{"id":26999,"date":"2015-10-26T15:25:09","date_gmt":"2015-10-26T15:25:09","guid":{"rendered":"http:\/\/healthmedicinet.com\/news\/a-polynomial-time-algorithm-for-computing-the-area-under-a-gdt-curve\/"},"modified":"2015-10-26T15:25:09","modified_gmt":"2015-10-26T15:25:09","slug":"a-polynomial-time-algorithm-for-computing-the-area-under-a-gdt-curve","status":"publish","type":"post","link":"http:\/\/healthmedicinet.com\/news\/a-polynomial-time-algorithm-for-computing-the-area-under-a-gdt-curve\/","title":{"rendered":"A polynomial time algorithm for computing the area under a GDT curve"},"content":{"rendered":"

Definition of the GDT function<\/h4>\n

The GDT<\/em> function is a mapping that relates each distance cutoff ?<\/em> to the percentage of model residues that can be placed at distance ??<\/em> from the corresponding residues in the experimentally determined structure. The graph
\n of a GDT<\/em> function provides a valuable insight into the quality of a protein model (Fig.\u00c2\u00a01). More specifically, the closer the graph runs to the horizontal axis (in other words,
\n the smaller the area under the graph), the better the model.<\/p>\n

Fig.\u00c2\u00a01.<\/strong> CASP 8 example. a<\/strong>Red lines<\/em> represent GDT<\/em> plots of different 3D models of the target protein T0482 submitted during the CASP8
\n experiment. The model submitted by the group TS208 is represented in blue<\/em>. b<\/strong> Structural alignment of the model submitted by the group TS208 (blue<\/em>) and the experimental structure (red<\/em>) over 67 residues evaluated by the CASP assessors\n <\/p>\n

As a single numerical measure of the model quality, GDT<\/em>_TS<\/em> is extensively used at CASP to rank different models for the same target 13], 14]. Since it represents the average of GDT<\/em>_P ?<\/sub><\/em>
\n at several distance cutoffs, GDT<\/em>_TS<\/em> is often viewed as an approximation of the area under the GDT<\/em> curve (GDT<\/em>_A<\/em>) 10]\u00e2\u20ac\u201c12]:\n <\/p>\n

\n

\n

(1)<\/span><\/p>\n

However, as we demonstrate below, such a sparse sampling of the values of GDT<\/em> function compromises the reliability of GDT<\/em>_TS<\/em>.\n <\/p>\n

In our first example, we analyze the protein model for the target T0482, submitted
\n by the group TS208 at CASP8 (Fig.\u00c2\u00a01). The GDT<\/em>_TS<\/em> score of this particular model was not even among the best dozen at CASP8, despite
\n the fact that it fits the largest number of residues at distance <\/span> from the corresponding residues in the experimental structure. In fact, the blue
\n model (Fig.\u00c2\u00a01a) can be superimposed onto the experimental structure so that all of its residues
\n are at distance <\/span> from the residues in the experimental structure (Fig.\u00c2\u00a01b), while no such superposition exists for any other model, even for the distance
\n cutoff of 10\u00c3\u2026. Interestingly, according to the MAMMOTH algorithm 15], the blue model is the best model for this particular target, while the DALI 16] algorithm ranks it as the second best.\n <\/p>\n

Although it is impossible to tell whether #13 GDT<\/em>_TS<\/em> rank is more or less fair than #1 and #2 rank assigned by MAMMOTH and DALI, respectively,
\n it is also not difficult to see that the ranking by the area under the GDT<\/em> plot (GDT<\/em>_A<\/em>) would serve as a good compromise between these extremes.\n <\/p>\n

The next two examples illustrate further disadvantages of GDT<\/em>_TS<\/em>. As seen in Fig.\u00c2\u00a02, better GDT<\/em>_TS<\/em> scores can be assigned to obviously worse models. Moreover, as demonstrated in Fig.\u00c2\u00a03, very similar models can have significantly different GDT<\/em>_TS<\/em> scores. <\/p>\n

Fig.\u00c2\u00a02.<\/strong> Insensitivity of GDT_TS. This theoretical example shows no sensitivity of GDT<\/em>_TS<\/em> to large variations in model quality. Surprisingly, the red model<\/em> has a better GDT<\/em>_TS<\/em> score than the better blue model<\/em>, even though it is worse by all standards. Notice that, unlike GDT<\/em>_TS<\/em>, the GDT<\/em>_A<\/em> measure is not skewed by the values at the cutoff points 1, 2, 4 and 8\u00c2\u00a0\u00c3\u2026. In fact,
\n the GDT<\/em>_A<\/em> score of the blue model is twice as good as that of the red model<\/em><\/p>\n

Fig.\u00c2\u00a03.<\/strong> Oversensitivity of GDT_TS. a<\/strong> A four helix bundle-like (toy) protein (dashed grey line<\/em>) along with two of its, almost identical, models (red<\/em> and blue<\/em>). A realistic example of such a target protein (PDB ID: 1JM0A) is shown on the right<\/em> (b<\/strong>). In this example, we assume that the protein and its models are extended to the
\n right to include 100 or more residues. Note that, if d<\/em>\u00c2\u00a0?\u00c2\u00a0{1\u00c2\u00a0\u00c3\u2026, 2\u00c2\u00a0\u00c3\u2026, 4\u00c2\u00a0\u00c3\u2026, 8\u00c2\u00a0\u00c3\u2026} then the GDT<\/em> score of the blue model<\/em> is significantly higher than that of the red model<\/em>. For instance, if d<\/em>\u00c2\u00a0=\u00c2\u00a02\u00c2\u00a0\u00c3\u2026, then the blue model has the GDT_TS<\/em> score of about 87.5 since ~<\/strong>50\u00c2\u00a0% all of its residues can be fit at distance ?1\u00c2\u00a0\u00c3\u2026 and 100\u00c2\u00a0% under each distance
\n 2, 4 and 8\u00c2\u00a0\u00c3\u2026 from the corresponding residues in the experimental structure (dashed grey<\/em>). On the other hand, the GDT_TS<\/em> score of the red model<\/em> is only about 75, since only ~50\u00c2\u00a0% of the red model\u00e2\u20ac\u2122s<\/em> residues can be placed under 1 and 2\u00c2\u00a0\u00c3\u2026 and 100\u00c2\u00a0% under 4 and 8\u00c2\u00a0\u00c3\u2026. In fact, no matter
\n how close the red model<\/em> gets to the blue model<\/em>, its GDT_TS<\/em> score will never improve. Note also that the blue<\/em> and red models<\/em> have almost identical GDT_A<\/em> scores, since GDT_A<\/em> is not sensitive to small coordinate changes\n <\/p>\n

Mathematical formalism<\/h4>\n

Strictly speaking, the GDT<\/em> function is not well-defined. Zooming into the plot of the model highlighted in Fig.\u00c2\u00a01a, we see a set of many small vertical segments, meaning that each point on the horizontal
\n axis is mapped to zero or more points on the vertical axis (Fig.\u00c2\u00a04). On the other hand, the inverse function (mapping each distance cutoffs ?<\/em> to the percentage of residues in the model structure that can be fit under the distance
?<\/em> from the corresponding residues in the experimental structure) is obviously well
\n defined. This allows us to define the area under the GDT<\/em> plot as the complement of the area under the inverse function:\n <\/p>\n

\n

\n

(2)<\/span><\/p>\n

where Total<\/em>_Area<\/em> represents the area of the rectangular region under consideration (100\u00c2\u00a0\u00c3\u2014\u00c2\u00a010). We
\n start our mathematical formalism by first defining a protein structure.<\/p>\n

Fig.\u00c2\u00a04.<\/strong> A closer look at the GDT_TS function. Zooming into the GDT<\/em> plot of the model highlighted in Fig.\u00c2\u00a01. What appears to be the graph of a continuous function is, in fact, a set consisting
\n of many separated vertical line<\/em> segments\n <\/p>\n

Definition 1<\/strong><\/h4>\n

A protein structure a<\/em> is a sequence of points in the three dimensional Euclidean space <\/span><\/p>\n

\n

\n

(3)<\/span><\/p>\n

The sequence elements a i<\/sub><\/em>
\n can represent individual atoms, but it is more typical (in particular in protein structure
\n prediction experiments) to assume that each point a i<\/sub><\/em>
\n corresponds to the alpha-carbon atom of the protein\u00e2\u20ac\u2122s ith<\/em> amino acid.\n <\/p>\n

In what follows, we formally define the GDT<\/em> function 17]. For simplicity of presentation, we will modify the codomain of GDT<\/em> to represent the \u00e2\u20ac\u0153fraction of residues\u00e2\u20ac\u009d (ranging from 0 to 1) instead of \u00e2\u20ac\u0153percentages
\n of residues\u00e2\u20ac\u009d (ranging from 0 to 100). We note that this simple rescaling of the ordinate
\n values will have no effects on the results obtained in our study.\n <\/p>\n

Definition 2<\/strong><\/h4>\n

Let <\/span> be a protein structure consisting of n<\/em> amino acids, let <\/span> be a 3D model of a<\/em>, and let <\/span> be a positive constant. The Hubbard function (or GDT function)<\/em> is the function <\/span>, defined by <\/span>, where denotes the Euclidean norm on <\/span> and ?<\/em> is a rigid transformation (a composition of a rotation and a translation).\n <\/p>\n

Theorem 1<\/strong><\/h4>\n

H b<\/sub> is a stepwise function with finitely many steps<\/em><\/span>, 1\u00c2\u00a0?\u00c2\u00a0k<\/em>\u00c2\u00a0?\u00c2\u00a0n<\/em>\u00c2\u00a0?\u00c2\u00a01.\n <\/p>\n

Proof<\/em><\/h4>\n

Since H b<\/sub><\/em>
\n is monotony non-decreasing and since the range of H b<\/sub><\/em>
\n is a finite subset of (0,1], it follows that H b<\/sub><\/em>
\n must be a stepwise function. To complete the proof, we note that the number of steps
\n in H b<\/sub><\/em>
\n matches the size of its range, which does not exceed n<\/em>\u00c2\u00a0?\u00c2\u00a01, where n<\/em> is the length of b<\/em>.\n <\/p>\n

For simplicity of presentation, from now on (and whenever the model b<\/em> is implied), we will omit the subscript in H b<\/sub><\/em>
\n and denote the Hubbard function only by H<\/em>.\n <\/p>\n

Algorithm for GDT_A<\/h4>\n

The area under H<\/em> is the sum of the areas of the rectangular regions (? i<\/sub><\/em>
\n )(? i<\/sub><\/em>
\n \u00c2\u00a0?\u00c2\u00a0? i<\/sub><\/em>?1<\/sub>
\n ):\n <\/p>\n

\n

\n

(4)<\/span><\/p>\n

where ?<\/em>0<\/sub>
\n \u00c2\u00a0=\u00c2\u00a00 and <\/span> (Fig.\u00c2\u00a05). It would be trivial to compute Area<\/em> had we known all ? i<\/sub><\/em>
\n and all function values H<\/em>(? i<\/sub><\/em>
\n ). Unfortunately, even if we knew the step points ? i<\/sub><\/em>
\n , it would be computationally very difficult to compute the function values at them,
\n since the best to date algorithm for computing H<\/em>(? i<\/sub><\/em>
\n ) runs on the order of O<\/em>(n<\/em>7<\/sup>
\n ) 7]. Hence, we resort to using the Riemann sums to approximate (instead of to compute
\n exactly) the area under the graph of H<\/em>.<\/p>\n

Fig.\u00c2\u00a05.<\/strong> The general shape of the Hubbard function. Notice that the values ? i<\/sub><\/em>
\n along with the function values H b<\/sub><\/em>
\n (? i<\/sub><\/em>
\n ), <\/span>, uniquely determine the area under the graph of H b<\/sub><\/em>
\n . At the biannual CASP experiment, <\/span><\/p>\n

The following definition and an accompanying theorem can be found in virtually any
\n mathematical analysis textbook.\n <\/p>\n

Definition 3<\/strong><\/h4>\n

If <\/span> is a function then <\/span> where a<\/em>\u00c2\u00a0=\u00c2\u00a0x<\/em>0<\/sub>
\n \u00c2\u00a0\u00c2\u00a0x<\/em>1<\/sub>
\n \u00c2\u00a0\u00c2\u00a0\u00c2\u00b7\u00c2\u00b7\u00c2\u00b7\u00c2\u00a0\u00c2\u00a0x n<\/sub><\/em>
\n \u00c2\u00a0=\u00c2\u00a0b<\/em> is the partition of the interval [a<\/em>,\u00c2\u00a0b<\/em>] and v i<\/sub><\/em>
\n denotes the supremum of f<\/em> over [x i<\/sub><\/em>?1<\/sub>
\n ,\u00c2\u00a0x i<\/sub><\/em>
\n ], is called the upper Riemann sum<\/em> of f<\/em> on [a<\/em>,\u00c2\u00a0b<\/em>].\n <\/p>\n

Theorem 2<\/strong><\/h4>\n

Let f be a real, non-decreasing, Riemann integrable function on an interval<\/em> [a<\/em>,\u00c2\u00a0b<\/em>]. Then<\/em><\/p>\n

\n

\n

(5)<\/span><\/p>\n

where<\/em><\/p>\n

\n

\n

(6)<\/span><\/p>\n

is the upper Riemann sum of f the and<\/em><\/span><\/p>\n

Observe that, since H b<\/sub><\/em>
\n is piecewise continuous, it must be integrable on <\/span>. Thus, the area under the graph of H<\/em> is\n <\/p>\n

\n

\n

(7)<\/span><\/p>\n

To approximate Area<\/em> with a Riemann sum, one can define the partition points <\/span>, where <\/span> (Fig.\u00c2\u00a06) and then compute an estimate Area<\/em>(<\/span>) of Area<\/em> as\n <\/p>\n

\n

\n

(8)<\/span><\/p>\n

Fig.\u00c2\u00a06.<\/strong> Approximation of the Hubbard function by the Riemann sum. Area<\/em>(<\/span>) is the sum of the areas of all rectangular regions\n <\/p>\n

The error |Area<\/em>\u00c2\u00a0\u00e2\u20ac\u201c\u00c2\u00a0Area<\/em> (<\/span>)| in the estimate (8) is below 2<\/span>. Up to a half of this error is due to the error in the Riemann sum with the remaining
\n error being due to the possible placement of the last partition point m<\/em><\/span> outside the interval <\/span>.\n <\/p>\n

Unfortunately, computing the area estimates according to (8) is still a challenging problem, because (as we mentioned above), there is no computationally
\n effective procedure for finding the function values H<\/em>(i<\/em><\/span>). To circumvent the problem, we utilize an efficient algorithm capable of computing
\n the lower bound estimates H i<\/sub><\/em>
\n of H<\/em>(i<\/em><\/span>), satisfying <\/span>, <\/span>. We then compute an estimate <\/span> of Area<\/em> as\n <\/p>\n

\n

\n

(9)<\/span><\/p>\n

Since <\/span>, it follows that <\/span> is a <\/span>-approximation of Area<\/em>. Below we show how to compute all H i<\/sub><\/em>
\n \u00e2\u20ac\u2122s<\/em>, and, in turn, <\/span> in time <\/span>, where n<\/em> is the length of b<\/em>. Our algorithm takes advantage of an efficient procedure for computing near optimal
GDT<\/em>_TS<\/em> values 5].\n <\/p>\n

Let T<\/em>(b<\/em>) denotes the image of the model structure b<\/em> under the transformation T<\/em>. Denote by <\/span> the largest fraction of residues from T<\/em>(b<\/em>) that are at distance ??<\/em> from the corresponding residues in the experimental structure a<\/em>. To find each H i<\/sub><\/em>
\n , it is enough to compute a rigid body transformation T i<\/sub><\/em>
\n satisfying <\/span>.\n <\/p>\n

Denote by T ?<\/sub><\/em>
\n a transformation that places a largest subset b ?<\/sub><\/em>
\n of residues from b<\/em> at distance ??<\/em> from the corresponding residues in the experimental structure. Given T ?<\/sub><\/em>
\n , one can easily compute b ?<\/sub><\/em>
\n by calculating all n<\/em> distances between the residues a i<\/sub><\/em>
\n and T ?<\/sub><\/em>
\n (b i<\/sub><\/em>
\n ). Note that P<\/em>(T ?<\/sub><\/em>
\n ,\u00c2\u00a0?<\/em>)\u00c2\u00a0=\u00c2\u00a0H<\/em>(?<\/em>). We approximate the transformation T ?<\/sub><\/em>
\n by a so-called \u00e2\u20ac\u0153near-optimal\u00e2\u20ac\u009d transformation i.e., a transformation that places at
\n least as many residues from the model structure under distance <\/span> as the optimal transformation T ?<\/sub><\/em>
\n places under the distance ?<\/em>. From now on, we will use <\/span> to denote a \u00e2\u20ac\u0153near-optimal\u00e2\u20ac\u009d transformation and the corresponding set of residues will
\n be denoted by <\/span>. Observe that <\/span>.\n <\/p>\n

Building upon any procedure for computing <\/span>, one can develop an algorithm for <\/span> by substituting <\/span> for H i<\/sub><\/em>
\n in (10), where <\/span>. Several existing methods can be modified and made suitable for finding <\/span>. The most efficient such method relies on the concept of \u00e2\u20ac\u0153radial pair\u00e2\u20ac\u009d 5].\n <\/p>\n

Definition 4<\/strong><\/h4>\n

Let <\/span> be a set of points in the three-dimensional Euclidean space. An ordered pair of points
<\/span> is called a radial pair<\/em> of S<\/em> if s j<\/sub><\/em>
\n is the furthest point from s i<\/sub><\/em>
\n among all points in S<\/em>.\n <\/p>\n

Theorem 3<\/strong><\/h4>\n

Let T<\/em>1<\/sub> and T<\/em>2<\/sub>be two transformations and let<\/em><\/span> be a radial pair of S<\/em>. If<\/em><\/span>and<\/em><\/span> then there exists a rotation R around the line through T<\/em>1<\/sub>
\n (s k<\/sub><\/em>
\n ) and<\/em><\/span>such that<\/em><\/span>, for any s p<\/sub> in S<\/em>. The rotation R can be found in time O<\/em> (nlogn<\/em>), where n is the size of S<\/em>.\n <\/p>\n

A proof of the above theorem can be found in 5]. The algorithm for finding R<\/em> is fairly straightforward and it relies on the so-called plane<\/em>\u00e2\u20ac\u201csweep<\/em> approach 18].\n <\/p>\n

The Theorem 3 implies that one choice for the near-optimal transformation <\/span> is the transformation R<\/em>\u00c2\u00a0?\u00c2\u00a0T<\/em>, where T<\/em> is any transformation that maps the points b k<\/sub><\/em>
\n and b l<\/sub><\/em>
\n from the radial pair (b k<\/sub><\/em>
\n ,\u00c2\u00a0b l<\/sub><\/em>
\n ) of b ?<\/sub><\/em>
\n to the <\/span> neighborhoods of T ?<\/sub><\/em>
\n (b k<\/sub><\/em>
\n ) and T ?<\/sub><\/em>
\n (b l<\/sub><\/em>
\n ), respectively, and R<\/em> is the rotation around the radial axis <\/span> that maps the remaining points from T<\/em>(b ?<\/sub><\/em>
\n ) to the <\/span>-neighborhoods of the corresponding points from T ?<\/sub><\/em>
\n (b ?<\/sub><\/em>
\n ).\n <\/p>\n

In search for a radial pair of b ?<\/sub><\/em>
\n , the algorithm in 5] explores all n<\/em>2<\/sup>
\n possible pairs of residues in b<\/em>. For each candidate radial pair <\/span>, the algorithm generates a finite, representative set of transformations that map
b k<\/sub><\/em>
\n and b l<\/sub><\/em>
\n into <\/span> neighborhoods of a k<\/sub><\/em>
\n and a l<\/sub><\/em>
\n , respectively (see the paragraph below for more details). For every such transformation
T<\/em>, a plane-sweep algorithm 18] is used to find a rotation R<\/em> around the axis <\/span> that maximizes the number of residues from R<\/em>(T<\/em>(b<\/em>)) that can be placed at distance <\/span> from the corresponding residues in a<\/em>.\n <\/p>\n

A finite set of transformations that map the residues b k<\/sub><\/em>
\n and b l<\/sub><\/em>
\n into the <\/span> neighborhoods of a k<\/sub><\/em>
\n and a l<\/sub><\/em>
\n , respectively, is constructed in such a way to ensure that for at least one of those
\n transformation T<\/em>, <\/span> and <\/span>. This can be achieved by partitioning <\/span> into small cubes of side length slightly smaller than <\/span> and then collecting the vertices of the cubes that are inside the open ball of radius
<\/span> around a k<\/sub><\/em>
\n (Fig.\u00c2\u00a07). The elements of this set, denoted by A k<\/sub><\/em>
\n , are the candidate points T<\/em>(b k<\/sub><\/em>
\n ). The number of points in A k<\/sub><\/em>
\n is <\/span> and at least one of them must be at distance <\/span> from T ?<\/sub><\/em>
\n (b k<\/sub><\/em>
\n ) (Fig.\u00c2\u00a07). For each point a k<\/sup><\/em>
\n \u00c2\u00a0?\u00c2\u00a0A k<\/sub><\/em>
\n , the set A l<\/sub><\/em>
\n (a k<\/sup><\/em>
\n ) of possible images of b l<\/sub><\/em>
\n under T<\/em> is computed by discretizing the spherical cap <\/span>, where S<\/em>(a<\/em>,\u00c2\u00a0r<\/em>) and B<\/em>(a<\/em>,\u00c2\u00a0r<\/em>) denote the sphere and the open ball in <\/span> with center a<\/em> and radius r<\/em>, respectively, in such a way that at least one point from A l<\/sub><\/em>
\n (a k<\/sup><\/em>
\n ) is found at distance <\/span> from T ?<\/sub><\/em>
\n (b l<\/sub><\/em>
\n ) (Fig.\u00c2\u00a07). We note that size of A l<\/sub><\/em>
\n (a k<\/sup><\/em>
\n ) is <\/span>. Hence, the total number of candidate pairs of points (T<\/em>(b k<\/sub><\/em>
\n ),\u00c2\u00a0T<\/em>(b l<\/sub><\/em>
\n )) is <\/span>.<\/p>\n

Fig.\u00c2\u00a07.<\/strong> Discretizing the space of rigid body transformations. 2D illustration of A k<\/sub><\/em>
\n (the set of the vertices of the squares<\/em> shown on the left<\/em>) and the set A l<\/sub><\/em>
\n (a k<\/sup><\/em>
\n ) generated for one a k<\/sub><\/em>
\n \u00c2\u00a0?\u00c2\u00a0A k<\/sub><\/em><\/p>\n

An obvious to compute <\/span> is to run the just described algorithm m<\/em> times in succession, for <\/span>. However, such an approach results in many unnecessary repeated calculations as the
\n area around a k<\/sub><\/em>
\n and the corresponding spherical cap in the neighborhoods of a l<\/sub><\/em>
\n are discretized over and over again. Moreover, all transformations T<\/em> and R<\/em>, generated and inspected during the procedure for finding <\/span>, are inspected again during the procedure for finding <\/span>, for each j<\/em>\u00c2\u00a0\u00c2\u00a0i<\/em>. <\/p>\n

We show that all transformations <\/span> and the corresponding values H<\/em>1<\/sub>
\n ,\u00c2\u00a0\u00e2\u20ac\u00a6,\u00c2\u00a0H m<\/sub><\/em>
\n can be computed, at once, during the procedure of finding the last transformation,
\n namely <\/span>. As demonstrated in the pseudocode above, the transformation T<\/em> is generated only once for each pair of points <\/span> and a sweep-plane algorithm for finding R<\/em> is called only once for each i<\/em> satisfying <\/span> and <\/span>. The values of H i<\/sub><\/em>
\n are updated on the fly.\n <\/p>\n

Running time analysis<\/h4>\n

To analyze the algorithm\u00e2\u20ac\u2122s running time, we note that the number of iterations of
\n the first for<\/em> loop is equal to the number of candidate radial pairs (b k<\/sub><\/em>
\n ,\u00c2\u00a0b l<\/sub><\/em>
\n ), which is <\/span> The number of iterations of the second for<\/em> loop matches the number of pairs of grid points around a k<\/sup><\/em>
\n and a l<\/sup><\/em>
\n , which is <\/span>. Each one of <\/span> iterations of the third for<\/em> loop calls a O<\/em>(nlogn<\/em>) plane-sweep procedure to compute an optimal rotation and (if needed) to update the
\n value H i<\/sub><\/em>
\n . Hence, the asymptotic time complexity of the three nested for<\/em> loops is <\/span><\/p>\n","protected":false},"excerpt":{"rendered":"

Definition of the GDT function The GDT function is a mapping that relates each distance cutoff ? to the percentage of model residues that can be placed at distance ?? from the corresponding residues in the experimentally determined structure. The graph of a GDT function provides a valuable insight into the quality of a protein […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[],"tags":[],"class_list":["post-26999","post","type-post","status-publish","format-standard","hentry"],"_links":{"self":[{"href":"http:\/\/healthmedicinet.com\/news\/wp-json\/wp\/v2\/posts\/26999","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/healthmedicinet.com\/news\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/healthmedicinet.com\/news\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/healthmedicinet.com\/news\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/healthmedicinet.com\/news\/wp-json\/wp\/v2\/comments?post=26999"}],"version-history":[{"count":0,"href":"http:\/\/healthmedicinet.com\/news\/wp-json\/wp\/v2\/posts\/26999\/revisions"}],"wp:attachment":[{"href":"http:\/\/healthmedicinet.com\/news\/wp-json\/wp\/v2\/media?parent=26999"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/healthmedicinet.com\/news\/wp-json\/wp\/v2\/categories?post=26999"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/healthmedicinet.com\/news\/wp-json\/wp\/v2\/tags?post=26999"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}