- Methodology article
- Open Access
Computing H/D-Exchange rates of single residues from data of proteolytic fragments
- Ernst Althaus^{1}Email author,
- Stefan Canzar^{2},
- Carsten Ehrler^{3},
- Mark R Emmett^{4, 5},
- Andreas Karrenbauer^{6},
- Alan G Marshall^{4, 5},
- Anke Meyer-Bäse^{7},
- Jeremiah D Tipton^{4} and
- Hui-Min Zhang^{4}
https://doi.org/10.1186/1471-2105-11-424
© Althaus et al; licensee BioMed Central Ltd. 2010
- Received: 16 January 2009
- Accepted: 11 August 2010
- Published: 11 August 2010
Abstract
Background
Protein conformation and protein/protein interaction can be elucidated by solution-phase Hydrogen/Deuterium exchange (sHDX) coupled to high-resolution mass analysis of the digested protein or protein complex. In sHDX experiments mutant proteins are compared to wild-type proteins or a ligand is added to the protein and compared to the wild-type protein (or mutant). The number of deuteriums incorporated into the polypeptides generated from the protease digest of the protein is related to the solvent accessibility of amide protons within the original protein construct.
Results
In this work, sHDX data was collected on a 14.5 T FT-ICR MS. An algorithm was developed based on combinatorial optimization that predicts deuterium exchange with high spatial resolution based on the sHDX data of overlapping proteolytic fragments. Often the algorithm assigns deuterium exchange with single residue resolution.
Conclusions
With our new method it is possible to automatically determine deuterium exchange with higher spatial resolution than the level of digested fragments.
Keywords
- Exchange Rate
- Deuterium
- FK506
- Integer Linear Program
- Deuterium Exchange
Background
In the solution-phase Hyrdogen/Deuterium ex-change (sHDX) experiment, protein surface accessibility is probed by exchange of labile hydrogen for deuterium. Simply speaking, hydrogens located at solvent exposed sites exchange at a higher rate with deuteriums from the solution than others. From these exchange rates one can therefore deduce information about protein solvent accessibility and thus protein conformation.
There is controversy surrounding the effect of D_{2}O solvent on the conformation of proteins. Sheu et al. [1] used molecular dynamic modeling of a small peptide to illustrate compaction of the peptide conformation in D_{2}O versus H_{2}O. This small compaction of the conformation occurs when the pep-tide is fully deuterated (which is never observed in the sHDX experiments). Since sHDX monitors the incorporation of deuterium over time the resulting slight compaction of the structure is minimized. Other methods used for the study of protein/protein interaction or protein conformation such as cross-linking [2, 3] or hydroxyl radical addition [4–6] result in large conformational change of the protein structure; leaving sHDX as the method of choice for probing protein conformational changes in solution.
NMR spectroscopy has been the gold standard for determination of protein structure, but it has limitations on protein solubility and molecular weight (<50 kD). Solution-phase HDX with mass spectrometry analysis has higher sensitivity and is not limited by molecular weight, but sHDX is hampered with a major difficulty. One only obtains exchange data for peptic fragments and assigning exchange rates to single residues has to be done by manual interpretation.
We provide an automated method to resolve this problem. More precisely, we present an algorithm that enumerates all possible exchange rates for single residues that explain the observed data of the peptic fragments. As the number of possibilities is often very large, we combine sets of assignments to equivalence classes which are easily interpreted such that the number of equivalence classes is typically very small.
The assignment of exchange rates to single residues from the data of the peptic fragments is a combinatorial problem. Hence, we apply methods from combinatorial optimization to it, i.e. we show how to formalize the problem as an integer linear program and propose methods to solve the problem.
Biochemical Background
Concerning the determination of protein-protein interaction, X-ray crystal diffraction and NMR [7] pro-vide the highest resolution of the sites of interaction. On the downside, both methods require large (milligram) quantities of protein. Other techniques rely on chemical or photo-induced reactions with MS analysis [8, 9] to reveal functional groups that are ex-posed to the solvent. These methods also suffer from physical limitations.
Another method utilizes hydroxyl radical reactions with alkyl CH bonds. The OH tends to re-act mainly with surface-exposed residues providing a good footprint of the solvent exposed surface of the protein(s) [4, 6]. The modification is covalent and thus irreversible, but each modification can potentially change the conformation of the protein, thus skewing results.
Exchange of labile hydrogens for deuteriums (sHDX) as a probe of protein surface accessibility does not change the conformation of the protein. Advantages of MS over NMR and X-ray crystallography structural determination are the ability to work at low concentration and high molecular weight.
The experiment is initiated by dilution of the protein solution into a biological buffer made with D_{2}O. Solvent accessible hydrogens are exchanged with deuterium. The exchange is quenched (greatly slowed) by dropping the pH to between pH 2.3 and pH 2.5 and lowering the temperature to approximately 0°C. The protein complex is digested with a protease that is active under quench conditions (such as pepsin) and on-line liquid chromatography is performed directly to the FT-ICR MS. Deuterium in corporation is monitored by the increase in mass of each peptic fragment as the deuterons are added.
Mathematical Abstraction
Overview of terms and symbols
Term/Symbol | Meaning |
---|---|
fragment | A fragment is a set of consecutive residues resulting from the digestion of the protein. |
ℱ | The set of all (possibly overlapping) fragments. |
color | We divide the exchange rates into classes and associate a color with each class. |
$\mathcal{S}$ | The set of the K distinct colors. |
part | A part is a maximal set of residues contained in the same set of fragments, i.e., an inclusion-wise maximal subset of a fragment that is either contained in or disjoint from any other fragment. |
$\mathcal{P}$ | The partition of the residues into parts as defined above. |
subproblem | An instance decomposes into independent subproblems if there is no overlap between the fragments of the different subproblems. |
The mathematical notion introduced above is illustrated in Figure 1. There the residues, numbered from 1 to 28, are spanned by 9 peptic fragments, i.e. |ℱ = 9|. The third peptic fragment "VWGKVEAD" will then be represented by the integer interval (12; 19). From the experimental data we know that 5 out of the 8 residues contained in this fragment exchanged slowly (s), two at medium rate (m), and the last remaining residue exchanged fast (f) and thus ${b}_{(12,19)}^{s}=5,\phantom{\rule{0.5em}{0ex}}{b}_{(12,19)}^{m}=2$, and ${b}_{(12,19)}^{f}=1$.
Determining the exchange rate of single residues from the experimentally found data for the peptic fragments then translates into finding a "consistent" assignment of colors from $\mathcal{S}$ to the integer points from {1, ..., n}, representing the residues of the protein, that complies with the constraints imposed by the "requirements" ${b}_{\left(i,j\right)}^{k}$. More precisely, we have to determine an assignment π : {1, ..., n} ↦ $\mathcal{S}$ such that |{i ≤ l ≤ j : π(l) = k}| = ${b}_{\left(i,j\right)}^{k}$ for all given fragments (i, j) ∈ ℱ and all possible colors k ∈ $\mathcal{S}$. We call such an assignment feasible.
We say that two fragments (i, j) and (i', j') over-lap, if they share at least one common residue, i.e. (i, j) ∩ (i', j') ≠ ∅. The partition of the set of fragments ℱ into a maximum number of subsets, such that no two fragments from different subsets overlap, defines independent subproblems; an assignment of exchange rates to the residues spanned by the fragments of one subset does not affect the solution of a subproblem corresponding to any other subset of fragments.
Furthermore, we denote by $\mathcal{P}$ the partition of the set of residues {1, ..., n} into maximal subsets such that residues from the same subset are spanned by exactly the same set of fragments. More precisely, for all residues i and j in the same part of $\mathcal{P}$ and for all fragments f ∈ ℱ it holds i ∈ f ⇔ j ∈ f. Hence, for each part p ∈ $\mathcal{P}$ and each fragment f ∈ ℱ either p ⊆ f or p ∩ f = ∅. In Figure 1 for example, residues number 7 (Q), 8 (Q), and 9 (V), are all contained in fragments number 1, 2, 6, and 8 and thus form an element $\stackrel{\xaa}{p}$ of partition $\mathcal{P}$. Note that for the two neighboring residues the set of containing fragments differs from {1, 2, 6, 8} and therefore part$\stackrel{\xaa}{p}$ = {7, 8, 9} is maximal.
In Figure 1 the colors green, yellow and red encode an optimal assignment π* of the exchange rates slow, medium and fast, with respect to objective (2). Under this assignment, fragment 3 contributes an error of 1 both w.r.t. color yellow (medium exchange rate)and color red (fast rate) to the total error of 17, while it satisfies the requirement for color green (slow rate, numbered 1) ${b}_{\left(12,19\right)}^{1}=5$ exactly.
Results and Discussion
In the following, we present different approaches to tackle the assingment problem that we have derrived from the mathematical abstraction mentioned before.
Integer linear programming formulation
First, we formulate the idealized version of the problem assuming error-free experimental data as an integer linear program (ILP). That is, we give an ILP whose feasible solutions correspond one-to-one to the feasible assignments of colors to residues.
We denote by ${x}^{k}=\left({x}_{1}^{k},{x}_{2}^{k},\mathrm{...},{x}_{n}^{k}\right)$ the vector of binary variables modeling the assignment of color k and let $x={({x}^{k})}_{k\in \mathcal{S}},\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}x\in {\{0,1\}}^{Kn}$.
Since every residue is assigned exactly one color, it must hold ${\sum}_{k\in \mathcal{S}}{x}_{i}^{k}}=1$ for all i ∈ {1, ..., n}. Conversely, every 0-1 assignment to variables x satisfying ${\sum}_{k\in \mathcal{S}}{x}_{i}^{k}}=1$ for all i ∈ {1, ..., n} corresponds to an assignment of colors to residues. A 0-1 assignment to x corresponds to a feasible color assignment π, if and only if furthermore ${\sum}_{l=i}^{j}{x}_{l}^{k}}={b}_{\left(i,j\right)}^{k$ holds for all (i, j) ∈ ℱ and k ∈ $\mathcal{S}$.
We refer to this integer linear program as basic-ILP.
In our experiments, it turns out that finding a single solution is very fast, whereas enumerating all solutions takes quite some time due to their large number. This large number can be explained as follows: Recall that $\mathcal{P}$ is the partition of {1, ..., n} into a minimal number of parts, such that for each element p ∈ $\mathcal{P}$ and each fragment f ∈ F either p ⊆ f or p ∩ f = ∅. In other words, no fragment starts or ends within such a part. Therefore, from an assignment π we can derive further assignments π' exhibiting the same total error, by simply permuting the colors within these parts, i.e. if i, j ∈ p for p ∈ $\mathcal{P}$ and the total error of an assignment π is e_{1}, than π' with π' (i) = π (j), π' (j) = π (i) and π' (l) = π (l) for l ≠ i, j has total error e_{2} with e_{2} = e_{1}. We call two assignments equivalent, if one can be obtained from the other by iteratively applying this rule.
where P is the vector that contains |p| for each component p ∈ $\mathcal{P}$and $y={({y}^{k})}_{k\in \mathcal{S}}$ We refer to this integer linear program as improved-ILP. We compute all solutions within a certain error bound by following basically the same approach as described above. However, the number of solutions now is just a fraction of the number of solutions of the original basic-ILP yielding a significant speed-up
Although there is commercial software for integer programming which quickly solves instances of reasonable size, there is no algorithm that is guaranteed to find an optimum solution in polynomial time, since integer programming is NP-complete in general. However, the problem of assigning exchange rates to residues in a way that is conform with the experimentally found bulk data exhibits a certain combinatorial structure. In the next section, we exploit this fact to derive an exact polynomial-time algorithm for the case of two colors and use it as a building block for approximation algorithms for more than two colors subsequently.
A Combinatorial Approach
For three or more colors the complexity is open. The totally unimodularity of the constraint matrix is destroyed, i.e. there are instances with fractional vertices, e.g. the one from Figure 2 with the appropriate right hand sides. Moreover, there is an instance which has a positive error, but the value of the LP is 0. Hence the integrality gap is infinite. If the number of colors is not fixed but part of the input, the problem is NP-complete [13].
A Simple and Efficient Heuristic for the General Case
We present an algorithm that uses our combinatorial approach for the 2-color case (K = 2) from previous section as a subroutine to provide solutions that approximate (without performance guarantee) a coloring, i.e. an assignment of colors to residues, with minimum total error for instances with arbitrary but fixed number of colors. The general idea is to reduce the problem to the 2-color case by merging all but one color, say color i, to a single color and solve the resulting problem by an algorithm for the minimum cost circulation problem, as described in the section about the Combinatorial Approach. We remove residues colored i by the obtained solution and solve the coloring problem on the remaining residues using K - 1 colors recursively.
Residues assigned color k in an optimal solution to this problem will be colored k in the final solution too, the assignment of the remaining colors $\mathcal{S}$\{k} to the remaining residues is computed recursively.
Note that the order in which colors are selected to be the next fixed color k in the recursive computation can be arbitrary. Nevertheless, they might lead to solutions of different total error. As we have only three different colors in our experimental data, we evaluate all six orderings and return the best solution found.
In the next section we present a Lagrangian relaxation method to compute, based on our combinatorial approach for the 2-color case, a bound on the minimum total error, which is exploited in a branch-&-bound manner to determine all optimal colorings.
A Lagrangian Relaxation Approach
This linear program differs from LP (7) only in the right-hand sides of the equality constraints.
Conclusions
We applied our methods to process data from typical biochemical experiments. We report our results for four proteins: Calcium-binding protein (Cabin), Cytochrome P450 (CytoC), FK506 binding protein(FKBP), with two different digests (pepsin and XIII), and myoglobin. As a preprocessing step, the single fragments were analyzed with our integer linear programming based technique [14], except for FKBP V2 (MEM) which was analyzed with the MEM-method [15] and is based on the same data as FKBP V1 (ILP). We analyzed FKBP with only the xiii digestion (V3) and combined the datasets from the two digestions (V1, V2 and V4). The number discretized exchange rates per fragment obtained in this preprocessing step serves as input to the algorithm.
Results & Runtime
Instance | Improved-ILP | Lagrange | Heuristic | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Name | n | $\mathcal{P}$ | n / $\mathcal{P}$ | ℱ | ϵ | T _{ 1 } | T _{ all } | # -Sol | T _{ 1 } | T _{ all } | T _{ 1 } |
Cabin | 78 | 26 | 3.0 | 34 | 128 | 0.02 | 3.25 | 36 | 1.36 | 8.35 | 0.02 |
CytoC | 74 | 18 | 4.1 | 17 | 40 | 0.03 | 0.37 | 1980 | 0.27 | 6.10 | 0.01 |
Subproblem 1 | 27 | 5 | 5.4 | 6 | 6 | 0.01 | 0.01 | 1 | 0.01 | 0.01 | 0.003 |
Subproblem 2 | 26 | 5 | 5.2 | 6 | 30 | 0.01 | 0.32 | 110 | 0.17 | 5.81 | 0.004 |
Subproblem 3 | 15 | 6 | 2.5 | 5 | 4 | 0.01 | 0.04 | 18 | 0.09 | 0.28 | 0.004 |
FKBP V1 (ilp) | 101 | 34 | 3.0 | 31 | 47 | 0.04 | 1.18 | 37800 | 1.03 | 137.36 | 0.017 |
Subproblem 1 | 35 | 15 | 2.3 | 12 | 15 | 0.01 | 0.45 | 126 | 0.57 | 32.83 | 0.009 |
Subproblem 2 | 16 | 5 | 3.2 | 5 | 4 | 0.01 | 0.02 | 4 | 0.04 | 0.05 | 0.003 |
Subproblem 3 | 36 | 12 | 3.0 | 14 | 28 | 0.02 | 0.71 | 75 | 0.42 | 104.48 | 0.005 |
FKBP V2 (mem) | 101 | 34 | 3.0 | 31 | 46 | 0.03 | 13.82 | 1160040 | 2.03 | 560.56 | 0.02 |
Subproblem 1 | 35 | 15 | 2.3 | 12 | 16 | 0.01 | 4.41 | 840 | 1.26 | 305.4 | 0.007 |
Subproblem 2 | 16 | 5 | 3.2 | 5 | 2 | 0.01 | 0.01 | 1 | 0.01 | 0.01 | 0.002 |
Subproblem 3 | 36 | 12 | 3.0 | 14 | 28 | 0.01 | 9.4 | 1381 | 0.76 | 255.15 | 0.007 |
FKBP V3 (xiii) | 103 | 34 | 3.0 | 47 | 38 | 0.05 | 0.16 | 6 | 0.14 | 0.13 | 0.026 |
Subproblem 1 | 22 | 10 | 2.2 | 16 | 12 | 0.01 | 0.04 | 1 | 0.02 | 0.03 | 0.008 |
Subproblem 2 | 10 | 4 | 2.5 | 4 | 2 | 0.01 | 0.02 | 3 | 0.02 | 0.02 | 0.002 |
Subproblem 3 | 11 | 5 | 2.2 | 4 | 0 | 0.01 | 0.01 | 1 | 0.02 | 0.01 | 0.003 |
Subproblem 4 | 25 | 10 | 2.5 | 22 | 24 | 0.01 | 0.08 | 2 | 0.07 | 0.06 | 0.008 |
Subproblem 5 | 3 | 1 | 3.0 | 1 | 0 | 0.01 | 0.01 | 1 | 0.01 | 0.01 | 0.001 |
FKBP V4 (both) | 105 | 43 | 2.4 | 56 | 58 | 0.05 | 0.96 | 1536 | 0.88 | 7.15 | 0.032 |
Subproblem 1 | 49 | 20 | 2.5 | 24 | 18 | 0.02 | 0.55 | 24 | 0.8 | 6.08 | 0.012 |
Subproblem 2 | 11 | 5 | 2.2 | 4 | 0 | 0.01 | 0.01 | 2 | 0.02 | 0.01 | 0.003 |
Subproblem 3 | 25 | 12 | 2.1 | 26 | 40 | 0.01 | 0.39 | 16 | 0.5 | 1.05 | 0.009 |
Subproblem 4 | 4 | 3 | 1.3 | 2 | 0 | 0.01 | 0.01 | 2 | 0.01 | 0.01 | 0.002 |
Myoglobin | 152 | 49 | 3.1 | 48 | 42 | 0.1 | 0.98 | 1121760 | 1.13 | 13.25 | 0.023 |
Subproblem 1 | 17 | 9 | 1.9 | 10 | 14 | 0.02 | 0.16 | 20 | 0.22 | 1.98 | 0.004 |
Subproblem 2 | 12 | 2 | 6.0 | 4 | 2 | 0.01 | 0.01 | 2 | 0.01 | 0.01 | 0.002 |
Subproblem 3 | 22 | 8 | 2.8 | 8 | 8 | 0.01 | 0.26 | 82 | 0.34 | 8.56 | 0.005 |
Subproblem 4 | 37 | 14 | 2.6 | 17 | 14 | 0.01 | 0.49 | 38 | 0.45 | 2.51 | 0.009 |
Subproblem 5 | 3 | 1 | 3.0 | 1 | 0 | 0.01 | 0.01 | 1 | 0.01 | 0.01 | 0.002 |
Subproblem 6 | 21 | 6 | 3.5 | 6 | 4 | 0.02 | 0.03 | 9 | 0.08 | 0.16 | 0.003 |
Subproblem 7 | 4 | 1 | 4.0 | 1 | 0 | 0.01 | 0.01 | 1 | 0.01 | 0.01 | 0.001 |
Subproblem 8 | 7 | 1 | 7.0 | 1 | 0 | 0.01 | 0.01 | 1 | 0.01 | 0.01 | 0.002 |
Comparison with NMR
Dataset | Single solution | Majority Vote | Arithmetic Mean |
---|---|---|---|
CytoC | 77.87 | 69.45 | 69.45 |
FKBP V1 (ilp) | 58.03 | 67.09 | 74.69 |
FKBP V2 (mem) | 67.09 | 67.68 | 67.09 |
FKBP V3 (xiii) | 75 | 71.88 | 70.32 |
FKBP V4 (both) | 58.03 | 62.97 | 64.20 |
In our solutions, the resolution is significantly increased compared to the input data, i.e., the length of fragments obtained from the sHDX experiments. The parts are typically small (see Table 2), between2 and 4 residues. 75% of the parts are smaller than 8 residues. For 46% of the amino-acids, we get single-residue resolution on the data.
The results for the real instances are very promising, as the small number of easily interpretable classes of equivalent solutions can be used in protein structure prediction tools and for manual inspection.
Methods
In this Section, we describe the computational methods, which we use to solve the different formulation, as well as the biochemical methods to obtain the experimental data.
Solution of the integer linear program
We implemented our approach using the C++-Library SCIL [18] to solve integer linear programs. SCIL uses the libraries LEDA [19] and SCIP [20].SCIP uses CPLEX [21] or SoPlex [22] as solver for linear programs. The underlying solution method is branch-&-bound, that is described in detail in [23].
is added to the integer linear program and hence we are faced with the problem of computing all feasible solutions of an integer linear program. We do this with a branching-approach similar to the classical branch-&-bound method for finding an optimal solution: First, the linear relaxation is solved. If the linear relaxation is infeasible, the search on this branch terminates. If the solution is integral, it is stored (provided the solution was not found yet). If there is a binary variable which was not fixed so far (i.e. not set to 0 or 1), one such variable ${x}_{l}^{k}$ is picked and the two subproblems, in which the variable is fixed o 0 or 1 recursively, are solved. Notice that it is possible that we branch on a variable which already has an integral value. In this case, the solution of the linear relaxation of the subproblem will be the same as in problem itself. Nevertheless, we will terminate, as there are only a finite number of variables to branch on.
Solving the Combinatorial Problem
We may use any algorithm that solves the Minimum Cost Circulation problem, e.g. Cycle Canceling or Successive Shortest Path (see [12] for further reference). Both approaches have their advantages. The former always maintains a feasible circulation, i.e. we start with the zero flow and augment flow along negative cycles in the residual network until no negative cycle remains. Since the residual network with respect to an optimal circulation does not contain a directed negative circuit, we can find node potentials, i.e. a corresponding dual solution, using the Bellman-Ford algorithm in $O\left(\left|\mathcal{P}\right|\phantom{\rule{0.5em}{0ex}}\cdot \phantom{\rule{0.5em}{0ex}}\left|\mathcal{F}\right|\right)$ time. The difference between the potential of two neighboring nodes then yields the value of the corresponding y-variable. The errors are determined straight forward. If there is a solution without error this approach yields a solution within the running time of Bellman-Ford. On the other hand, the Successive Shortest Path algorithm maintains similar node potentials such that the arc-weights remain non-negative. Since the total excess is bounded by |$\mathcal{P}$| in our case, the running time of that algorithm is $O\left(\left|\mathcal{P}\right|\xb7\left|\mathcal{F}\right|+\left|\mathcal{P}{|}^{2}\mathrm{log}\right|\mathcal{P}|\right)$.
Solving the Lagrangian Dual
Instead of a minimum cost circulation problem(right-hand side is 0), we have to solve the more general minimum cost flow problem [12] where the supplies and demands $\stackrel{\xaa}{\lambda}$ of the nodes are determined by the difference of Lagrangian multipliers, i.e. $\stackrel{\xaa}{\lambda}$ is of dimension |$\mathcal{P}$| + 1 and ${\stackrel{\xaa}{\lambda}}_{i}={\lambda}_{i}-{\lambda}_{i-1}$ for $2\le i\le \phantom{\rule{0.5em}{0ex}}\left|\mathcal{P}\right|,{\stackrel{\xaa}{\lambda}}_{1}={\lambda}_{1}$ and ${\stackrel{\xaa}{\lambda}}_{\left|\mathcal{P}\right|+1}=-{\lambda}_{\left|\mathcal{P}\right|}$. A feasible flow of minimum cost can be computed efficiently by, e.g., the cycle-canceling algorithm and the successive shortest path algorithm, as well as variants of them, like the capacity scaling algorithm[12]. In our implementation (C++) we used the LEDA library [19] to solve the Lagrangian subproblem by analgorithm based on capacity scaling and successive shortest path computation [12].
We improve the resulting bounds by the subgradient optimization method described in the following and incorporate the overall approach into a branch-&-bound algorithm as the lower bounding scheme.
where UB is a previously computed upper bound on z* and θ is a step size parameter assuming values in {x ∈ ℝ | 0 < x ≤ 2}. In the experiments it turns out, that initializing the vector of Lagrangian multipliers λ^{0} to the length P of the corresponding intervals in $\mathcal{P}$ increases the convergence rate dramatically. We also experienced a fast convergence to near-optimal Lagrangian multipliers when following the classical Held-Karp method to choose the step size scalar θ: We start with θ_{0} = 2 and half θ_{ℓ} whenever the best Lagrangian bound v(IP(λ)) found so far has not increased in a certain number of iterations. As soon as the step size scalar falls below a specified threshold or the number of iterations exceeds a certain limit (which is adaptive with respect to the depth of the branch-&-bound node), we branch on a variable ${y}_{p}^{k},\phantom{\rule{0.5em}{0ex}}k\in S,\phantom{\rule{0.5em}{0ex}}p\in \mathcal{P}$, such that${\stackrel{\xaa}{y}}_{p}^{k}-\lfloor {\stackrel{\xaa}{y}}_{p}^{k}\rfloor $ is close to 0.5, where ${\stackrel{\xaa}{y}}_{p}^{k}$ is the average value of variable ${y}_{p}^{k}$ in the last h = 10 Lagrangian solutions. Since we aim to find all optimal colorings, we also branch on variables that are integral. Incorporating the Lagrangian approach as a lower bounding scheme into a branch-&-bound frame work gives an alternative algorithm that does not depend on commercial software packages.
Experimental Setting
The entire sHDX experiment was automated with a LEAP robot (HTS PAL, Leap Technologies, Carrboro, NC).
Automation of the experiment reduces human error and reduces deuterium for hydrogen back-exchange. All time points where interlaced and performed in triplicate to ensure experimental reproducibility. After digestion, the protein digest was injected from a 10 μ L loop to either a 1 mm × 50 mm C5 column (Phenomenex) or a Pro-Zap Pro-sphere HP C18 HR 1.5u 10 mm × 2.1 mm (All-tech). A rapid gradient 2% B to 95% B in 1.5 min(A: acetonitrile/H_{2}O/formic acid 5/94.5/0.5, B: ace-tonitrile/H_{2}O/formic acid 95/4.5/0.5) was used to elute peptides. The eluent was post-column split and infused by microelectrospray ionization into a custom built 14.5 T LTQ FT-ICR mass spectrometer. The extraction of the peptic fragments and their deuterium uptakes from these data was done by an in-house analysis package [25]. Then we compute the cumulative exchange rates from the deuterium uptakes with either the MEM-method [15] or a new approach based on integer linear programming [14].
A current limitation for implementation of this software is back exchange of deuterium-to-hydrogen during the separation of the samples. It has been reported that different peptides have a different percentage of back exchange due to the sequence of amino acids [26, 27]. Furthermore, the peptide sequence overlap will limit the ability to map single amino acid rate kinetics. Thus, reduction of backexchange has been investigated [28, 29], along with multiple acid proteases to increase sequence coverage [30]. The sHDX experiment is continually being improved, but in its current state the sHDX experiment does not take away from the integrity of the algorithm to discern single amino acid exchange kinetics.
Declarations
Acknowledgements
Work supported by NSF (DMR-00-84173), NIH (GM78359), Florida State University, the National High Magnetic Field Laboratory in Tallahassee, FL, and by Deutsche Forschungsgemeinschaft (DFG) within Priority Programme 1307 "Algorithm Engineering".
Authors’ Affiliations
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