Challenges in mathematical cognition: a collaboratively-derived research agenda

This paper reports on a collaborative exercise designed to generate a coherent agenda for research on mathematical cognition. Following an established method, the exercise brought together 16 mathematical cognition researchers from across the fields of mathematics education, psychology and neuroscience. These participants engaged in a process in which they generated an initial list of research questions with the potential to significantly advance understanding of mathematical cognition, winnowed this list to a smaller set of priority questions, and refined the eventual questions to meet criteria related to clarity, specificity and practicability. The resulting list comprises 26 questions divided into six broad topic areas: elucidating the nature of mathematical thinking, mapping predictors and processes of competence development, charting developmental trajectories and their interactions, fostering conceptual understanding and procedural skill, designing effective interventions, and developing valid and reliable measures. In presenting these questions in this paper, we intend to support greater coherence in both investigation and reporting, to build a stronger base of information for consideration by policymakers, and to encourage researchers to take a consilient approach to addressing important challenges in mathematical cognition

Combining Physicochemical and Evolutionary Information for Protein Contact Prediction

<div><p>We introduce a novel contact prediction method that achieves high prediction accuracy by combining evolutionary and physicochemical information about native contacts. We obtain evolutionary information from multiple-sequence alignments and physicochemical information from predicted <i>ab initio</i> protein structures. These structures represent low-energy states in an energy landscape and thus capture the physicochemical information encoded in the energy function. Such low-energy structures are likely to contain native contacts, even if their overall fold is not native. To differentiate native from non-native contacts in those structures, we develop a graph-based representation of the structural context of contacts. We then use this representation to train an support vector machine classifier to identify most likely native contacts in otherwise non-native structures. The resulting contact predictions are highly accurate. As a result of combining two sources of information—evolutionary and physicochemical—we maintain prediction accuracy even when only few sequence homologs are present. We show that the predicted contacts help to improve <i>ab initio</i> structure prediction. A web service is available at <a href="http://compbio.robotics.tu-berlin.de/epc-map/" target="_blank">http://compbio.robotics.tu-berlin.de/epc-map/</a>.</p></div

Definition of graphs used to model the neighborhood of the contacting residues <i>i</i> and <i>j</i>: Nodes represent residues (circles), edges represent contacts (solid black lines).

<p><b>A:</b> The neighborhood graph for residue contains all residues in contact with residues , and (dark grey). <b>B:</b> The neighborhood graph . <b>C:</b> The shared neighborhood graph for the contact between residues and is defined by the intersection of and . Residues that belong to are shown in blue. Shared neighborhood graphs capture the local context of the shared neighborhood of the contacting residues. <b>D:</b> The immediate neighborhood graph is defined by all residues that are in contact to or . Residues that belong to are shown in blue. Immediate neighborhood graphs capture the direct neighborhood of the contacting residues.</p

Alignment depth composition of the CASP9-10_hard, EPC-map_test, D329 and SVMCON_test data sets.

<p>Proteins are grouped into bins based on their number of sequences in the alignment. Colors correspond to a particular bin, from dark blue (few sequences) to red (many sequences). Data sets are sorted from difficult (CASP9-10_hard) to easy (SVMCON_test). The last panel shows the pooled results.</p

Dependence of prediction accuracy on sequence length.

<p>EPC-map is more accurate or on par with GREMLIN, irrespective of sequence length. The performance increase over GREMLIN is most pronounced for proteins smaller than 250 residues. Counting performs better on smaller proteins. The SVM component of EPC-map consistently improves the contact prediction from decoys over Counting by leveraging physicochemical information.</p

Prediction performance for proteins with increasing sequence alignment depth.

<p>Results are shown for all proteins pooled from the CASP9-10_hard, EPC-map_test, D329 and SVMCON_test data sets. Different methods are shown as color coded violin plots. The lower and upper end of the black vertical bars in each violin denote the accuracy at the 25 and 75 percentile, respectively. White horizontal bars indicate the median, red horizontal bars the mean accuracy. The distribution of the prediction accuracies for individual proteins is indicated by the shape of the violin. EPC-map is consistently more accurate than the other tested methods, regardless how many sequences are available.</p

Contribution of the SVM component to contact prediction.

<p>Contribution of the SVM component to contact prediction.</p

Flowchart overview of EPC-map, combining evolutionary information (upper box) and physicochemical information (lower box).

<p>For evolutionary contact prediction, multiple-sequence alignments are constructed by searching the Uniprot20 database with HHblits. GREMLIN is then used to predict contacts from the alignments. For physicochemical contact prediction, decoys are generated with Rosetta. From each decoy, contact graphs are constructed and feature input vectors computed. An SVM ensemble predicts the contact probability from each feature vector. The SVM probability and occurrence statistics predict physicochemical contacts. Lastly, evolutionary and physicochemical contact prediction are combined to form the output of EPC-map.</p

Comparison of <i>ab initio</i> structure prediction of 132 proteins from EPC-map_test with and without predicted contacts: each data point corresponds to the GDT_TS of the lowest-energy structure generated with and without the use of EPC-map predicted contacts.

<p>EPC-map increases the average prediction accuracy by 7.8% from 33.1 to 40.9 GDT_TS (paired Student's <i>t</i>-test <i>p</i>-value).</p

Prediction performance overview for the CASP9-10_hard, EPC-map_test, D329 and SVMCON_test data sets.

<p>The figure shows the long-range contact prediction performance of the top scoring L/5 contacts. Different methods are shown as color coded violin plots. The lower and upper end of the black vertical bars in each violin denote the accuracy at the 25 and 75 percentile, respectively. White horizontal bars indicate the median, red horizontal bars the mean accuracy. The distribution of the prediction accuracies for individual proteins is indicated by the shape of the violin. Data sets are sorted from difficult (CASP9-10_hard) to easy (SVMCON_test). The last panel shows the pooled results for all proteins from these data sets.</p