome-wide binding and control data come from identical cellular and processing conditions. We evaluated the ability of DynaSeq-predicted ensembles to classify TFBS from genomic controls by creating sequence windows at different positions with respect to the motif start position and using their conformational ensembles to create cross-validated classifiers.

We placed a sliding window covering a fixed number of nucleotides at different distances from the motif start positions (motif start site is defined as per the PIQ annotations) and trained an elastic-net regularized logistic regression model to classify binding sites from non-binding sites (see Methods) using 10-fold cross validation. All the binding sites of a TF and the control regions are collected individually and prediction models are trained for each pair of binding and control regions of the TF. The AUC of ROC for a classifier in this 10-fold cross-validated predictive modeling is used to evaluate the performance of DynaSeq features to predict TFBS compared to DNAshape features. AUC is determined at all positions for all TFs individually and the entire set of 1312 values is averaged to estimate the performance levels across all TFs.

Prediction models were developed in two ways to draw an unbiased comparison. First, predicted ensemble bin occupancies for only one conformational feature was used and 13 cycles of such predictions were made to assess which of the 13 parameters had the most accurate predictions. Secondly, all the ensembles from all of the 13 parameters were put together to create an all-feature model, primarily to compare it with models trained with the smaller number of static features from DNAshape.

TFBS predictions from single parameter ensembles

We examined the performance of each feature by feeding ensembles of only one shape parameter into TFBS prediction models. We observed that even though the ensembles of all the 13 parameters considered here lead to comparable performance levels (Fig. 4a,b), two of the four parameters, Roll and Prop-Tw used in DNAshape are among the top five (Fig. 4d). This validates the choice of the parameters in DNAshape and highlights the critical nature of these shape properties. Surprisingly, however, the top ranked feature in our analysis was “base-pair opening”, a feature not used by DNAshape. It is intuitive to think that base-pair opening may represent the DNA base accessibility better than other shape parameters (Supplementary Table ST1). Although detailed experiments will be needed to determine the physical effect of base-pair opening on TFBS, its ability to predict TFBS better than any other single conformational parameter is a surprising outcome of our analysis. It is possible that base-pair opening is indeed a critical mechanism of strand separation required for TF binding as argued in earlier works31.

From the perspective of accurate prediction of binding sites, it is evident that DynaSeq can provide an alternative approach for modeling of genome-wide TFBS.

Comprehensive modeling of genome-wide binding sites in 1312 TFs

To further evaluate how DynaSeq-derived shape ensembles can model TFBS dynamics, we created their predictive models from an exhaustive set of features in our models. Figure 5 shows the comparison between prediction models trained using four DNAshape features versus those which employ all the 65 bin occupancies used by DynaSeq for a range of window sizes. Cross-validation and regularization in the elastic net attempts to ensure that the models are not over-fitted for DynaSeq (See Supplementary Methods SM2).

We observe that DynaSeq could more accurately distinguish binding sites from control as compared to DNAshape at the motif position as well as its flanking regions, especially for smaller sliding windows. However, at large window sizes, difference in DNAshape and DynaSeq performances starts to disappear, presumably because of the use of a strict cross-validation and elastic net regularization, both of which penalize models using more features. This is supported by the fact that AUC values for neither DNAshape, nor DynaSeq based models increase significantly beyond this point. More work would be needed to enable utilization of very large sequence windows for modeling TFBS using DynaSeq, although encouragingly DynaSeq performs at the same level as DNAshape for a window size of 7 nucleotides or smaller. This window size is used for evaluating contributions from an exhaustive set of conformational parameters only few of which are covered in DNAshape. Recently, another study also addressed the analysis of the DREAM5 by employing DNAshape, they also reported that the models combining k-mers with DNA shape are more successful at smaller windows, presumably due to the arguments we present above29.

With a window size of 7 nucleotides, even though on average the performance levels appear similar, there is still a statistically significant difference in favor of DynaSeq in comparison to DNAshape. While for reasons discussed above this difference does not hold for larger window sizes, the TF-wise scatterplot of performances (Fig. 6(b)) shows that the general trends are very similar across various TFs. However, 91 TFs show improved performance of more than 5% AUC with DynaSeq compared to DNAshape, whereas only 12 of them showed the opposite trend. The identities of the two TFs groups are presented in Table 2. A quick look suggests that many of the TFs whose binding sites are better predicted by DynaSeq are minor groove binders such as TATA box binding and those known for causing large scale DNA bending in their targets. A comprehensive analysis of their functional implications is being carried out and will be reported separately.

A comparison with GC-content based predictions

Some shape features such as propeller-twist and opening have been reported to be highly related with GC-content30,31,32. One wonders if some of the predictive ability in the above models is caused by the GC-content of the binding sites. To address this question, we trained models to predict TFBS using GC content in a 7-nt window for all the considered positions similar to DNAshape and DynaSeq. Figure 6(c,d) shows the results obtained by DynaSeq, DNAshape and GC-content in the same plot both in terms of the absolute and signed distances from the motif sites. It is clear that at, and even as far as 200 bases from the motif location, the shape-based models carry significantly more information than the GC-content alone. Thus, we conclude that the analysis based on shape and dynamics in this work goes beyond the standard considerations of compositional biases in genomic TF targets. This holds true not only for the composite models with integrated feature sets but also for those based on single conformational parameters as in section 2.3.1 as the latter shows an AUC close to 60%, which is about 5% better than the GC-content based model shown in Fig. 6(c,d).

Discussion

DNA conformational dynamics is known to play a crucial role in its recognition by proteins. Several approaches to model it from sequence have been developed resulting in increasingly deeper insights33,34,35,36,37,38,39. The only method for genome-wide prediction of DNA shape available in public domain provides its estimates in terms of static values, which cannot directly capture significant DNA properties such as sequence-dependent polymorphism38, 39. In this work, we have shown how the static equilibrium values of DNAshape parameters may be augmented with the introduction of a shape ensemble instead of shape parameter values. Further, we include additional parameters that enable modeling the complete atomic structure of DNA for a give sequence.

The exact nature of conformational dynamics in TF recruitment, target search and complex stabilization is not well understood even though the role of binding site proximal and contiguous regions at genomic scales has been recently reported40. This study provides support for the results reported in ref. 40 from a different perspective and goes on to look at a more comprehensive description of structure. Conformational dynamics of TFs’ genomic targets has been elusive partly due to the lack of methods to perform large-scale simulations. In this work, we attempt to bridge this gap and show that predicted conformational dynamics provide important biological insights into TF recognition of its genomic targets.

TFs in general show highly redundant sequence-specific DNA binding41, yet they can exhibit highly specific cell-type activity42. Here we show that DNA regions much larger than the well-known TF binding sequence motifs encode shape and specificity information for TFs, indicating that genomic DNA is not just a ‘passive observer’ of TF binding. Instead, TF-DNA interaction is a mutual event between the DNA sequence and the TF, which acts in unison to bring about specific biological activity, as highlighted earlier43. This also reiterates the significance of allostery and cooperativity in protein-DNA recognition as implied from our previous works44,45,46,47. The allosteric effect in DNA targets in the recognition process is a subject of great interest 44,45,46, 48,49,50,51. An analysis at a scale done in this work is not available and our results suggest a role for allosteric control in target recognition of most TF targets.

In this work, we present a novel approach to predict sequence-dependent DNA-conformational ensembles directly from sequences, which does not require detailed simulations of their structures. Models were trained and cross-validated on MD simulations of all unique tetrameric DNA sequences and perform well in various evaluation tests. Model systems representing genome-wide binding preferences of TFs were analyzed. Genome-wide binding preferences of 1312 TFs could be modeled using features derived from predicted conformational ensembles. Together, these results suggest the cooperation of much larger chromatin regions and potential modularity between them for many TF-target associations than realized so far.

Base-pair opening is a fundamental molecular process in strand separation which is essential in transcription, DNA replication and recombination52. It has been suggested that negative supercoiling of DNA or intermolecular DNA-DNA interactions33 induces the strand separation. Thermal fluctuations leading to base-pair opening, also called DNA breathing are known to be sequence-dependent53, which seems to be well-represented by our conformational ensemble. In this study, we found that the base-pair opening feature contributed the most in predicting TFBS, indicating that the energy barrier in the strand separation is lower at TFBS as well as within the 200 upstream and downstream of TFBS. The genomic positions where the strand separation should occur might actually be encoded in the DNA-sequence itself, so that transcription can smoothly start.

In summary, this work presented a novel approach to predict DNA shape in the form of a conformational ensemble and depicted its practical applicability in modeling large scale TFBS data from multiple sources. In particular, the role of DNA shape in protein-DNA interactions is vindicated and a step in understanding them better is made possible.

Methods

The detailed methods for each one of the three components of this study as outlined in Fig. 1 are explained in the following.

MD trajectory to conformational ensemble

Molecular Dynamic (MD) simulation of 12-mers

DynaSeq predictions are based on conformational ensembles obtained by molecular dynamics (MD) simulations. MD simulations were performed for 100 ns on each of the 136 unique (12 mer) DNA sequences with an explicit solvent model. Each of these 12 mers has a unique tetrameric DNA sequences at the center and flanked by GCGC on terminal positions. The detailed simulation conditions in this study were the same as described in our previous work18. This method of collecting MD trajectory data to analyze sequence-dependent effects is consistent with our previously reported works18, 54 and one by the ABC project55. The latter has grown in size and scope over a period of time, but due to practical reasons of data accessibility we retained our method, which has been shown to produce sound scientific insights into sequence-dependent DNA dynamics.

However, as an advance over our previous works, in the current study, we have opted for bsc0 force field to describe nucleic acid atomic interactions56 (After this study was completed, another version of this force field parambsc1 57 was released, which addresses some of the limitations observed in bsc0. Since the approach presented here is based on a high degree of coarse-graining of ensemble populations, we do not expect a dramatic change by using the new version. Notwithstanding, we plan on detailed comparisons that will be reported in the future.) Snapshots were saved every 1 ps (500 steps), yielding 100,000 snapshots for each tetrameric sequence.

Computation of conformational features of all poses in central tetramers

From the snapshots obtained above, 13 conformational features at each base position in the central tetramer were computed, leading to 136 × 4 = 544 sets of 13 such values each from a snapshot. A list of 12 of these features (Supplementary Table ST1) and their definitions is provided in Supplementary Materials, while the minor groove width (MGW) forms the 13th feature (not shown in the table). Conformational feature values were computed using a local installation of the 3DNA program25. Both base pair helical and base pair step parameters are adopted from the bp_step.par file generated by the “analyze module” of 3DNA program. As per the convention of this software, base step parameters between the i-1 th and i th positions are assigned to the i th position in a nucleotide sequence.

Definition of global conformational ensemble using 5 equal-frequency bins

Each of the 13 conformational features from the snapshots were pooled together to define five equal frequency bins. Conformational feature values were split into five ranges using four break points such that each range was occupied by 20% of the snapshots or poses from all base positions (Figure SF1(a)). For example, all values of twist were pooled together from all positions of all tetramers and sorted in ascending order. Between the extreme observed values and four break points are defined to create five bins, each of which is expected to have 20% poses in a global data. These ranges were termed as global ensemble bins. These occupancies differ from the 20% when data from only specific positions are considered, as seen in (Figure SF1(b–d)). This specificity of different base environments needs to be captured by a training model and forms the basis of most analyses in this work.

Mapping MD trajectory data of all nucleotides to the global ensemble bins

As stated above, the individual positions of the 12-mer MD trajectories do not have the same occupancy profile as the global ensemble. To describe the specific divergences from the global values, the ensemble occupancies at each of the 544 base positions in the 136 12-mers were computed. These occupancies were computed in reference to the same global ensemble bins as defined above (employing pooled data) but a 20%-occupancy of in each bin is no longer guaranteed (Figure SF1(b–d)).

Sequence to ensemble predictor: training and validation of DynaSeq

For the purpose of training and cross-validating DynaSeq, we used 65 ensemble features on the 544 positions, which form the desired output of a trained model. We described the sequence environment of each of the 544 base positions by the identity of the base at that location and its flanking bases, which formed the input for our training model. Each of the corresponding 65 values can be predicted by one support vector regression (SVR) model. Thus, we created 65 SVR models, each of which takes the sparse-encoded DNA sequence as the input and returns the ensemble occupancy for the central nucleotide position as the output. Once a trained model is ready, a strategy to convert ensemble into averaged structure and comparing predicted values of DNAshape and DynaSeq for an exhaustive list of 6-mers was developed. These steps are summarized below.

Sparse-encoding of DNA sequence

Machine-readable, unique representations of DNA sequences are sparse encoded, as in our previous works on nucleic acids and proteins24, 58, 59. In this work each base position (or its n th neighbor) is occupied by one of the four bases A, C, G or T. A four-dimensional vector- in which all but one dimension denoting the identity of the base are zero- represents these four possibilities. A DNA sequence segment is a systematic concatenation of these vectors encoding the occurrence of a given base at individual positions.

Training and cross-validation of DynaSeq

To ensure that the 544-training instances are not over-fitted, cross-validation was performed by separating trajectories from each of the 136 sequences in the MD data and training the data over 135 cases, and testing how well such models can make predictions for the left-out set of ensemble occupancies. All SVR models were implemented using e1071 package60 in the R programming environment, using RBF kernel and default cost and gamma values61. Finally a trained model of window size 5 can be thought of as a dictionary of 1024 5-mer sequences mapped to their corresponding 65-dimensional ensemble populations of 13 parameters. This dictionary of the current implementation of DynaSeq can be accessed from http://dynaseq.sciwhylab.org as stated in the abstract.

Computing average structure from predicted ensembles

DNAshape predicts 4 static values, whereas DynaSeq is trained to predict 65 values of the ensemble bin occupancies. To convert ensemble bin occupancies of a given base in a DNA sequence, we used a simple approach of computing the statistical average using the following formula.

$$\langle {C}_{ij}\rangle =\frac{\sum ({M}_{k}{C}_{ijk})}{\sum {C}_{ijk}}$$

where 〈Cij〉 is the predicted mean structure for the conformational parameter C at i th base-position in the sequence j and M k and C ijk are the mid-point values of the k th bin and their predicted occupancies respectively.

Comparing corresponding features in DNAshape and DynaSeq

To estimate how the predicted values from DNAshape compare with DynaSeq derived static values defined in this way, we created an exhaustive list of 6-mers (46 in number) and obtained predictions from both tools (see Results). Pearson’s coefficients of correlation between the two predicted values were used to assess the agreement between DNAshape and DynaSeq.

DynaSeq benchmarks on PDB structures and DREAM5 data on TF specificities

In contrast to DNAshape, DynaSeq predicts all conformational parameters sufficient to rebuild complete three-dimensional structure of DNA. We evaluated if the DynaSeq predicted structure of the ‘original’ or ‘native’ sequence from Protein Data Bank (PDB) is closer to its crystal structure, compared to the predictions from random sequences of the same length. Similarly, DynaSeq was evaluated if a model trained on DREAM5 training data could correctly recall TF-binding specificities in its test data. Detailed methods to perform both these step and corresponding additional results are provided in the Supplementary Material.

Evaluating predictability of TFBS from DynaSeq-derived features

Extracting genome-wide TFBS for 1312 TFs in ES cells from PIQ study

Genome-wide binding sites of 1312 TFs are available from a recent study establishing the directionality of binding in some TFs26. Binding site coordinate data was taken from the same study as available from related online resource located at (http://piq.csail.mit.edu/data/v1.3.calls/140906.mES.calls.tar.gz) (data was downloaded on October 1, 2014 and has been reorganized on the authors’ website since then). In the study authors eventually examined the binding sites of only 733 TFs after post processing to discard TFs with insignificant profile strength and merging sets of motifs with similar binding patterns. However, for the present study we have utilized the entire data set of 1312 TFs. Binding sites from both forward and reverse strands are selected from “calls” data and equally sized corresponding controls are sampled at low binding scores assigned by PIQ (0.25 or lower scores; with cutoffs adjusted if the number of reads at this cutoff was too small). Typically the number of binding sites and control in a TF ranges from around 100 to as many as tens of thousands. In the latter case, maximum number of binding sites considered was fixed at 2000, selected by random sampling.

Computing DynaSeq -predicted ensembles at each binding site and in the 200 upstream and downstream sequence positions

PIQ data consists of TFs with different motif sizes and there is no natural way to align all of them with respect to one another to develop a cumulative understanding of position dependent predictability of TFBS from DNA shape or dynamics. Nonetheless, motif start site provides a good reference point as regions following this are enriched in motif residues and away from it are depleted in it. Using motif start site as a reference point, we predicted 65-dimensional DynaSeq features at all DNA positions within 200 bases upstream and downstream from the start site. At any given sequence position, where we wish to determine the prediction performance, we utilized these predicted 65 DynaSeq features each for all the positions within a window. These features for a window are then concatenated to form the inputs of a new cross-validated prediction model employing elastic nets. Similar treatment was given to four-dimensional predictions from DNAshape i.e. prediction models for DNAshape-based features were also based on the predict shape features for all nucleotide positions within a window.

Evaluating DynaSeq performance on PIQ data

Using feature sets and DNA positions as described above, we created new cross-validated prediction models, whose inputs are shape or ensemble features of DNA sequence in a window and outputs are class labels indicating if the sequence is derived from a TFBS or a genomic control. For all such binding site prediction models, we used elastic net regularized logistic regression models, implemented in the R-package glmnet 62. Elastic nets provide robust models free from over-fitting with appropriate penalties for large number of features via its adjustable parameter alpha, which was selected to be 0.1 in all models being discussed here. In all cases a 10-fold cross-validation was used to estimate strictly independent performance of these models61. Elastic net implementation in glmnet produces multiple prediction models by adjusting a tunable parameter lambda. Out of all such models in each case, we have used the one with highest AUC for all comparisons (see Supplementary Methods SM2).

Training TFBS predictor with one parameter ensemble at a time

To assess the extent to which each of the 13 shape features studied in this work contribute to predictions, we repeated the PIQ TFBS prediction steps described above by creating multiple prediction models at each position of each TF. One model each was created for 5-dimensional ensemble of one conformational parameter at one time at each base position in the TFBS of each TF. Average AUC of all TFs for positions enriched in motif (motif start site and a fixed distance on 5′ sequence neighborhood) and remaining regions are considered. In addition, at each base position relative to motif start site, the name of the parameter whose ensemble gave the best prediction performance was retained and finally a frequency of occurrences of each of the 13 feature names corresponding to conformational parameters were compared.

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Travers, A. & Muskhelishvili, G. A common topology for bacterial and eukaryotic transcription initiation? EMBO reports 8, 147–151 (2007).

53. 53.

Krueger, A. & Protozanova, E. & Frank-Kamenetskii. Sequence-dependent base pair opening in DNA double helix. Biophysical Jounal 90, 3091–3099 (2006).

54. 54.

Arauzo-Bravo, M. J., Fujii, S., Kono, H., Ahmad, S. & Sarai, A. Sequence-dependent conformational energy of DNA derived from molecular dynamics simulations: toward understanding the indirect readout mechanism in protein-DNA recognition. J Am Chem Soc 127, 16074–16089, doi:10.1021/ja053241l (2005).

55. 55.

Beveridge, D. L., Cheatham, T. E. 3rd & Mezei, M. The ABCs of molecular dynamics simulations on B-DNA, circa 2012. Journal of biosciences 37, 379–397 (2012).

56. 56.

Perez, A. et al. Refinement of the AMBER force field for nucleic acids: improving the description of α/γ conformers. Biophysical Journal 92, 3817–3829 (2007).

57. 57.

Ivan Ivani et al. Parmbsc1: a refined force field for DNA simulations. Nature methods 13, 55–58, doi:10.1038/nmeth.3658 (2016).

58. 58.

Ahmad, S., Gromiha, M. M. & Sarai, A. Analysis and prediction of DNA-binding proteins and their binding residues based on composition, sequence and structural information. Bioinformatics 20, 477–486 (2004).

59. 59.

Andrabi, M., Mizuguchi, K., Sarai, A. & Ahmad, S. Prediction of mono- and di-nucleotide-specific DNA-binding sites in proteins using neural networks. BMC Structural Biology 9, 30 (2009).

60. 60.

e1071: Misc Functions of the Department of Statistics (e1071), TU Wien (2010).

61. 61.

R: A Language and Environment for Statistical Computing (R Foundation for Statistical Computing, Vienna, Austria, 2014).

62. 62.

Friedman, J., Hastie, T. & Tibshirani, R. glmnet: Lasso and elastic-net regularized generalized linear models. R package version 1 (2009).

Acknowledgements

This work has been supported by a grant by University for Potential of Excellence (UPoE-II) scheme of the University Grants Commission, India to S.A. This work has also been supported by a grants-in-aid (kaken-hi #15K00419) program of Japanese Society for Promotion of Science to S.A. This work has also been supported by Grants-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan [JP25116003 and JP26330339 to H.K.] and Platform for Drug Discovery, Informatics, and Structural Life Science from Japan Agency for Medical Research and Development (AMED) to H.K. This study was in part supported by Grants-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science, and Technology (Grant Numbers 25430186 and 25293079) and from the Ministry of Health, Labor, and Welfare to K.M. This project has been funded in whole or in part with Federal funds from the National Cancer Institute, National Institutes of Health, under contract number HHSN261200800001E. The content of this publication does not necessarily reflect the views or policies of the Department of Health and Human Services, nor does mention of trade names, commercial products, or organizations imply endorsement by the U.S. Government. This research was supported (in part) by the Intramural Research Program of the NIH, National Cancer Institute, Center for Cancer Research.

Author information

Author notes

• Munazah Andrabi

Present address: Faculty of Biology,Medicine and Health, Michael Smith Building, The University of Manchester, Dover Street, Manchester, M13 9PT, UK

Affiliations

1. National Institutes of Biomedical Innovation Health and Nutrition, 7-6-8, Saito-Asagi, Ibaraki, Osaka, 5670085, Japan

• Munazah Andrabi
• , Kenji Mizuguchi
2. Department of Biology, Southern University of Science and Technology of China, Shenzhen, 518055, China

• Andrew Paul Hutchins
3. World Premier International (WPI) Immunology Frontier Research Center (IFReC), Osaka University, 3-1 Yamadaoka, Suita, 565-0871, Osaka, Japan

• Diego Miranda-Saavedra

• Diego Miranda-Saavedra
5. Department of Computer Science, University of Oxford Wolfson Building, Parks Road, OXFORD, OX1 3QD, United Kingdom

• Diego Miranda-Saavedra
6. Molecular Modeling and Simulation (MMS) Group, National Institutes for Quantum and Radiological Science and Technology, 8-1-7, Umemidai, Kizugawa, Kyoto, 619-0215, Japan

• Hidetoshi Kono
7. National Cancer Institute, Cancer and Inflammation Program, Leidos Biomedical Research, Inc. Frederick, Maryland, USA

• Ruth Nussinov
8. Department of Biochemistry and Human Genetics, Sackler School of Medicine, Tel Aviv University, Tel Aviv, Israel

• Ruth Nussinov

Contributions

S.A. conceived and designed the main components of this study and implemented it together with M.A. H.K. provided MD. data and helped in analyzing other results. S.A., M.A. and A.P.H. prepared the manuscript with discussions and critical comments from R.N., D.M.S., H.K. and K.M. All authors participated in discussions on all aspects of the manuscript and read, improved and approved the final version of the manuscript.

Competing Interests

The authors declare that they have no competing interests.


text:
# Chapter 1 - Review: 21

False.

#### Work Step by Step

The set of all integers (set $C$) IS a subset of all rational numbers (set $E$).

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.


text:
# DistributedDeque¶

Usage

use DistributedDeque;

or

import DistributedDeque;

## Summary¶

A parallel-safe scalable distributed deque. A deque is a double-ended queue that supports insertion and removal from both ends of the queue, effectively supporting both FIFO, LIFO, and a Total ordering, where the order in which you add them will be the exact order you remove them in; for emphasis, a Deque can be used as a Queue, a Stack, and a List respectively.

Note

This module is a work in progress and may change in future releases.

## Usage¶

First, the DistDeque must be initialized before use by calling its initializer.

var deque = new DistDeque(int, cap=maxElem, targetLocales=ourLocales);

The deque can be used as a queue by using the DistributedDequeImpl.enqueue and DistributedDequeImpl.dequeue convenience methods or inserting from one end to remove from another…

deque.enqueue(1);
var (hasElem, elem) = deque.dequeue();

The deque can be used as a stack by using the DistributedDequeImpl.push and DistributedDequeImpl.pop convenience methods, or insertion and removing from the same ends…

deque.push(1);
var (hasElem, elem) = deque.pop();

The deque can be used as a list by using the DistributedDequeImpl.pushBack, DistributedDequeImpl.pushFront, DistributedDequeImpl.popBack, and DistributedDequeImpl.popFront methods. While the deque is not indexable, the ability to append or prepend is powerful enough to allow a total ordering, allowing the user to define the order by letting them insert and remove at whichever ends they so choose.

var deque = new DistDeque(int);
forall i in 1 .. N {
if i % 2 == 0 then deque.pushFront(i);
else deque.pushBack(i);
}

The deque supports both serial and parallel iteration, and a means to iterate in a particular order (currently only FIFO and LIFO) using the Ordering enumerator.

for elt in deque.these(Ordering.FIFO) {
// ...
}

for elt in deque.these(Ordering.LIFO) {
// ...
}

The deque can also be used in a reduction, although currently reduction only used parallel-iteration, hence reduction will be performed in an unordered fashion. In the future, a specific function may be created to allow reduction in a certain ordering…

deque.addBulk(1..100);
var result = + reduce deque;

## Bugs and Known Issues¶

1. It is not safe to call other methods while iterating, as it will lead to deadlock. It is an open question whether using a snapshot approach is better to allow concurrent operations at the expense of elevated memory consumption, and iterating directly over elements while holding locks, which strangles potential concurrency.
2. Reduction cannot be performed in any ordered way. This may be fixed in the near future, either by adding pseudo-parallel iterators that merely yield sequentially in order, or by creating a method to perform reduction for the user in a specified ordering.
3. This data structure requires network atomic support for scalability, and without it will result in degrading performance. It is another open question whether a specific implementation that is more friendly for remote-execution atomic operations should be provided.
4. The ordered serial iterators currently do not work when the globalHead or globalTail are negative, which is a result of iteration being an after-thought. This will be improved upon soon, but for now if you use DistributedDequeImpl.pushBack or DistributedDequeImpl.pushFront methods, I would advise against using them for now.

## Planned Improvements¶

1. Double the size of each successor up to some maximum, similar to DistributedBag for unroll blocks. Currently they are fixed-sized, but it can benefit from improved locality if a lot of elements are added at once.

## Methods¶

config param distributedDequeBlockSize = 8

Size of each unroll block for each local deque node.

enum Ordering { NONE, FIFO, LIFO }

The ordering used for serial iteration. NONE, the default, is the most performant and is algorithmically similar to parallel iteration.

record DistDeque

A parallel-safe scalable distributed double-ended queue that supports both insertion and removal from either end of the queue. Can be used as a Queue, Stack, or even a List.

type eltType
var _impl: unmanaged nilable DistributedDequeImpl(eltType)

The implementation of the Deque is forwarded. See DistributedDequeImpl for documentation.

proc init(type eltType, cap = -1, targetLocales = Locales)
class DistributedDequeImpl: CollectionImpl
var cap: int

Capacity, the maximum number of elements a Deque can hold. A cap of -1 is considered unbounded.

var targetLocales: [targetLocDom] locale

Locales to distribute the Deque across.

proc init(type eltType, cap: int = -1, targetLocales: [?locDom] locale = Locales)
override proc add(elt: eltType): bool

Syntactic sugar for pushBack.

override proc remove(): (bool, eltType)

Syntactic sugar for popFront.

proc enqueue(elt: eltType): bool

Syntactic sugar for pushBack.

proc dequeue(): (bool, eltType)

Syntactic sugar for popFront.

proc push(elt: eltType): bool

Syntactic sugar for pushBack.

proc pop(): (bool, eltType)

Syntactic sugar for popBack.

proc pushBack(elt: eltType): bool

Appends the element to the tail.

proc popBack(): (bool, eltType)

Removes the element at the tail.

proc pushFront(elt: eltType): bool

Appends the element to the head.

proc popFront(): (bool, eltType)

Removes the element at the head.

override proc getSize(): int

Obtains the number of elements held by this queue.

override proc contains(elt: eltType): bool

Performs a lookup for the element in the data structure.

iter these(param order: Ordering = Ordering.NONE): eltType

Iterate over all elements in the deque in the order specified.

iter these(param order: Ordering = Ordering.NONE): eltType
iter these(param order: Ordering = Ordering.NONE): eltType
iter these(param order: Ordering = Ordering.NONE, param tag: iterKind)
iter these(param order: Ordering = Ordering.NONE, param tag: iterKind, followThis)


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 The impact of low oil prices on VenezuelaJan. 25, 2015 — Caracas, Venezuela Featured on RTÉ World Report Venezuela’s economy is in tatters. With severe shortages, there are are now queues of hundreds, thousands of people outside supermarkets. Annual inflation is running at more than 64%. On the black market, the local currency has lost 65% of its value against the US dollar in the last year. At a supermarket line in Caracas, I found 60-year-old lab assistant Antonia Rodríguez waiting for milk. “This government is filth,” she told me. “There’s no food! If Chávez were alive, this wouldn’t be happening.” Former president Hugo Chávez lubricated his socialist revolution with the world’s largest oil reserves. Some 96% of the country’s foreign income is derived from selling oil. When Chávez came to power in 1999, he toured the nations of OPEC and invited their heads of state to Caracas for a major summit. Prices went up and stayed there for years. But Chávez is dead, and his successor Nicolas Maduro is paying the bill. International oil prices have halved in the last few months. In his annual State of the Nation speech on Wednesday, Maduro admitted the country was suffering. He blamed the problems on an economic war being waged against Venezuela, part of which is the fall in oil prices. “There’s less foreign currency,” the president told Congress. “The oil barrel has gone from $96 to$40 but God won’t fail us. God will provide.” That line, “God will provide,” angered many in Venezuela, evidence to them that there is little hope left for the country’s spluttering economy. Outside a pharmacy in Caracas the following morning, hundreds stood in line hoping to buy basics such as diapers. Maureen Garban, 55, bakes and sells cakes for a living. Though she struggles to find sugar. “Maduro is waiting for a miracle but it’s not going to happen,” she told me. Last year, Maduro faced the biggest anti-government protests in Venezuela in more than a decade. Now, with oil revenues in free fall, Maduro’s popularity is in the low twenties and looks to be falling. Pockets of protest have begun to form across the country. Even those who once supported Chávez, like Rodriguez in the supermarket line, are unhappy with Maduro’s handling of the economy. The opposition has called for further protests. Cheap oil puts the president in a tough position. Bringing in pragmatic economic policies would hurt poor Venezuelans, who form the base of his support. Currency controls, enacted over a decade ago by Chávez, have created a black market on which the dollar sells for 30 times its strongest official value. But devaluing the currency, which would provide the government more local currency, would exacerbate inflation and lower the value of Venezuelans’ savings. During his speech to Congress, Maduro did slightly revamp the currency controls though not to any great extent. Another option for him is to end a \$12 billion-a-year subsidy on petrol at the pump, which allows Venezuelans to fill up their cars for just a few cents. In the speech, Maduro described the subsidy as a distortion that needed to end. However, he made no concrete pronouncements. Politically, a price rise could spark major riots—just as it did when a similar move was tried in 1989. In a recent cartoon by Venezuelan satirist EDO, Maduro is shown as a sweating snake charmer trying to persuade a barrel of oil to rise. So far, the magic hasn’t worked. For World Report, this is Girish Gupta in Caracas


text:
## The Annals of Mathematical Statistics

### On the Test of Independence Between Two Sets of Variates

#### Abstract

In this paper an asymptotic expansion is derived for the power function of the likelihood ratio criterion for testing independence between two sets of variates for the case when the population canonical correlation coefficients are small. The method used can theoretically give the expansion up to any order of $N$ where $N$ is the sample size. Here the expansion is given up to $N^{-3}$ and is an extension of an expansion obtained independently by Sugiura [10] using a different method. The theorem in Section 3 summarizes the final result. In Section 4 the expansion is compared numerically with a different approximation obtained by Sugiura and Fujikoshi [11] and with exact results obtained by Pillai and Jayachandran [9].

#### Article information

Source
Ann. Math. Statist., Volume 43, Number 5 (1972), 1491-1497.

Dates
First available in Project Euclid: 27 April 2007

https://projecteuclid.org/euclid.aoms/1177692381

Digital Object Identifier
doi:10.1214/aoms/1177692381

Mathematical Reviews number (MathSciNet)
MR348922

Zentralblatt MATH identifier
0261.62015

JSTOR


text:
you are viewing a single comment's thread.

[–] 13 points14 points  (3 children)

sorry, this has been archived and can no longer be voted on

$\epsilon$

[–] 6 points7 points  (2 children)

sorry, this has been archived and can no longer be voted on

I'm not sure about that's a 0 liner, can someone give me a $\delta$ to be sure?

[–] 11 points12 points  (1 child)

sorry, this has been archived and can no longer be voted on

δ


text:
# Combine a grid of images into a single figure WITHOUT subfigures and labels, captions, etc

MWE:

\begin{figure*}
\subfloat[\label{fig:test1}]
{\includegraphics[width=.45\linewidth]{one}}\hfill
\subfloat[unwanted caption\label{fig:test2}]
{\includegraphics[width=.45\linewidth]{two}}\hfill
\caption{Something something, etc. a desired overall caption}
\end{figure*}

This will produce a figure with an overall (desired) caption, and two subfigures, oriented left and right. The left figure will have a "caption" which just says

(a)

and the right figure will have caption

(b) unwanted caption

1. How do I remove these captions entirely so that the two images appear to be simply part of one overall figure?

2. How can I generalize this simply to allow for a 2x2 grid of 4 images, combined with just one overall caption and no individual (a),(b), (c), (d) or captions?

Also, generically, the subcaption package doesn't work for me, so I really hope you can suggest a way to accomplish this without that...

• If you don't need subfloats, don't use subfloats. \includegraphics{a}\hfill\includegraphics{b}\par\includegraphics{c}\hfill\includegraphics{d}. Instead of \par you can add a blank line. – Torbjørn T. May 2 '16 at 16:23
• Please always provide a complete example which can be copy-paste-compiled to reproduce the issue. In this case, just don't use \subfloat or subfigure or sub anything. Just use \includegraphics or, if necessary, minipages or similar. – cfr May 2 '16 at 16:23
• @TorbjørnT. See below. – cfr May 11 '16 at 18:14

The comments above basically gave me the answer, and here it is

\documentclass{article}
\usepackage{graphicx}
\begin{document}
\begin{figure*}
\includegraphics[width=.48\linewidth]{example-image-a} \hfill
\includegraphics[width=.48\linewidth]{example-image-b} \\
\vspace{5mm}

\includegraphics[width=.48\linewidth]{example-image-c}\hfill
\includegraphics[width=.48\linewidth]{example-image}
\caption{Something something, overall}
\end{figure*}
\end{document}

• Asking a question, getting help and then taking credit for it is somewhat discourteous. It would be better - and more usual - to ask one of the commenters to write up an answer you could accept so that they get the credit. In this case, you should definitely ask Torbjørn T. to write the answer. Also, answers - like questions - should typically include a complete example. – cfr May 2 '16 at 16:34
• That's fair. I don't mean to take credit for the solution. If Torbjørn T. wants to take the time to write a complete example solution, like you say, I will gladly "accept it as the genuine solution" with a checkmark. I wrote this answer for other readers who come here from google in the future and just want to copy paste this solution. – Steve May 11 '16 at 16:43
• Nevermind. As @cfr says it is customary to ask commenters to make answers, but no need for any more fuzz here. Part of the reason I only commented in the first place was that I was sure there was a duplicate somewhere, but I wasn't able to find it, and then I forgot about it. If you feel bad, find an answer of mine and give it an upvote :). I also edited your code a bit. – Torbjørn T. May 11 '16 at 19:45


text:
# Decoupling of charm beyond leading order

19 Nov 2018

We study the effective theory of decoupling of a charm quark at low energies. We do this by simulating a model, QCD with two mass-degenerate charm quarks. At leading order the effective theory is a pure gauge theory. By computing ratios of hadronic scales we have direct access to the power corrections in the effective theory. We show that these corrections follow the expected leading behavior, which is quadratic in the inverse charm quark mass.


text:
# Efficient way to do congruent transformation using matrix inverse?

I know a square self-adjoint matrix $S_{vv}$ and I want to find:

$S_{rr} = HS_{vv}H^{\dagger}$

where $\dagger$ denotes conjugate transpose.

I do not know $H$ but I do know $H^{-1}$.

What is the most efficient way to solve something of this form?

I have considered rewriting it as:

$S_{rr} = H(HS_{vv})^{\dagger}$

Finding the LU decomposition of $H^{-1}$, and then doing two lots of forward and back substitutions to solve for $S_{rr}$. But it feels like there might be a better way?

Any help much appreciated.

• $H^{-1}$ is not singular, right? – Jeff Hammond Feb 11 '14 at 15:57
• As a general rule, any time you are tempted to compute the explicit inverse of a matrix, there is a better way. I'm not sure what it's called when solves $R=A^{-1}V(A^{-1})^{\dagger}$ for $V$, given known $R$ and $A$, but it has at least one obvious solution. For example, I might case this as $AX^{\dagger}=V$ and solve for $X$, then solve $X=AR$ for $R$. You can use LU for this, of course. – Jeff Hammond Feb 11 '14 at 16:00
• Thanks Jeff, yes $H^{-1}$ is not singular. – James Feb 11 '14 at 17:10
• I agree that it can be done that way, as I indicated in my question. But I'm wondering what "the" way is to approach this problem, if such a thing exists. The thing is, like you, I don't know what to call this so it is difficult to search for a good reference. – James Feb 11 '14 at 17:14
• If you're looking for a congruent transform, LDL is probably a better choice. – Geoff Oxberry Feb 11 '14 at 20:16


text:
How to use a dataset with only one category of data

I am performing a classification task, to try to detect an object. A picture of the environment is taken, candidates are generated of this possible object using vision algorithms, and once isolated, these candidates will be passed through a CNN for the final decision on whether the object has been detected or not. I am attempting to use transfer learning on InceptionV3 but am having difficulty training it, as I only have one set/class of images.

The dilemma is that I only have one class of data and when I pass it through the network, I get a 100% accuracy (because there is nothing to compare it to). How should I overcome this? Should I find more categories online to add to my dataset? What should these categories be?

Just to clarify, as an example, I have class "cat".

Not "cat" and "dog".

Not "cat" and "no cat".

Just "cat". That is what my dataset consists of at the moment.

The Model learns to match the weights as per the image and feedback from label data.

If you will feed a few Image classes as "Not Cat", it will learn to classify similar features as "Not Cat". But might fail for a new Class.
e.g. if it is trained on "Car/Furniture/Dog" as "Not Cat", then chances are high that a Wild Cat will be classified as Cat.
Dumping all the Imagenet dataset will definitely provide quite a good variance to the "Not Cat" class and may work most of the time but that is not the appropriate solution for the problem.

Such type of problem will fall under One-Class-Classification.
Core idea is to use CNN to extract features then use some specialized models e.g. one-class SVM, Gaussian Mixtures, etc. to define a boundary for "Cat"

This problem, as defined by the one-class SVM approach, consists of identifying a sphere enclosing all (or the most) of the data. The classical strategy to solve the problem considers a simultaneous estimation of both the center and the radius of the sphere.

Hackernoon blog
Arxiv
Researchgate

There are other approaches too i.e. based on Auto-encoder. Here, we try to put a Threshold on reconstruction error.
References-
Quora
SO
Keras blog

Also, may look here to check an idea which generated Random images for "No Cat" Class.
Here

It seems that you're looking for One-Class classification approaches. There are several approches, such as isolation forest, one class SVM, recontruction error of autoencoder (trained only by your positive class), and so on... All those classifiers learn from one class.

EDIT below

About creating a "no cat" class

Must to know: when commonly training classifiers to distinguish cat from no-cat, you should interpret their predictions as follow:

If it says it is a cat, that means that it looks more similar to a cat than to a no cat. Nothing more.

If one day your classifier sees an input that it has never seen in your "no cat" training dataset, it could choose that it looks more similar to a cat.

Conclusion: Be careful/aware when creating "no cat" class.

A first understanding of one-class classification

The objective of one class classification is not to differentiate multiple classes anymore but to find the best descriptive boundaries of your single class.

One easy-to-understand example with a distance approach:

1. Take some features that represent your one-class input data.
2. In this features space, compute the maximum distance $$d_{max}$$ between 2 nearest neigbors.
3. Project any new input in this features space and compute its distance from its nearest neighbor.
4. If this distance is more than $$d_{max}$$, it is not your class. Otherwise, it is.

Of course this is a primary example but it might give you an idea of what one-class classification does.

One difficulty of one-class classification is to find the right set of features. To go further from this example, anything that bounds a cluster (such as some clustering algorithms) could be used to create a one-class classifier.

Going further

One-class classification problems has draw more and more attention in recent years. You could have a look at those articles:

The model can't be trained if all it has ever known is just one thing. There must be examples of "not cat" for it to learn.

To fix this you can find anything to train it on as long as it is "not cat" and beef up your training dataset. The training set should be a mix of "cat" and "not cat". Once you do then you'll start the process of refining your model's accuracy.

• So I should go for binary classification instead of multi label? I was thinking of using imagenet data. So I should just dump all the imagenet images into the "not" category? Sep 7 '20 at 6:35
• @FinnWilliams whether to dump everything in the "not" category or to add specific categories has to be tested experimentally. It's probably not useful to teach your model to distinguish between a plane and a car, if all you care about are cats and not-cats. But if the ImageNet dataset contains animals that look a lot like cats, it might help your model to explicitly teach it to recognize these particular animals as a different class.
– Stef
Sep 8 '20 at 12:27
• As @Stef mentioned, all imagenet images is huge and wouldn't make sense to train on. We'd need a little more details to know exactly what you're trying to achieve first because training your model to recognise "cat" may only need just a couple of reasonable "not cats" to train with. Training to recognise "cat" images from "dog" images makes more sense than training with "spider" images.
– Noel
Sep 8 '20 at 14:53
• @Noel Hey Noel sorry for the long reply, you didn't tag me in your comment so I didn't get a notification. I am training the network on pictures of an object called a cache; this is a green cylindrical object the length of a fork. It will be buried in sand, and I am writing algorithms which will allow a robot to detect it, so it can travel to it and pick it up Sep 10 '20 at 18:13
• @Noel I have a vision algorithm to generate candidates - objects that COULD be the cache. Now i'm building a neural network which will take these candidates and check to see if it actually is a cache. My plan was to use transfer learning - the expertise of an already trained model to better recognise the cache, but I ran into the problem described above of only having pictures of the cache. There are a few things I could do from here, so I thought i'd take to Stack Overflow to get some more input. Sep 10 '20 at 18:16


text:
Rope between inclines

A rope rests on two platforms which are inclined at an angle $$\theta$$ (which you are free to pick, as shown. The rope has uniform mass density, and its coefficient of friction with the platforms is 1.The system has left right symmetry. What is the largest possible fraction of the rope that does not touch the platforms ?

×