New Approaches to in silico Design of Epitope-Based Vaccines

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24 CHAPTER 3. ALGORITHMIC BACKGROUND The optimization problem for an ε-SVR is: min w∈H, b∈R, ξ,ξ ∗ ∈R m This yields the following dual problem: max α,α ∗ ∈R m 1 2 ||w||2 + C m i=1 (ξi + ξ ∗ i ) subject to yi − 〈w, xi〉 − b ≤ ε + ξi − ε − 1 2 m i=1 〈w, xi〉 + b − yi ≤ ε + ξ ∗ i ξi, ξ ∗ i ≥ 0 for all i = 1, . . . , m. m i,j=1 (α ∗ i + αi) + m i=1 (α ∗ i − αi)yi (α ∗ i − αi)(α ∗ j − αj) 〈xi, xj〉 subject to 0 ≤ αi, α ∗ i ≤ C for all i = 1, . . . , m and m i=1 The corresponding prediction function is: Non-linear SVMs (αi − α ∗ i ) = 0 . f(x) = 〈w, x〉 + b m with w = i=1 (αi − α ∗ i ) xi . (3.14) (3.15) (3.16) So far we have only considered linear SVMs. However, depending on the problem at hand, a non-linear function might perform better. The data in Figure 3.4A, for example, cannot be classified properly by a linear function. In SVMs non-linearity is achieved by mapping the instances into another, commonly higher dimensional, dot product space, typically referred to as feature space, in which they can be separated linearly. For instance, applying the feature map φ : R 2 → R 3 with φ (x1, x2) = (x 2 1, x 2 2, √ 2x1x2) (3.17) to the data points in Figure 3.4A yields a distribution of the two classes in 3D that can be separated linearly (Figure 3.4B). Thus, mapping the input data into a suitable feature space prior to the application of the SVC algorithm permits non-linear separation. However, there is major drawback to this approach: For realistically sized problems the feature space can be very high-dimensional. Working in such a high-dimensional space is expensive with respect to computation time and memory usage. A solution to this problem was proposed by Boser et al. [63]. Consider the feature map in (3.17). The dot product can be computed as ′ φ(x), φ(x ) = (x 2 1, x 2 2, √ 2x1x2), (x ′2 1, x ′2 2, √ 2x ′ 1x ′ 2) = x, x ′2 . (3.18)
3.2. MACHINE LEARNING 25 Figure 3.4: Input space vs. feature space. A) Input space: The data points cannot be separated properly by a linear function. B) Feature space: Mapping the two-dimensional input data given in A to a three-dimensional feature space using the feature map φ (x1, x2) = (x 2 1, x 2 2, √ 2x1x2) enables linear separation. Based on [46]. Note that the dot product in the feature space corresponding to φ can be determined without explicitly mapping into it (cf. right hand side of equation 3.18). This observation is useful since SV algorithms via their duals only depend on dot products between instances xi. Hence, it is sufficient to know the function K(x, x ′ ) = 〈φ(x), φ(x ′ )〉 to perform nonlinear classifications. Knowledge of the corresponding feature map φ is not essential. The soft margin SVC (3.13) can be restated as follows: and accordingly max α∈R m m i=1 αi − 1 2 m αiαjyiyjK(xi, xj) i,j=1 subject to 0 ≤ αi ≤ C for all i = 1, . . . , m m and αiyi = 0 i=1 f(x) = (3.19) m yiαiK(x, xi) + b (3.20) i=1 with K(x, x ′ ) corresponding to the dot product in some feature space. If the underlying feature map is non-linear, then so is the SVC. Non-linearity in SVR is achieved analogously. Kernel Functions The question remains how to determine whether a function corresponds to the dot product in some feature space. According to Mercer’s Theorem [64] this is the case if the function is symmetric and if its Gram matrix is positive semi-definite. Such a function is called kernel function or kernel. Kernels represent similarity measures.
Page 1 and 2: New Approaches to in silico Design
Page 3: Abstract Traditional trial-and-erro
Page 7 and 8: Acknowledgments First of all, I wou
Page 9 and 10: Contents 1 Introduction 1 2 Biologi
Page 11 and 12: 7.3 Design of String-of-Beads Vacci
Page 13 and 14: Chapter 1 Introduction Motivation T
Page 15 and 16: prediction of T-cell epitopes is a
Page 17 and 18: systemic property but employ peptid
Page 19 and 20: Chapter 2 Biological Background In
Page 21 and 22: 2.2. CELLULAR IMMUNE RESPONSE 9 A B
Page 23 and 24: 2.2. CELLULAR IMMUNE RESPONSE 11 [3
Page 25 and 26: 2.2. CELLULAR IMMUNE RESPONSE 13 Ho
Page 27 and 28: 2.3. VACCINES 15 the immune system
Page 29 and 30: Chapter 3 Algorithmic Background In
Page 31 and 32: 3.1. COMBINATORIAL OPTIMIZATION 19
Page 33 and 34: 3.2. MACHINE LEARNING 21 Figure 3.2
Page 35: 3.2. MACHINE LEARNING 23 Figure 3.3
Page 39 and 40: 3.2. MACHINE LEARNING 27 Figure 3.5
Page 41 and 42: Chapter 4 Epitope Discovery After h
Page 43 and 44: 4.1. INTRODUCTION 31 Figure 4.2: Sc
Page 45 and 46: 4.2. IMPROVED KERNELS FOR MHC BINDI
Page 51 and 52: 4.3. MHC BINDING PREDICTION FOR ALL
Page 57 and 58: 4.4. T-CELL EPITOPE PREDICTION 45 T
Page 59 and 60: 4.4. T-CELL EPITOPE PREDICTION 47 i
Page 61 and 62: 4.4. T-CELL EPITOPE PREDICTION 49 F
Page 63 and 64: 4.4. T-CELL EPITOPE PREDICTION 51 t
Page 65 and 66: Chapter 5 Epitope Selection The pre
Page 67 and 68: 5.3. MATHEMATICAL ABSTRACTION 55 im
Page 69 and 70: 5.3. MATHEMATICAL ABSTRACTION 57 fo
Page 71 and 72: 5.4. EXPERIMENTAL RESULTS 59 Let G
Page 73 and 74: 5.6. IMPLEMENTATION 61 number of no
Page 75 and 76: Chapter 6 Epitope Assembly The math
Page 77 and 78: 6.2. APPROACH 65 Figure 6.2: Graph
Page 79 and 80: 6.3. INCORPORATION OF PROTEASOMAL C
Page 81 and 82: 6.4. EXPERIMENTAL RESULTS 69 Figure
Page 83 and 84: 6.5. DISCUSSION 71 epitope sets wit
Page 85 and 86: Chapter 7 Applications In the previ
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7.1. OPTITOPE - A WEB SERVER FOR EP
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7.2. DESIGN OF A PEPTIDE COCKTAIL V
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7.3. DESIGN OF STRING-OF-BEADS VACC
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Chapter 8 Discussion & Conclusion P
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selection of an optimal epitope set
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Appendix A Abbreviations AA Amino a
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Appendix B Epitope Discovery Table
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Table B.3: Regression performances
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Table B.5: Overall performance of M
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Table B.7: Gene expression omnibus
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Appendix C Applications Table C.1:
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Appendix D Contributions Chapter 4
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Appendix E Publications Published M
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Bibliography [1] M. Moutschen, P. L
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BIBLIOGRAPHY 115 [19] T. Sturniolo,
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BIBLIOGRAPHY 117 [44] F. Morein and
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BIBLIOGRAPHY 119 [72] C. Widmer, N.
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BIBLIOGRAPHY 121 [97] H. Parkinson,
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BIBLIOGRAPHY 123 [123] NCBI dbMHC d
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BIBLIOGRAPHY 125 [150] World Health
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New Approaches to in silico Design of Epitope-Based Vaccines

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