Bernard Brogliato
email: bernard.brogliato@inria.fr

pages 6/7, equations (1.12) and (1.13): the equality (1.13) is equivalent to the (perhaps) more familiar:

). The equivalence can be shown using Fact 8.4.21 in Bernstein's third edition, as follows. From (1.12) we have:

(1) Now (M (q) -1 2 J(q) ⊤ ) † = (J(q) ⊤ ) † (M (q) -1 2 ) † = (J(q) ⊤ ) † M (q)

1 2 , hence we obtain (1.12) and (1.13) since (J(q) ⊤ ) † = (J(q)J(q) ⊤ ) -1 J(q).

• page 21: Figure 1.3 (b) has a severe shortcut in the right curved arrow indicating "Yosida approximation". A better view of the transformations is depicted in Figure 1 below. In the same vein it is worth reading [START_REF] Curnier | Unilateral Contact: Mechanical Modelling[END_REF].

• page 33: line 12, inside the second brackets, this is q1 = -e n q2 .

• page 38: the representation in (1.56) (1.57) holds when the pair (E, A) is regular, that is λE -A is invertioble for some real λ. This is a standard assumption in DAE theory.

• page 49, section 1.4.3.1: other studies using the Zhuravlev-Ivanov nonsmooth transformation may be found in [START_REF] Feng | Stochastic responses of vibro-impact Duffing oscillator excited by additive Gaussian noise[END_REF][START_REF] Feng | Stochastic responses of Duffing-Van der Pol vibro-impact system under additive and multiplicative random excitations[END_REF][START_REF] Rong | Subharmonic response of a single-degree-of-freedom nonlinear vibroimpact system to a ramdomly disordered periodic excitation[END_REF][START_REF] Li | Response probability density functions of Duffing-Van der Pol vibro-impact system under correlated Gaussian white noise excitation[END_REF]. • page 60: add BF (t) in the third line of the right-hand side of equation (2.13). Two lines below equation (2.13), change conv to conv and add also the missing term BF (t).

• pages 71-72: A first comment is that the damping coefficients that incorporate e n (as those listed page 71), may be seen as extensions to the nonlinear spring-dashpot model, of (2.9) page 56 which also involves the CoR. We note that in reference 1220, the coefficient p in (2.24) is found heuristically to be p = (compare with the expression in (2.9) dropping km outside the parantheses). In [START_REF] Hu | On the determination of the damping coefficient of nonlinear spring-dashpot system to model Hertz contact for simulation by discrete element method[END_REF] it is proposed to enlarge the right-hand side of (2.24) to -γx(t) p ẋ(t)kx(t) w , and γ is chosen as αm ẋ(t 0 ) k m ẋ(t 0 ) 2 1+p 1+w . They find that p = 1-w 2 yields α =ln(e n ) 2(1+w) π 2 +ln(en) 2 . According to [START_REF] Hu | On the determination of the damping coefficient of nonlinear spring-dashpot system to model Hertz contact for simulation by discrete element method[END_REF]Figure 2], their model has a loading-unloading curve similar to Figure 2.4 (a) page 70, with no negative contact force near zero indentation. Several nonlinear spring-dashpot models (Kuwabara-Kono, Hu et al, Tsuji et al) are further compared in [START_REF] Kacianauskas | Comparative evaluation of normal viscoelastic contact force models in low velocity impact situations[END_REF] in terms of variation of e n with respect to the damping coefficient, acceleration histories, acceleration/identation loading/unloading curves, etc, for the case of two spheres colliding. Then [START_REF] Kacianauskas | Comparative evaluation of normal viscoelastic contact force models in low velocity impact situations[END_REF] investigates the influence of the spring-dashpot models on a multiple impact process in a chain of aligned balls. To this aim they rely on the experimental data in reference 625. In particular [START_REF] Kacianauskas | Comparative evaluation of normal viscoelastic contact force models in low velocity impact situations[END_REF]Figure 11] is exactly Figure 20 in [START_REF] Nguyen | Shock dynamics in granular chains: numerical simulations and comparisons with experimental tests[END_REF] (reference 928) and Figure 29 in [START_REF] Nguyen | Multiple Impacts in Dissipative Granular Chains[END_REF] (reference 629), where the numerical results are obtained with the LZB model (section 6.3). More work is necessary to determine the domains of validity (in terms of applications, ease of numerical simulation, etc) of these models.

• page 75: footnote 21 should be on page 74, at the end of the framed paragraph.

• page 81: the models shown in Figure 2.8 are used in modeling of rail/wheel contact and sometimes called Piotrowski's models [START_REF] Piotrowski | Model of the UIC link suspension for freight wagons[END_REF][START_REF] True | On the numerical and computational aspects of nonsmoothness that occur inrailway vehicla dynamics[END_REF].

• page 99, line 5: replace (B.5) by (B.6). The calculations after (3.13) are in fact the proof in this particular case, that the normal cone can also be written as in (B.5) page 549.

• page 102, line -5: replace n i ∈ IR 3 by n ∈ IR 3 (the subscript i in F n,i refers to the contact point number).

• page 103, line 4: • page 113: equation (3.41) is obtained assuming that at an impact instant all virtual displacements and velocities are allowed. This indeed results in zero impact because M (q(t k ))( q(t + k )q(t - k )) = 0 ⇒ q(t + k )q(t - k ) = 0 (the mass matrix is assumed to be full rank). In reference 724 it is supposed that the generalized momentum satisfies p(t + k )p(t - k ) ∈ -N Φ (q(t k )), as a modelling assumption since constraints are perfect during impacts (see (iii) page 104). It is shown in reference 724 that this inclusion is equivalent to the variational inequality p(t + k )p(t - k ), δq ≥ 0 for all δq ∈ T Φ (q(t k ), where the virtual displacements δq are continuous (we can therefore see these developments as the extension of the material in (3.13) and below, to the case with elastic impacts). This equivalence is not surprizing when we consider the normal cone definition in (B.5), recalling that this definition remains true even for nonconvex sets, see Remark B.1 page 550. Using that p(t) = M (q(t)) q(t) and Moreau's set inclusion in section B.2.2, we may impose the stricter inclusion M (q(t k ))( q(t + k )q(t - k )) ∈ -N T Φ (q(t k )) ( q(t + k ) which is equivalently rewritten as the variational inequality: find q (5.53) and (5.60) in section 5.2). Thus we obtain this time the extension of (3.7) where virtual velocities are considered, and the extension of (5.45) for the case of impacting motions. We note in passing that the arguments that yield [13, Equation (1a)] (which is the same as p(t + k )p(t - k ) ∈ -N Φ (q(t k )) with missing mass matrix and minus sign in the right-hand side) are spurious: there is a shortcoming in the reasoning in [START_REF] Fetecau | Nonsmooth Lagrangian mechanics and variational collision integrators[END_REF] because of the use of the condition δq(t k ) + q(t k )δt k ∈ T q(t k ) bd(Φ), while q(•) jumps at t k , and T q(t k ) bd(Φ) denotes the tangent plane at q(t k ) (and not the tangent cone) to the boundary of the admissible domain (the correct way to derive the material page 385 in [START_REF] Fetecau | Nonsmooth Lagrangian mechanics and variational collision integrators[END_REF], is in section 6 of reference 724 by Leine et al).

P i = v n,i F n,i = ....
(t + k ) ∈ T Φ (q(t k )) such that M (q(t k ))( q(t + k )-q(t - k )), v -q(t + k ) ≥ 0 for all v ∈ T Φ (q(t k )) (see
The above basic assumption yields Theorem 3 in reference 724 which is a Hamilton principle in strong norm, or strong Hamilton principle. There exists a weak norm for a weak Hamilton's principle which somewhat relaxes the assumption on the generalized momenta at impact times, see Theorem 4 and condition (99) in reference 724.

• page 130, about the calculation of the normal and tangential vectors at the contact point A (local kinematics): the starting point is that if a 3D surface is defined by two parameters u and v and a differentiable function r(u, v), r : IR 2 → IR 3 , then ∂r ∂u (u 1 , v 1 ) ∈ IR 3 and ∂r ∂v (u 1 , v 1 ) ∈ IR 3 span the tangent plane at the point A 1 parameterized by u 1 and v 1 , and one can then define the normal vector as the cross product n 1 = ∂r ∂u T (u 1 , v 1 ) × ∂r ∂v T (u 1 , v 1 ) (one should take care of the correct order to get the right orientation of the normal).

• page 133, about the time-derivative of the right-hand side of (4.17): Let us denote the Galilean frame as L 0 = G and the local frame as L. The angular velocity vector between both frames is denoted as

Ω L/L 0 . We obtain d dt [(A 2 A 1 ) T n] = d dt [(A 2 A 1 ) T ]n + (A 2 A 1 ) T d dt n = d dt [(A 2 A 1 ) T ]n + (A 2 A 1 ) T d
dt n. Assume that the vectors are expressed in L 0 . Then basic kinematics say that

d dt n = d dt n| L + Ω L/L 0 × n = Ω L/L 0 × n since n is constant in L. Thus (A 2 A 1 ) T d dt n = (A 2 A 1 ) T (Ω L/L 0 × n) = 0 since the vector product is orthogonal to n and the vector (A 2 A 1 ) is along n. Therefore d dt [(A 2 A 1 ) T n] = d dt [(A 2 A 1 ) T ]n. The vector d dt [(A 2 A 1 ) T ]
is the time derivative of (A 2 A 1 ) in L 0 and may be named the relative velocity between both bodies. The sentence "In particular..., except if n is fixed in G" is meaningless and should be deleted. It becomes true if stated in terms of the acceleration d dt V A i and vi,n . • page 162, section 4.2.3 (also page 169, section 4.2.5.2): [START_REF] Xiao | On the contact stiffness and nonlinear vibration of an elastic body with a rough surface in contact with a rigid flat surface[END_REF] study the force/identation relation f (δ) = kδ n for an elastic body with a rough surface in contact with a rigid flat surface. A three-dimensional rough surface is constructed using a modified two-variable Weierstrass-Mandelbrot fractal function. Results in [25, Table 2] show that n can vary between 2.11 and 1.19 depending on some parameters (like fractal dimension, fractal roughness, root-mean-square roughness and arithmetic average height R a ). Thus the elasticity is found to be superlinear and even sometimes super-Hertz. Compared with the study described in section 4.2.3 where n = 1 in (4.61), it seems that surface roughness increases n. The interest of such study is to show that Hertz' fundamental assumption (ii) page 147 (introduction of section 4.2.1.1) on smooth contacting surfaces, may play a role in the elasticity coefficient. However we should also note that the surfaces in contact considered in [START_REF] Xiao | On the contact stiffness and nonlinear vibration of an elastic body with a rough surface in contact with a rigid flat surface[END_REF] are conforming in addition to be rough, and that contrary to (4.61) the flat is considered non-deformable.

• page 164, section 4.2.4: see a comment below (page 384).

• page 165: Hurmuzlu's analysis of the micro-impact phenomenon (section 4.3.10.2) seems to contradict Love's criterion, because micro-collisions excite transversal modes in the beam (page 228) and these vibrations dissipate energy.

• pages 165-166: estimations of the CoR for harmonic chains of aligned beads colliding a wall, and taking into account sequences of repeated impacts as well as the vibrational energy trapped in the chain, are given in [START_REF] Basile | Coefficient of restitution for one-dimensional harmonic solids[END_REF][START_REF] Nagahiro | Collision of one-dimensional nonlinear chains[END_REF].

• page 184, equation (4.92): either write D(θ(t k )), or drop the argument t k in A(θ), B(θ), C(θ) as done in Proposition 4.4.

• page 184, line 13: one should read "Then its inverse multifunction is g -1 (y) = ..." .

• pages 184-185: in fact it is not necessary to invert the mappings to analyze the well-posedness of the generalized equation ax + b|x| + c ∈ -(1 + d)sgn(x) (though as said in the book, inverting allows one to recover the framework of generalised equations as in Facchinei-Pang's book, and apply general results on existence and uniqueness of solutions to such GEs). It can be studied by inspection, enumerating the different cases. In Figure 2, dashed lines are the graph of

x → ax + b|x| + c = (a -b)x + c if x ≤ 0 (a + b)x + c if x ≥ 0
, and solid lines are the graph of the set-valued

mapping x → -(1 + d)sgn(x).
The solution x * is at the intersection of the graphs.

-Figure 2 (a): a -b > 0, a + b > 0, 1 + d > 0 (unique solution). 1. Case 1: -(1 + d) ≤ c ≤ 1 + d: x * = 0. 2. Case 2: c < -(1 + d): x * = -(1+d)-c a+b > 0. 3. Case 3: c > (1 + d): x * = (1+d)-c a-b < 0. -Figure 2 (b): a -b < 0, a + b < 0, 1 + d < 0 (unique solution). 1. Case 1: (1 + d) ≤ c ≤ -(1 + d): x * = 0. 2. Case 2: c > -(1 + d): x * = -(1+d)-c a+b > 0. 3. Case 3: c < 1 + d: x * = 1+d-c a-b < 0. -Figure 2 (c): a -b > 0, a + b < 0, 1 + d < 0. 1. Case 1: c < 1 + d: no solution. 2. Case 2: c = 1 + d: one solution x * = 0. 3. Case 3: 1 + d < c ≤ -(1 + d): two solutions x * 1 < 0 and x * 2 = 0. 4. Case 4: c > -(1 + d): two solutions x * 1 < 0 and x * 2 > 0. -Figure 2 (d): 1 + d < 0. 1. Case 1: a -b = 0, a + b < 0, 1 + d < c < -(1 + d): one solution x * = 0. 2. Case 2: a -b < 0, a + b = 0, 1 + d < c < -(1 + d): one solution x * = 0. 3. Case 3: a -b < 0, a + b = 0, c = -(1 + d): an infinity of solutions x * ≥ 0 (for x > 0 the GE becomes in this case c = -(1 + d)sgn(x) = -(1 + d)). 4. Case 4: a -b < 0, a + b = 0, c > -(1 + d): no solution. 5. Case 5: a -b = 0, a + b < 0, c < 1 + d: no solution. 6. Case 6: a -b = 0, a + b < 0, c = 1 + d: an infinity of solutions x * ≤ 0 (for x < 0 the GE becomes in this case c = -(1 + d)sgn(x) = 1 + d).
This analysis can be used to study the GE in (4.92), with

a = A(θ(t k )), b = B(θ(t k )), c = C(θ(t k )), d = D(θ(t k )).
• page 185, proof of Proposition 4.5, line 2:

C(θ) < -(1 + D(θ)),
• page 200, equation (4.124): the last term is dp dt ′ , . • page 245, line 6: ... + w b (q, q, t)).

• pages 247-248, Proposition 5.3: The equality h(t ′ )h(t) = t ′ t ḣ(s)ds used in the proof means that h(•) is absolutely continuous (not just continuous) and that what is denoted as ḣ(•) is its almost-everywhere derivative. The same holds for (ii), where ḣ(•) has to be absolutely continuous as well.

• page 270, line above (5.61): [780,781].

• page 280, in Theorem 5.3, line 3: replace T Φ (q(t 0 )) by V (q(t 0 )), as it is not guaranteed in general that both tangent cones are equal. Same page 282 in Remark 5.13.

• page 285, about Amontons's work: Guillaume Amontons, De la résistance causée dans les machines. Mémoires de l'Académie Royale, 4:257-282, 1699. See [START_REF] Blau | Amontons' laws of friction[END_REF].

• page 286, in (5.86) this is

F t ∈ IR 3 , with F t = F t 1 t 1 + F t 2 t 2 , while F =   F n F t 1 F t 2   ∈ IR 3
, and • page 290, change R to F in the line just above (5.94).

F t = F t 1 F t 2 ∈ IR 2 .
A,t = v t 1 v t 2 ∈ IR 2 . (c) (d) c c c c c c c c c c x x 1 + d 1 + d -(1 + d) 1 + d x -(1 + d)
• page 293, line 5: ...sgn(ẍ(t + )) when ẋ(t) = 0,...

• page 293, after (5.96): ...and Fext µFn > 1 ⇒ ...

• page 295: in case of non trivial mass matrix, equation (5.97) becomes u(t [START_REF] Paoli | Vibro-impact problems with dry friction-Part I: Existence result[END_REF]. Proposition 3.3 in [START_REF] Paoli | Vibro-impact problems with dry friction-Part II: Tangential contacts and frictional catastrophes[END_REF] extends (5.97), and Figure 5.12 which depicts the algorithm in case of a trivial mass matrix. Additional figures which complete Fig. 5.12 page 295, are in Figure 3. Case (a) shows that if the mass matrix is trivial (the identity matrix) then when u(t -) is tangent to the constraint boundary, there is no velocity jump (no impact without collision). Case (b) shows that when the mass matrix is not trivial and the generalized friction cone dips below the tangent hyperplane, then the velocity may jump. But case (c) shows that depending on the tangent velocity signum, the velocity may remain continuous. It is noteworthy that the zero vector in (5.97) refers to velocities, not positions.

+ k ) = proj(0, [u(t - k ) + M (q(t)) -1 C(q(t))] ∩ T (q(t))
u + = u - n u - u + = 0 n u + = u - 0 0 0 n (c) (b) (a) T (q)
T (q) T (q)

Figure 3: Sweeping process with friction.

• page 298, last line: replace x 2,n by (A 2 A 1 ) T n.

• page 299, more details about P-matrices and the difference with positive definite matrices.

Principal minors are the determinants of the principal submatrices, they are sometimes also called principal subdeterminants. Principal submatrices of a matrix M ∈ IR n×n are constructed as follows: let the index set I = {1, 2, ..., n}, and consider any subset J ⊆ I. Let us denote J = {i 1 , i 2 , ..., i m }, with m ≤ n, and i k ∈ I for all 1 ≤ k ≤ m. Delete all rows and columns of M which are indexed by i with i ∈ J . The obtained matrix M J ∈ IR m×m is a principal submatrix of M . A leading principal submatrix of M is obtained by considering i k = k, k = 1, 2, ..., m. The determinant of a leading principal submatrix is a leading principal minor.

Theorem 5.5 states that a P-matrix has to have all its principal submatrices with positive determinant, i.e. all positive principal minors. A (real) matrix M (nor necessarily symmetric) is positive definite if and only if M T + M has all its leading principal minors positive (and also its leading principal submatrices are positive definite matrices). If in addition M = M T , then M ≻ 0 if and only if all its principal minors are positive (and another characterization is: if and only if all its leading principal minors are positive, showing in passing that a positive definite matrix is a P-matrix). A (real) matrix M (nor necessarily symmetric) is positive semidefinite if and only if M T + M has all its principal minors non negative. Thus for positive definiteness it is sufficient to check the leading principal submatrices, while for positive semi definiteness all principal submatrices have to be checked.

Another characterization of P-matrices is as follows [2, Lemma 16, Theorem 59]: if M is a Pmatrix, the inequalities M x ≤ 0 and x ≥ 0 have only the trivial solution x = 0. Also there exists

x > 0 such that Ax > 0.

• section 5.4.2: I said nothing on cone linear complementarity problems of the form K ∋ z ⊥ w = M z + r ∈ K ⋆ where K is a closed convex cone. See Theorem 8 and Corollary 5 in reference 23 for the existence of solutions to CLCP.

• section 5.4.2. Another result that completes this section concerns the number of solutions to LCPs for which uniqueness fails. This is tackled in [START_REF] Kojima | On the number of solutions to a class of linear complementarity problems[END_REF]. A matrix is said to be an N -matrix if its all principal minors are negative. For such matrices, the LCP (5.105) has either 0, 1, 2 or 3 solutions. A solution z * is degenerate if z * i = 0 and w i = (M z * ) i + r i = 0 for some i.

Lemma 1 [14, Lemma 2.4, Theorem 3.2, Theorem 3.3, Theorem 3.4] (i) Let M be an N -matrix and M ≺ 0. Then for each r ≥ 0, the LCP in (5.105) has a solution, and has no solution for r ≥ 0 (componentwise inequality). The LCP has exactly two solutions for r > 0. (ii) Let M be an N -matrix and r ∈ IR n . Then if M is not negative definite and r ≥ 0, the LCP (5.105) has a unique solution. (iii) Let M be an N -matrix, non negative definite, and r > 0. Then if all solutions of the LCP (5.105) are non degenerate, the LCP has exactly three solutions. Otherwise it has exactly two solutions. (iv) Let M be an N -matrix, non negative definite, and 0 = r ≥ 0, with r i = 0 for some i. Then the LCP (5.105) has exactly two solutions, with one solution degenerate.

• page 300: about copositive matrices: there is a seminal paper by Hiriart-Urruty and Seeger [START_REF] Hiriat-Urruty | A variational approach to copositive matrices[END_REF] which gives necessary and sufficient conditions on a matrix entries so that it is copositive on IR 2 + or IR 3 + . An extension of the Schur complement theorem is also given in [38, Theorem 3.5, Corollary 3.6].

• page 312: after (5.127), change x 2 (t) = t 0 i 1 (s)ds by x 2 = Cu c , where u c is the voltage across the capacitor in Fig. 5.15. Another choice is x 2 = u c , where C uc = i 1 . The dynamics in (5.127) is correct, but the choice of the state variable x 2 is not correct.

• page 313: replace u 2 (t)u 1 (t) by u 2 (t), and replace u 2 (t) by u 2 (t)u 1 (t), in the dynamical equations (5.130).

• page 319: in (3.145), the cone CP (third line) is true only if K(t + ) is a cone (in addition to be convex and non empty).

• pages 320-321: further studies on the stability of nonsmooth circuits (characterization of equilibria, Lyapunov stability) may be found in [START_REF] Adly | Stability analysis and attractivity results of a DC-DC Buck converter[END_REF][START_REF] Adly | Variational analysis and generalized equations in electronics[END_REF].

• page 322: in the paragraph after Theorem 5.12: ε > 0.

• page 323: in (5.150) it happens that V (t + ) -V (t -) ≤ 0 provided that 0 ∈ S(t), with S(t) in (5.139). See reference 480, Lemma 3.

• page 328, line 1: delete the "and".

• page 331, equation (5.156):

In the case of planar friction, the dissipation function is:

D(q, q) = -qT ∇f (q)λ n,u + m b i=1 |λ b,i |µ i q⊤ H t,b,i (q)sgn(H ⊤ t,b,i (q) q) + m b +mu i=m b +1 |λ u,i |µ i q⊤ H t,u,i (q)sgn(H ⊤ t,u,i (q) q) ≥ 0. (2)
• page 332, Proposition 5.24 and equation (5.158): the matrix P µ (q) is defined as

-∇f (q) T M (q) -1 H t,u (q)[µ i sgn(v t,u,i )].
• page 352, section 5.6.4. Mathematical results on the existence of solutions (velocity u(•) of local bounded variation, q(t) = q(0) + t 0 u(s)ds) which extend results in both references 867 and 1141, 1142, can be found in [START_REF] Paoli | Vibro-impact problems with dry friction-Part I: Existence result[END_REF][START_REF] Paoli | Vibro-impact problems with dry friction-Part II: Tangential contacts and frictional catastrophes[END_REF]. It incorporates Painlevé paradoxes, in the sense that the generalized friction cone may dip below the tangent cone boundary at the considered contact point (as pointed out in footnote 31 page 295, and in a paragraph above Figure 5.24 page 353). An example of calculation of the generalized friction cone for the Painlevé sliding rod system, may be found in [START_REF] Génot | New results on Painlevé paradoxes[END_REF]Section 2.2]. Using the notation of equation (5.1) and of section 6.2, the Painlevé sliding rod example of section 5.6.1 may be written as q = n q λ n + M -1 H t (q)λ t . The generalized friction cone is the cone generated by n q = 1 m (0, 1, -3 l cos(θ)) T and ±µM -1 H t (q), H t (q) = M (1, 0, 3 l sin(θ)) T , M = diag(m, m, I), with the edges e ± = n q ± µM -1 H t (q). It is not symmetric around n q , and n T q M e + = ∇f (q) T e + = B(θ, µ) in (5.175) so that the generalized cone may dip below the tangent hyperplane T (q), this reflects normal/tangential couplings as in equation (6.30). This is depicted in Figure 4.

• page 352, section 5.6.4, line 18: replace [755, Theorem 1] by [1328, Theorem 1].

• page 352, section 5.6.4: the article [START_REF] Champneys | The Painlevé paradox in contact mechanics[END_REF] is worth reading.

• pages 367 and 430 (sections 5.7.3.5 and 7.3.2): a (θ, γ) time stepping scheme is used in [START_REF] Sessa | A complementarity approach for the computation of periodic oscillations in piecewise linear systems[END_REF] to calculate periodic solutions of set-valued Lur'e systems (as in Figure 2.2 but with a feedback setvalued nonlinearity of the more general form (w, λ) ∈ R). After discretization a mixed quadratic complementarity problem (MQCP) is constructed. State jumps are incorporated in the problem (remind that LCS may have state discontinuities, see section 5.4.4.3). The period, the state and the multiplier λ are unknowns of the MQCP. A MQCP is a complementarity problem of the form: Find z, w, v such that

nq = M -1 ∇f (q) ∇f (q) q T (q) in
ϕ(z) + M z + q = w -v l ≤ z ≤ u, (z -l) T w = 0, (u -z) T v = 0 (3)
where ϕ(•) is a vector of quadratic forms in z. The solver path is used to solve the discretized MQCP.

• page section 5.7.3.2: an interesting application of the NSCD method is in [START_REF] Lancioni | Seismic vulnerability of ancient stone arches by using a numerical model based on the Non-Smooth Contact Dynamics method[END_REF] that deals with collapse mechanism of ancient stone arches. Though the NSCD method (the Moreau-Jean algorithm in section 5.7.3.1) has several shortcomings (like a simplistic impact law as explained in Example 5.6 page 275, see also section 6.3.3 and the comments page 411), it is interesting for complex systems with many degrees of freedom and unilateral contacts and seems to encapsulate essential modeling features in this case. For a finer analysis of stacked blocks dynamics under base excitation, one should certainly use more sophisticated models like LZB.

• page 384: it is shown that varying the stiffness ratio γ in the 3-ball system, changes a lot the impact outcome, and this is also true when the mass ratios are varied (the kinetic angle between the two constraints surfaces). This is also true for the 2-ball system hitting a rigid wall (take m 3 = +∞ in Figure 6.5 page 383). The analysis in [START_REF] Ruan | The unexpectedly small coefficient of restitution of a two-degree-of-freedom mass-spring system and its implications[END_REF] shows that the mass ratio and the stiffness ratio have a great influence on the kinematic CoR, which varies between 0.2 and 1 (the possibility of several impacts before definitive separation -a kind of micro-collisions effectbetween the first ball and the wall is taken into account 1 ). For the 2-ball system hitting a wall, the kinematic CoR is defined as -

v(t f ) v(t 0 ) where v = m 1 v 1 +m 2 v 2 m 1 +m 2
is the center of mass velocity, t 0 is the time of the first impact, t f is the time of the last impact before complete separation of the 2-ball system and the wall. Thus depending on these ratios, the system made of the two beads (which we can see as an approximation of the flexibilities in a rigid body like a rod) will rebound with a low or a high velocity. The apparently loss kinetic energy is in fact transformed into potential energy stored in the system's spring under the form of vibrations that persist after the impact is ended. This is therefore also quite related to sections 4.2.4 and 4.4 material. It is also interesting to compare this result to Theorem 4.1 page 238, which stipulates that an elastic rod that collides a wall with zero external force, has e n = 1. Thus the 2-ball system is not in general a good approximation of the infinite dimensional model. However the results in [START_REF] Ruan | The unexpectedly small coefficient of restitution of a two-degree-of-freedom mass-spring system and its implications[END_REF]Figure 2a] show that with equal masses and equal stiffnesses, then e n ≈ 1. This is extended to N -ball systems (see also [START_REF] Basile | Coefficient of restitution for one-dimensional harmonic solids[END_REF][START_REF] Nagahiro | Collision of one-dimensional nonlinear chains[END_REF]). One assumption that is made in these studies, and might make the analysed chains behaviour different from an elastic rod impacting axially a wall, is that it is assumed that the first (colliding) ball reverses its velocity instantaneously [START_REF] Nagahiro | Collision of one-dimensional nonlinear chains[END_REF].

• page 394 (relations between Moreau's impact law in (5.60) (5.61), and the restitution mapping in (6.44) (6.45)). Here I also refer to the reference [210, section 3.1.1], with some inaccuracies in equation (44). Let I(q(t)) be the index set of active constraints at position q(t). Moreau's law states that q(t + ) = q(t -) -(1 + e n )proj M (q(t)) [N (q(t)); q(t -)], see (5.60). Clearly if q(t -) is in the interior of N (q(t)) then q(t + ) = -e n q(t -). If this is not the case, one has to project q(t -) onto the normal cone, which is a polyhedral cone generated by the normals n q,i , i ∈ I(q(t)).

In case N (q(t)) is a so-called latticial cone, then the material in [210, section 3.1.1] is correct. We recall that N (q(t)) is a latticial cone if dim(q(t)) =card(I(q(t)), in other words the number of active constraints is equal to the dimension of q(t)) (the configuration space), and the active constraints (hence the vectors n q,i , i ∈ I(q(t))) are independent (see Németh and Németh, How to project onto an isotone projection cone, Linear Algebra and its Applications, vol.433, [START_REF] Blau | Amontons' laws of friction[END_REF][42][43][44][45][46][47][48][49][50][51]2010). In this case the projection of q(t -) onto N M Φu (q) in [210, section 3.1.1, line 8 after (43)], is correct according to Theorem 2 in Németh and Németh.

One difference between the impact law in (6.44) (6.45), and Moreau's impact rule, is first on the choice of the vectors qnorm and qtan . This explains some discrepancies between both models, as demonstrated for instance on the rocking block problem, see [228, section 3.4] which shows that with a specific choice of qnorm,1 and qnorm,2 , rocking motion is possible only if off-diagonal terms in the restitution matrix in (6.45) are introduced, whatever the kinetic angle value. On the contrary Moreau's law allows for rocking in some situations when the kinetic angle between the two active constraints, is larger than π/2, see [START_REF] Giouvanidis | Modelling contact in rocking structures with a nonsmooth dynamics approach[END_REF][START_REF] Giouvanidis | Nonsmooth dynamic analysis of sticking impacts in rocking structures[END_REF].

In practice one uses the equivalent form of Moreau's law in (5.66) and solves a LCP or a mLCP. Using (5.67) and assuming that D u (q(t k )) ≻ 0, it is easy to construct a LCP with unknown

W (t k ) ∆ = U n (t + k ) + E nn U n (t - k ): 0 ≤ W (t k ) ⊥ D u (q(t k )) -1 W (t k ) -D u (q(t k )) -1 (I m + E nn )U n (t - k ) ≥ 0. (4)
This LCP always has a unique solution.

If D u (q(t k )) -1 (I m + E nn )U n (t - k ) < 0, then U n (t + k ) + E nn U n (t -
k ) = 0 which gives Newton's law at each contact with CoR e n if E nn = diag(e n ) (see Proposition 5.15 for the link between Moreau's impact law and Newton's law at each contact with complementarity, see also the seminal reference [454]). The next step is to write down Moreau's law when q(t -) is not in the interior of N (q(t)), but has to be projected on it. When the constraints are not independent, one can still use a numerical solver that computes a solution for LCPs with positive semi-deifnite matrices (like Lemke's algorithm).

• page 398, line 2: choosing

• page 400, line just above Proposition 6.6: replace M (M (q, µ, v t ) by M (q, µ, v t ).

• page 406, line 7 after (6.68): δj (p f,j ) ≤ 0 (here we assume that compression occurs with positive identation velocity δj > 0 and expansion occurs with negative identation velocity δj < 0), in agreement with Figure 6.6. Thus indentation increases during compression and decreases during expansion (or restitution).

• page 407: Notice that equations (6.71) (6.72) can be rewritten in a differential form dE j dp j (p j ) = δj (p j )dp j and similarly for (6.72), so that the whole LZB dynamics is a first-order dynamics with augmented state variable (the potential energy becomes a state variable). Thus the calculation of the positions (assumed to be constant in the LZB approach) is not at all needed to integrate the system (see reference 929 equations (4.42) and (4.43) where a midpoint rule is used to approximate the potential energies). Notice further that (6.72) indeed stipulates that according to the energetic constraint W e,j = -e 2 ⋆,j W c,j (using (4.159)), then at the end of the impact E j (p f,j ) = 0.

• page 407 section 6.3.2: another interesting application of the LZB model is in [START_REF] Zhao | Docking dynamics between two spacecrafts with APDSes[END_REF].

• pages 414/415: a new kinematic approach has been proposed in [START_REF] Rakshit | Scalar generalization of Newtonian restitution for simultaneous impact[END_REF], this is based on solving a minimization of the kinetic energy of the system at the impact, with kinematic, kinetic constraints, and a new inequality that represents the restitution. This is a convex problem. Distance (non-local) effects are included which allow to model separation of balls in linear chains (and other systems undefoing multiple impacts). The energetic constraint is proved to be always satisfied from the optimization problem feasibility.

• page 415, line 8: drop the "be".

• page 419, Theorem 7.2: some constants a, b j , bj ∈ IR such that...

• page 422, lines 16 and 17 after equation (7.7): replace 1+en 1-en by 1 1-en . • page 429, equation (7.18): replace e in σ 1 and σ 2 by e n .

• page 452, equation (7.51): 0 ≤ f (q ⋆ ) ⊥ λ ⋆ n,u ≥ 0.

• page 456, equation (7.62), last two lines: Σ k∈[k 0 ,k 1 ] instead of Σ k∈{k 0 ,k 1 } which might let one think that only the two values k 0 and k 1 are taken into account.

• pages 457-458: The stability (finite-time convergence to a fixed point, plus Lyapunov stability) of simple systems with set-valued terms encompassing Coulomb's friction with constant normal force, is analysed in [START_REF] Adly | Finite time stabilisation of nonlinear oscillators subject to dry friction[END_REF]. The discretisation of the same dynamics is studied in [START_REF] Baji | An inertial proximal algorithm with dry friction: finite convergence results[END_REF], where it is shown that the sequence of discrete solutions converges in a finite number of steps to its limit. This is quite similar to the results mentioned in section 5.7.3.7 about implicit discretetime sliding mode control (though control has its own peculiarities like the fact that one wants to study robustness with respect to parameter uncertainties, unmatched disturbances, more complex attractive surfaces, etc). It may also be seen as a generalization of the case treated in Remark 5.33 pages 362-363.

• pages 457-458: it is clear that the fixed points of a system with unilateral contact and Coulomb friction are also solutions of a generalized equation, extending (7.51). As said page 458, in general uniqueness of the equilibrium is lost. Consider for instance (5.162) page 336. For this system the equilibria (q, q) = (q ⋆ , 0) satisfy the generalized equation:

       0 0 = sin(α) cos(α) λ ⋆ n + -cos(α) sin(α) λ ⋆ t + F x 0 0 ≤ f (q ⋆ ) ⊥ λ ⋆ n ≥ 0 λ ⋆ t ∈ -µλ ⋆ n sgn(0) (5)
with sgn(0) = [-1, 1]. It is noteworthy that usually the generalized equation for equilibria and the generalized equation for sticking contacts, are not the same (in this example they are the same because sticking contact implies that the system does not move, and conversely). Also this is different from the generalized equation in (5.165) which is obtained from the acceleration Coulomb's friction model.

• page 462: Chapter 7 could ne nicely completed with the study in [START_REF] Niculescu | Force measurement time-delays and contact instability phenomenon[END_REF] which analyzes the contact stability of a simple system, with a force feedback controller, and subject to delay in the force feedback. These theoretical results were experimentally validated by Tornambé in [START_REF] Tornambé | Discussion on "Force measurement time-delays and contact instability phenomenon[END_REF].

• page 531, section 8.5.2: controllability of juggling systems (with the important assumption that the robot dynamics in (8.74) has much bigger mass than the object, so that q2 is continuous at impacts) as well as their stabilization, is studied in [START_REF] Lynch | Recurrence, controllability, and stabilization of juggling[END_REF].

• page 548, in (B.3) (c) the "for µ n ց 0 and..." has to be understood as "for all µ n ց 0 and ...".

• page 548, in (B.4) it is more accurate to write "for some µ n ..." or even better: "there exists µ n ...". This definition of the contingent cone is the one given by Aubin and his co-authors in at least two books (Aubin-Frankowska and Aubin-Cellina). In other references one rather finds (B.3) (b). One can show that both formulations are equivalent as follows: in (B.3)(b) define y n = (q nq)/λ n and in (B.4) define q n = q + µ n y n (communicated by A. Saccon).

• • page 550, in (B.10), right-hand side, second line: bd(Φ).

• page 550, Definition B.8: A : IR n ⇒ IR n (that is, m = n as implied by the product in (B.11) to make sense).

• page 551, Lemma B.1: it is more appropriate to write f : dom(f ) ⊆ IR n → IR ∪ {+∞} (though one might understand that IR contains the infinity). • page 553: For better clarity replace the set Φ in the paragraph after (B.12), by C.

• page 558: let M be a P-matrix. Then the quadratic program (B.22) is equivalent to the LCP(q, M ), see reference [307, p.149]. The quadratic program may be non convex if M is not positive definite.

• page 558, in Theorem B.2, last line: ∂f (x) = A T ∂h(Ax).

• page 560, section B.2.2: line 3: replace (5.35) by (5.34); line 6: replace "Definition 5.34" by: "(5.34)".

• • page 563: reference 8 is 96(5), 585-603 (2016).

• page 570: reference 182 : Nonliner Dynamics, vol.77, no 3, pp.899-922, 2014.

• page 582: reference 460 appeared in IEEE Control Systems.

• page 622, index for G, add for Gauss's principle: systems with friction, 332

Figure 1 :

 1 Figure 1: Enhanced Fig. 1.3 (b).

1 4 .mK h x 1 4 ẋ

 44 Thus their model has a dissipative force equal to α √ if x is the normal indentation, with α =ln(e n ) 5 π 2 +ln(e 2 n

Figure 2 :

 2 Figure 2: Analysis of the generalised equation in (4.92).

Figure 4 :

 4 Figure 4: Painlevé paradox and generalized friction cone.

Figure 5 :

 5 Figure 5: Tangent, contingent, normal cones and Moreau's cone.

•

  page 551, Corollary B.1: Let A : IR m ⇒ IR m be a monotone multivalued operator. If the sector condition for a single-valued nonlinearity φ : IR m → IR m is defined as [φ(x)ax] T [bxφ(x)] ≥ 0 for all x, and a < b, a and b reals, then equivalently φ(x) T [x -φ(x) b ]b ≥ 0 for a = 0 and any b > 0. We infer letting b → +∞ that the nonlinearity belongs to the sector [0, +∞) if φ(x) T x ≥ 0 for all x. In the corollary we allow for b = +∞ so the sector is [0, +∞], which allows for vertical segments at the origin. • page 551, figure B.1, normal cone in the right-hand figure: • page 552, line 20 (last-but-one line in third paragraph): if |x| ≤ 1, x 2 2λ if |x| > 1,... • page 552, figure B.2: in the first figure on the left, replace f λ (x) by -f λ (x). The Moreau-Yosida approximation is a convex function. It becomes obvious also from this figure, that the function -f λ (•) represents the potential energy function of a unilateral spring, and has to be compared with the indicator function which represents the potential energy associated with complementarity conditions (which are a particular contact model).

H||H|| 2 ,

 2 pages 561-562, about prox-regular sets. Characterizations of finitely represented sets which are prox-regular are given in[START_REF] Adly | Preservation of prox-regularity of sets with applications to constrained optimization[END_REF] Theorems 3.1, 4.1], in addition to Theorem B.5. An interesting result about the preservation of prox-regularity under an inverse linear transformation S ′ = H -1 (S) = {z|Hz ∈ S}, is in [19, Lemma 2.7] (reference 1181 in the book's bibliography). If S ⊂ IR l is r-prox-regular, and if S is in the range space of H : IR n → IR l , then S ′ is r ′ -proxregular with r ′ = rσ + where σ + H is the least positive singular value of H and ||H|| is an induced matrix norm. An extension is in[START_REF] Adly | Preservation of prox-regularity of sets with applications to constrained optimization[END_REF] Corollary 6.5].

  • page 286, about Coulomb's friction: notice that in the sliding mode one has ||F t || = α||v t || = µ||F n || ⇔ α = µ||Fn|| ||vt|| . Hence we recover the equivalent classical way of expressing sliding Coulomb's friction as F t = -µ||F n || vt ||vt|| .

In[START_REF] Ruan | The unexpectedly small coefficient of restitution of a two-degree-of-freedom mass-spring system and its implications[END_REF] this is called multiple impacts, however in our terminology multiple impacts are simultaneous collisions, not a sequence of separated collisions.


Sayantika Chatterjee
email: sayantika21175@iiitd.ac.in

Angshul Majumdar
email: angshul@iiitd.ac.in

Emilie Chouzenoux
email: emilie.chouzenoux@inria.fr

Antibiotic Bacteria Interaction: Dataset and Benchmarking

Keywords:

This study introduces a dataset for drug-bacteria associations (DBA) that affects humans. Our contribution extends beyond merely curating the association matrix; we also conduct genomic similarity computations for all known bacteria impacting humans and assess structural similarities for all antibiotic drugs. This is a first of a kind dataset serving the fundamental goal of streamlining antibiotic repurposing for bacterial infections. we conducted thorough benchmarking on this dataset using state-of-the-art drug repurposing models. Furthermore, we applied this dataset to three case studies involving drug-resistant bacterial infections. Our results indicate that the dataset coupled with the algorithms are indeed effective for repurposing drugs with the potential for treating drug resistant bacterial infections.

Introduction

The central objective of this research is the curation of a comprehensive dataset named Drug-Bacteria Association (DBA). The dataset resource not only captures associations between drugs and bacteria but also provides essential information about the structural similarity of drugs and the genomic similarity of bacteria. The primary motivation behind curating this database is to harness its potential for drug repositioning, particularly in the context of antibiotics.

Drug repositioning, also referred to as drug repurposing or reprofiling, involves the exploration of new applications for existing drugs. In practice, it can be used in three scenarios:

• Investigating the effects of existing drugs on different diseases

• Identifying existing targets for new drugs

• Identifying existing drugs for new diseases Drug repositioning presents numerous advantages in contrast to traditional drug discovery methods. Firstly, as existing drugs have already undergone rigorous safety and efficacy testing, the need for extensive clinical trials is minimized. Consequently, the cost and time required for drug development can be significantly reduced. Secondly, since the safety of existing drugs for human use has already been established, the likelihood of encountering severe side effects is notably diminished.

Drug repositioning has witnessed numerous remarkable success stories, among which one of the most widely recognized is the utilization of Aspirin for heart protection. Originally developed as a headache reliever [START_REF] Duffy | Aspirin for cardioprotection and strategies to improve patient adherence[END_REF], this drug has demonstrated its efficacy in safeguarding the heart. Another notable example involves Propranolol, initially designed as a beta-blocker to address hypertension [START_REF] Lichstein | Effect of propranolol on ventricular arrhythmia. The beta-blocker heart attack trial experience[END_REF], but subsequently repurposed with great success in the prevention of migraines [START_REF] Linde | Propranolol for migraine prophylaxis[END_REF]. These instances highlight the potential of repurposing existing drugs to address new medical conditions.

The emergence of COVID-19 brought drug repositioning to the forefront of scientific exploration. Given the novelty of the virus, developing a completely new drug to combat this infection would have been a time-consuming process. Consequently, repurposing existing drugs became the primary option in the short term [START_REF] Guy | Rapid repurposing of drugs for COVID-19[END_REF][START_REF] Venkatesan | Repurposing drugs for treatment of COVID-19[END_REF][START_REF] Kumari | Discovery of new drug indications for COVID-19: A drug repurposing approach[END_REF]. The field of drug repositioning, particularly in the context of viral infections, experienced a surge in research efforts. New methodologies, such as networksMedicine [START_REF] Habibi | Topological network based drug repurposing for coronavirus[END_REF] and artificial intelligence [START_REF] Zhou | Artificial intelligence in COVID-19 drug repurposing[END_REF], were specifically developed to tackle this challenge. Additionally, an extensive database focusing on drug-virus associations was meticulously curated [START_REF] Mongia | A computational approach to aid clinicians in selecting anti-viral drugs for COVID-19 trials[END_REF].

Two extensive datasets are available for general-purpose drug repositioning, namely from sources [START_REF] Gottlieb | PREDICT: a method for inferring novel drug indications with application to personalized medicine[END_REF] (covering 313 diseases and 593 drugs) and [START_REF] Luo | Drug repositioning based on comprehensive similarity measures and bi-random walk algorithm[END_REF] (encompassing 313 diseases and 663 drugs). It is worth noting that the latter dataset is an expanded version of the former, including additional entries. These datasets are not specific to any particular branch of diseases or drugs, making them suitable for testing novel algorithms designed for drug repurposing purposes. However these datasets are not focussed and does not solve any specific problem, and can only be used for benchmarking algorithms.

According to the World Health Organization (WHO), viral and bacterial infections are predominantly categorized as neglected diseases 4 . As previously mentioned, a specific database tailored for viral infections has been curated in [START_REF] Mongia | A computational approach to aid clinicians in selecting anti-viral drugs for COVID-19 trials[END_REF]. Consequently, in this study, our focus lies in curating a comprehensive dataset specifically for drug-bacteria associations. The primary component of this database is a matrix that aligns bacteria along the columns and drugs along the rows. Within this matrix, a value of 1 indicates that a drug is effective against the corresponding bacterial infection. The information has been meticulously curated from the antibiotic policy established by the All India Institute of Medical Sciences5 . In addition to the main association matrix, two other matrices are associated with it. The first matrix represents the similarity between drugs and was curated using DrugBank6 and KEGG 7 . The second matrix depicts the similarity between bacteria, computed and curated using the Genome-Genome-Distance Calculator 8 .

The second major contribution of this study involves the rigorous benchmarking of drug repurposing algorithms on our curated dataset. It is worth noting that datasets for drug repurposing are often limited in size, typically consisting of only a few thousand entries. Consequently, deep learning approaches tend to yield suboptimal results due to the scarcity of data. Through our experimentation, we discovered that matrix completion-based techniques, which does not rely on training, yield by far the best results. This finding is corroborated by a recent survey conducted in the field [START_REF] Sadeghi | An analytical review of computational drug repurposing[END_REF]. It is important to recognize that contemporary network-based methods bear relevance to matrix completion techniques [START_REF] Chen | Link prediction based on non-negative matrix factorization[END_REF][START_REF] Zhao | A novel link prediction algorithm based on inductive matrix completion[END_REF][START_REF] Sadeghi | A network-based drug repurposing method via non-negative matrix factorization[END_REF]. Therefore, for the remainder of this study, we will adopt the nomenclature and approach of matrix completion.

The final contribution of this study centers on a set of case studies. While the use of antibiotics for non-complicated bacterial infections is widely acknowledged, challenges arise when bacteria develop resistance to these drugs. It is crucial to emphasize that antimicrobial resistance is a highly concerning global issue [START_REF] Hofer | The cost of antimicrobial resistance[END_REF][START_REF] Dadgostar | Antimicrobial resistance: implications and costs[END_REF]. Consequently, our research efforts concentrated on predicting appropriate antibiotics for drug-resistant isolates.

Methods

Data set: The information about the bacterial infections and the corresponding effective drugs are taken from the antibiotic guidelines 2018 of Christian Medical College, Vellore9 and the antibiotic policy manual of AIIMS (All India Institute of Medical Sciences). In these two manuals, bacterial infections including the causative organisms (bacteria) and the effective drugs or antibiotics are specified.

As mentioned in the introduction, our approach for drug-bacteria association is based on matrix completion. In this approach, antibiotics are represented along the rows, while bacteria are represented along the columns (or vice versa). Following the provided guidelines, we assigned a value of 1 to indicate the effectiveness of a drug against a particular bacterial infection, thus representing true positives in the matrix. However, it is important to acknowledge that the information regarding drugs known to be ineffective against specific bacterial infections were not available in the guidelines. This limitation is not exclusive to our data set, as it is a common challenge in various biological interaction problems, including drug-target interaction [START_REF] Shi | SRP: A concise non-parametric similarity-rank-based model for predicting drug-target interactions[END_REF], drug-disease interaction [START_REF] Gottlieb | PREDICT: a method for inferring novel drug indications with application to personalized medicine[END_REF][START_REF] Luo | Drug repositioning based on comprehensive similarity measures and bi-random walk algorithm[END_REF], drug-drug interaction [START_REF] Wishart | DrugBank 5.0: a major update to the DrugBank database for 2018[END_REF], drug-virus interaction [START_REF] Mongia | A computational approach to aid clinicians in selecting anti-viral drugs for COVID-19 trials[END_REF], and others. The resulting matrix has dimensions of 53 x 61, with 53 antibiotics and 61 bacteria. Standard matrix completion cannot incorporate metadata in its framework. However, there is a host of information available for both drugs and bacteria. Past research in biological interaction where metadata is available indictae that graphical matrix completion methods yield very good results. In this class of techniques, the graphs incorporate the associated metadata.

To incorporate relevant metadata for the drugs and bacteria, we obtain all the necessary information on drugs from the DrugBank vocabulary 10 . From there, we extract the unique DrugBank ID for the antibiotics. However, instead of utilizing the raw data for the drugs, we focus on the structural similarity between them. This follows from a prior work on drug virus association [START_REF] Mongia | A computational approach to aid clinicians in selecting anti-viral drugs for COVID-19 trials[END_REF][START_REF] Mongia | Deepvir: Graphical deep matrix factorization for in silico antiviral repositioning-application to covid-19[END_REF]. This similarity is computed using the SIMCOMP score, which requires access to the Kyoto Encyclopedia of Genes and Genomes (KEGG) [START_REF] Kanehisa | From genomics to chemical genomics: new developments in KEGG[END_REF]. Each KEGG object is assigned a unique identifier known as the KEGG ID (kid). The linking between KEGG and DrugBank can be found in the linking file 11 [START_REF] Wishart | DrugBank: a comprehensive resource for in silico drug discovery and exploration[END_REF]. In cases where certain drugs were missing from the file, we manually supplemented the information to establish the connection between each antibiotic and its corresponding DrugBank ID with the appropriate KEGG ID in the KEGG Compound/KEGG Drug database 12 of the KEGG.

Similarly, we also need to compute the similarity between bacteria, focusing on their genomic similarity. To accomplish this, we acquired the full genome sequences in FASTA format from the National Center for Biotechnology Information (NCBI) genome database 13 [22]. It is important to note that we specifically considered the primary strain for each bacteria, resulting in a total of 61 genome sequences. These sequences serve as input for calculating the similarity among bacteria.

Similarity computation

Analogies among drugs and bacteria are computed using the chemical information of drugs and the genome sequences of bacteria. These similarities prove valuable in designing the drug repurposing tool discussed in this study.

Structural similarity for drugs

The drug data includes DrugBank IDs and their corresponding KEGG IDs. These DrugBank IDs are then combined with the corresponding KEGG Compound IDs to create a new dataset that includes DrugBank ID, KEGG Compound ID, KEGG Drug ID, and the Drug Name. Each drug in the dataset is associated with either a KEGG Compound ID or a KEGG Drug ID that matches its DrugBank ID.

To calculate the similarity among the drugs, we consider their chemical significance or chemical structure. For this purpose, we utilize the SIMCOMP (SIMilar COMPound) server, which allows us to perform a chemical similarity search. The SIMCOMP score is employed to measure the chemical structure similarity between two drugs [START_REF] Hattori | SIMCOMP/SUBCOMP: chemical structure search servers for network analyses[END_REF]. This score is obtained using a graph-based method that identifies the most extensive common elements shared by the chemical structures of the compounds.

To conduct the sequence analysis, we utilize the GenomeNet Tools API 14 . We set a cutoff value of 0.001 for the SIMCOMP score calculation. If the SIMCOMP score for a drug falls below this cutoff or if a drug does not have any available KEGG IDs, we assign a similarity score of 1 to itself and a score of 0 to every remaining pharmaceuticals in the dataset.

Bacteria genomic similarity

To determine the similarity between two bacteria based on their genome sequences, we employ the Genome-Genome Distance Calculator (GGDC) [START_REF] Meier-Kolthoff | TYGS and LPSN: a database tandem for fast and reliable genome-based classification and nomenclature of prokaryotes[END_REF]. This method utilizes advanced in silico techniques to compare the genomes of different organisms, mimicking the conventional DNA-DNA hybridization (DDH) method with certain improvements and avoiding its limitations. The GGDC tool offers significantly faster performance compared to other techniques available.

We also explored the use of Stretcher, a global alignment tool from the Emboss suite, to determine the similarity between two genome sequences. However, this approach proved to be time-consuming, requiring approximately 10 hours for computing the sequence similarity. In contrast, GGDC delivers the most accurate and efficient results within a considerably shorter timeframe.

For the GGDC analysis, we submit multiple query genome sequences in FASTA format representing various bacteria, along with a single reference genome sequence in FASTA format. These query genomes and the reference genome are treated as inputs for the GGDC web server. The output of the comparison is can be obtained by emailing the first author; it will contain the relevant information derived from the analysis.

Benchmarked algorithms

Matrix completion is a relatively mature area of research. As mentioned before, we are solving the drug-bacteria-association problem as one of matrix completion. From the known entries, some portions are assumed to be unknown and are hidden (10% around in several experimental scenarios). Matrix completion is used to recover these. Once the recovery is done, the recovered entries are compared with the ground-truth (hidden) values.

Let X be the entire drug-bacteria association matrix of size m × n denoted by X m×n . Here m stands for the total count of drugs and n stands for the total count of bacteria. As mentioned before, the association matrix contains 1 indicating the drug's recognized efficacy against the bacteria. The goal is to estimate X based on its partially observed version Y . Consider M as a masking operator. with 1 indicating the known entries, 0 otherwise. Then,

Y = M (X). (1)
In the matrix completion paradigm, the assumption is that the underlying matrix to be recovered X, satisfies certain structural properties (low rank and uniformly distributed samples of known entries). Under such conditions, the number of free variables is reduced and one can expect to estimate the underlying matrix [START_REF] Sun | Guaranteed matrix completion via non-convex factorization[END_REF][START_REF] Candes | Matrix completion with noise[END_REF].

Matrix factorization (MF)

The simplest and most direct approach to complete a partially observed matrix is via matrix factorization. Here, matrix X is expressed as the result of multiplying two matrices V q×n and U m×q where q denotes the latent space dimension, so that

Y = M (U V ), (2)
using the factorized model,

X = U V. (3)
The factors are learned by finding the solution to the optimization problem outlined below: min

U,V ∥Y -M (U V )∥ 2 F , (4)
with ∥ • ∥ F the Frobenius norm on matrices. There are many variants to the MF approach. The most notable one is non-negative matrix factorization in which the elements of both matrices V and U are constrained to be positive. Other variants include different types of regularizations on U and V [START_REF] Wang | Nonnegative matrix factorization: A comprehensive review[END_REF].

Nuclear norm minimization based matrix completion

Matrix factorization builds upon a bilinear model, and as such, it necessitates a solution of a non-convex minimization task. Furthermore, one needs to have an estimate on the rank of X (i.e., the dimension of the latent space, q). Instead, one might prefer to directly estimate a low-rank matrix, by solving min X rankX such that Y=M(X) .

Unfortunately, solving this falls into the category of problems known to be NP-hard, which has the potential to be relaxed by its nearest convex equivalent, leveraging the nuclear norm [START_REF] Candes | Exact matrix completion via convex optimization[END_REF][START_REF] Candès | The power of convex relaxation: Near-optimal matrix completion[END_REF]. It leads to:

min X ||X|| * ensuring Y=M(X) . (6)
We define ∥ • ∥ * as the nuclear norm acting on matrices. The resolution of the above problem is achievable through the application of semi-definite programming, or proximal-based approaches [START_REF] Cai | A singular value thresholding algorithm for matrix completion[END_REF][START_REF] Mongia | McImpute: matrix completion based imputation for single cell RNA-seq data[END_REF].

Deep matrix factorization (DMF)

Fueled by its recent ubiquity in computational science, deep learning has transformed the classic technique of matrix factorization, endowing it with a powerful upgrade known as deep matrix factorization. Instead of factoring X into two matrices, one can factor it into more. Building on the achievements of deep dictionary learning [START_REF] Tariyal | Deep dictionary learning[END_REF], a deep factorization algorithm has been proposed. In this algorithm, X the data matrix is factorized into several-factor matrices enabling the extraction of intricate and potentially hidden structures within the data. For instance, in the case of three factors, we search for,

Y = M (U 1 U 2 V ), (7)
where,

X = U 1 U 2 V . (8)
Under positivity constraints, the minimization problem is formulated as:

min U1,U2,V ||Y -M (U 1 U 2 V )|| 2 F ensuring U 1 ≥ 0, U 2 ≥ 0. (9)
Efficient algorithms based on multiplicative updates [START_REF] Trigeorgis | A deep matrix factorization method for learning attribute representations[END_REF] and alternating least squares [START_REF] Mongia | deepmc: Deep matrix completion for imputation of single-cell rna-seq data[END_REF] can be used for solving the aforesaid optimization problem.

Binary matrix completion (BMC)

Drug-bacteria association problem is binary in nature. The aforementioned methods do not account for this, and return real outputs. AS crude way to binarize the results is to apply the previous algorithms followed by thresholding to binarise the output matrix. A more solution is to go for 1-bit / binary matrix completion [START_REF] Davenport | 1-bit matrix completion[END_REF]. The idea is to solve

min X ∥Y -M (X)∥ 2 F + λ||X|| * s.t. X ∈ {0, 1} n×m . ( 10
)
An efficient algorithm can be found in [START_REF] Davenport | 1-bit matrix completion[END_REF][START_REF] Gu | Collaborative filtering: Weighted nonnegative matrix factorization incorporating user and item graphs[END_REF].

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Graph regularised matrix factorization (GRMF)

All the aforementioned algorithms are designed for scenarios where the objective is to fill a sparse matrix without additional metadata about the rows or columns. In the specific context of drug-bacteria association, we have access to such metadata in the form of genomic sequence of the bacteria and chemical structure of the drugs. This information can be employed to regularize the matrix completion problem, as elaborated in [START_REF] Cai | Graph regularized nonnegative matrix factorization for data representation[END_REF]. Essentially, the model incorporates a graph Laplacian penalty into the cost function, derived from similarity weights among drugs/bacteria (i.e., rows/columns). The formulation is as follows:

min

U,V ||Y -M (U V )|| 2 F + µ 1 tr(U ⊤ L d U ) + µ 2 tr(V L b V ⊤ ), (11)
Here , i.e. the degree matrices. Such regularization minimizes the distance between latent feature vectors of two neighboring drugs (respectively, bacteria). A resolution technique for the above formulation has been presented in [START_REF] Ezzat | Drug-target interaction prediction with graph regularized matrix factorization[END_REF].

µ

Graph regularised matrix completion (GRMC)

Graph regularization can also be incorporated into the nuclear norm minimization framework. Again, this approach utilizes graph Laplacian penalties, which incorporate drug-drug and bacteria-bacteria similarity. The minimization problem can be written as:

min X ||Y -M (X)|| 2 F + λ||X|| * + µ 1 tr(X ⊤ L d X) + µ 2 tr(XL b X ⊤ ), (12)
Here µ 1 > 0, µ 2 > 0 and λ > 0 denote penalization weights. This problem can be solved using a primal-dual splitting technique, such as ADMM (alternating direction method of multipliers), exploiting the problem convexity [START_REF] Mongia | Drug-target interaction prediction using multi graph regularized nuclear norm minimization[END_REF].

Graph regularized deep matrix factorization (GRDMF)

Graph regularisation can also be incorporated into the deep matrix factorization approach [START_REF] Mongia | Deep matrix completion on graphs: Application in drug target interaction prediction[END_REF]. The formulation for the graph-regularized three-factor model is:

min U1,U2,V ||Y -M (U 1 U 2 V )|| 2 F + µ 1 tr(U 1 ⊤ L d U 1 ) + µ 2 tr(V L b V ⊤ ), (13)
Here µ 1 > 0 and µ 2 > 0 are penalization weights. In this context, it is assumed that U 1 corresponds to drugs and V corresponds to bacteria. However, unlike the two-factor model where there was a distinct one-to-one mapping between drugs, bacteria, and the factor matrices, the deeper model may encounter identifiability issues. This is because it is unclear whether U 1 exclusively pertains to drugs or if U 1 U 2 is related to drugs, and similarly, whether V exclusively pertains to bacteria or if U 2 V is associated with bacteria.

Graph regularised binary matrix completion (GRBMC)

A prior study has also introduced graph regularization into the binary matrix completion framework [START_REF] Mongia | Computational prediction of Drug-Disease association based on Graph-regularized one bit Matrix completion[END_REF]. The minimization problem reads as:

min X ||Y -M (X)|| 2 F + λ||X|| * + µ 1 tr(X ⊤ L d X) + µ 2 tr(XL b X ⊤ ), s.t X ∈ {0, 1} n×m (14)
January 6, 2024 7/22 with λ > 0, µ 1 > 0, µ 2 > 0 penalization weights. A proximal-based approach is introduced in [START_REF] Pustelnik | Parallel proximal algorithm for image restoration using hybrid regularization[END_REF] to address the above problem. We also refer the reader to [START_REF] Mongia | Computational prediction of Drug-Disease association based on Graph-regularized one bit Matrix completion[END_REF].

In this work we have concentrated on matrix completion based techniques for drug repurposing. This is driven by our own experience in prior studies, where we found that this class of techniques yields better results than others. However, we will briefly mention other classes of techniques for the sake of completeness. The simplest method involves application of neighbourhood models from collaborative filtering to this class of problems [START_REF] Zhang | Computational drug repositioning using collaborative filtering via multi-source fusion[END_REF]. A slightly more sophisticated approach is to use network diffusion based approaches [START_REF] Su | DTSEA: A network-based drug target set enrichment analysis method for drug repurposing against COVID-19[END_REF][START_REF] Zhang | NCH-DDA: Neighborhood contrastive learning heterogeneous network for drug-disease association prediction[END_REF]. Such methods are unfortunately not as accurate as matrix completion based techniques; furthermore current network based techniques are formulated as matrix completion and not network diffusion for the same reason. Current approaches solve the problem using deep learning [START_REF] Mei | Relation-aware heterogeneous graph transformer based drug repurposing[END_REF][START_REF] Doshi | A computational approach to drug repurposing using graph neural networks[END_REF]. Deep learning based techniques happen to be accurate; but they require large volumes of data which is typically absent in a problem such as ours. Hence, we have not considered this class of techniques.

Results

Within this segment we assess the effectiveness of several matrix completion algorithms. The Benchmarked algorithms subsection provides detailed descriptions of all these algorithms. Eight matrix completion algorithms or methods are used, which are divided into three categories listed here.

• Basic Frameworks (Matrix factorization(MF) [START_REF] Wang | Nonnegative matrix factorization: A comprehensive review[END_REF] and Matrix completion(MC) or Nuclear norm minimization [START_REF] Mongia | McImpute: matrix completion based imputation for single cell RNA-seq data[END_REF] and Binary matrix completion [START_REF] Davenport | 1-bit matrix completion[END_REF]).

• Deep Frameworks (Deep matrix factorization) [START_REF] Mongia | deepmc: Deep matrix completion for imputation of single-cell rna-seq data[END_REF] • Graph Regularized Frameworks (Graph regularized matrix factorization(GRMF) [START_REF] Ezzat | Drug-target interaction prediction with graph regularized matrix factorization[END_REF], Graph regularized matrix completion(GRMC) [START_REF] Mongia | Drug-target interaction prediction using multi graph regularized nuclear norm minimization[END_REF], Graph regularised deep matrix factorization(GRDMF) [START_REF] Mongia | Deep matrix completion on graphs: Application in drug target interaction prediction[END_REF], Graph regularized binary matrix completion(GRBMC) [START_REF] Mongia | Computational prediction of Drug-Disease association based on Graph-regularized one bit Matrix completion[END_REF])

Figure 1 presents a diagrammatic illustration of the proposed system's operation. The graph-regularised methods consider all three data sets (drug-bacteria association data, drug similarity, and bacteria similarity) to predict the associations or to reconstruct the data matrix.

Experimental protocols

To provide concrete examples and compare the effectiveness of various algorithms mentioned earlier in retrieving drug-infection associations from our curated dataset, we conducted a series of experimental protocols. These protocols involved implementing 10-fold cross-validation settings on three distinct scenarios.

The 10-fold cross-validations were performed on the training data, and we utilized eight matrix completion algorithms to predict the outcomes. Below are the details of the experimental protocols or scenarios that were employed:

• 10-fold Cross validation setting on scenario 1 (CV1):10 % of drug-bacteria association selected randomly are masked or left out for testing while the remaining ones are considered for training. This allows us to determine the ability of each algorithm to predict associations between existing drugs and bacteria. • 10-fold Cross validation setting on scenario 2 (CV2):10 % of randomly selected bacteria are masked or left out for testing while the remaining ones are considered. This helps to repurpose drugs for new bacterial infections.

• 10-fold Cross validation setting on scenario 3 (CV3):10 % of randomly selected drugs are masked or left out for testing while the remaining ones are considered. This helps find possible applications of a novel antibiotic.

Hyperparameter settings

The hyperparameters, encompassing the step size, regularization parameters, and latent factor dimensions utilized in the algorithms for various scenarios, underwent fine-tuning following the experimental protocols detailed above (see subsection "Experimental protocols" for more information). Optimal values for these hyperparameters were determined through a grid search conducted on the training data.

January 6, 2024 9/22

The hyperparameters obtained from the second cross-validation (CV2) were utilized for predicting drugs against multidrug-resistant bacteria. Similarly, the hyperparameters obtained from the third cross-validation (CV3) were utilized for predicting bacteria for any novel drug.

For each scenario and algorithm, the specific hyperparameters utilized can be found in Tables Table 1,Table 2, and Table 3. These tables illustrate a comprehensive overview of the selected hyperparameter values for each algorithm across different scenarios.

Experimental evaluation:

As explained in the subsection Experimental protocols, various experimental scenarios were employed by randomly concealing a subset of data. These hidden or withheld data served as the test set and were predicted by the mentioned algorithms. The performance of these algorithms was evaluated using two standard metrics: AUC (Area January 6, 2024 10/22

Under the Receiver Operating Characteristic curve) and AUPR (Area Under the Precision-Recall curve). Table 4 presents the results of AUC and AUPR for all the algorithms used in CV1. Similarly, Table 5 showcases the AUC and AUPR results for the algorithms utilized in CV2, and Table 6 showcases the AUC and AUPR results for the algorithms employed in CV3.

In this subsection, we evaluate and assess the performance of different algorithms in retrieving the drug-bacteria association matrix. In CV1, 10 % of randomly selected associations were masked, while in CV2 and CV3, 10 % of randomly chosen bacteria and 10 % of complete drug elements were masked, respectively.

To evaluate the models, two widely adopted metrics, namely AUC and AUPR, were utilized. AUC provides a quantitative assessment of the model's capability to separate negative and positive classes, assuming a balanced class distribution. However, since drug-bacteria association problems often exhibit imbalanced classes, AUPR is considered a more suitable metric for evaluation [START_REF] Ezzat | Drug-target interaction prediction with graph regularized matrix factorization[END_REF][START_REF] Burez | Handling class imbalance in customer churn prediction[END_REF].

The experimental results from CV1, CV2, and CV3 are presented in Table 4, Table 5, and Table 6, respectively. These tables display the AUC and AUPR values obtained for all eight algorithms employed in this study. Notably, the graph-regularized matrix factorization techniques outperformed the non-graph factorization techniques in terms of both AUC and AUPR. This outcome was expected, as the graphical versions incorporate additional information, such as the similarity among drugs and the similarity among bacteria, which is not available in the non-graphical versions.

The CV1 scenario focuses on the re-utilization of existing drugs to combat current infections. While this scenario may not hold significant practical importance, it serves as a standard benchmark in drug-disease association studies. Surprisingly, in this scenario, the matrix completion (MC) algorithm performs the worst, despite being a mathematically stronger technique. However, past observations have shown that matrix factorization (MF) tends to outperform nuclear norm minimization (MC) in practical applications [START_REF] Gu | Collaborative filtering: Weighted nonnegative matrix factorization incorporating user and item graphs[END_REF]. In our case, the binarized version of MC (BMC) achieves better results, likely due to the binary nature of our problem.

Both MF and DMF (deep matrix factorization) demonstrate substantial improvements over MC and BMC. However, MF performs slightly better than DMF, indicating that deeper models may lead to overfitting due to the small size of our dataset. Interestingly, the graph-regularized versions of the algorithms exhibit similar performance in this scenario, demonstrating comparable results to each other.

The CV2 scenario carries substantial practical significance as it simulates a situation where existing drugs can be employed to treat new bacterial infections. This scenario closely mirrors real-world challenges in addressing novel strains of bacterial infections. Expanding on the insights from CV1, we note similar trends in CV2 with some slight deviations. Consistent with the preceding scenario, the deeper versions of the algorithms generally outperform their shallower counterparts, indicating the potential advantages of integrating deeper architectures in predicting drug-bacteria associations for new infections.

In scenario CV3, the emphasis is on identifying which existing bacterial infections can be effectively treated with new antibiotics, holding practical importance, especially in the context of antibiotic development. The experimental results unveil intriguing findings. Surprisingly, the MC algorithm outperforms both MF and BMC, although the reason behind this anomaly remains unclear. As anticipated, the graph-regularized versions consistently yield better results compared to their non-graph-regularized counterparts. Additionally, the deeper versions of the algorithms demonstrate improved performance over the shallower ones. Notably, the GRBMC algorithm emerges as the top performer in this scenario, achieving an AUC of 0.8029 and an AUPR of 0.5825, January 6, 2024 11/22 indicating its effectiveness in predicting associations between new antibiotics and existing bacterial infections.

Case studies in drug resistant bacteria

Our initial case study focuses on the Klebsiella pneumoniae U25 isolate [START_REF] Rafiq | Whole genome sequence of Klebsiella pneumoniae U25, a hypermucoviscous, multidrug resistant, biofilm producing isolate from India[END_REF], a drug-resistant strain discovered in India that exhibits resistance to ciprofloxacin, norfloxacin, and other fluoroquinolones. Table 7 presents the top-6 recommendations suggested by the various algorithms. All algorithms provided consistent recommendations, with piperacillin/tazobactam being predicted as the top two choices. Previous reports from the last decade have indicated the effectiveness of these drugs, comparable to carbapenems, in treating Klebsiella infections [START_REF] Zha | Pragmatic Comparison of Piperacillin/Tazobactam versus Carbapenems in Treating Patients with Nosocomial Pneumonia Caused by Extended-Spectrum β-Lactamase-Producing Klebsiella Pneumoniae[END_REF][START_REF] Kang | Piperacillin-tazobactam as an initial empirical therapy of bacteremia caused by extended-spectrum β-lactamase-producing Escherichia coli and Klebsiella pneumoniae[END_REF]. Additionally, three of the algorithms predicted different carbapenems, which are also known to be effective against Klebsiella infections according to the aforementioned reports. Colistin, a last resort antibiotic for carbapenem-resistant strains, was correctly predicted by all algorithms and would have been a suitable treatment option for U25, considering it was not known to be resistant to colistin [START_REF] Karabinis | Colistin for Klebsiella pneumoniae-associated sepsis[END_REF]. Amikacin, which recent studies have shown to be effective against Klebsiella infections [START_REF] Rodrigues | Amikacin for the treatment of carbapenem-resistant Klebsiella pneumoniae infections: clinical efficacy and toxicity[END_REF][START_REF] Kuti | Defining the potency of amikacin against Escherichia coli, Klebsiella pneumoniae, Pseudomonas aeruginosa, and Acinetobacter baumannii derived from Chinese hospitals using CLSI and inhalation-based breakpoints[END_REF], was predicted by all four algorithms.

However, GRMF and GRBMC also predicted vancomycin, for which we could not find literature supporting its use in Klebsiella pneumoniae-induced infections. GRMF specifically predicted ceftriaxone (a cephalosporin), which is the first-line treatment for non-multi-drug-resistant Klebsiella infections. However, there is no clinical trial evidence to support its efficacy against U25 or other multi-drug resistant Klebsiella pneumoniae infections.

GRMC predicts sulbactam (ampicillin/sulbactam), we could not find any literature supporting its use in multi-drug-resistant Klebsiella infections. Rather, there is literature to support growing resistance to this drug [START_REF] Liu | Correlation between Drug Resistance of Klebsiella Pneumonia and Antimicrobial Drug Usage[END_REF]. January 6, 2024 12/22 The H041 strain of Neisseria gonorrhoeae, isolated in Japan [START_REF] Ohnishi | Is Neisseria gonorrhoeae initiating a future era of untreatable gonorrhea?: detailed characterization of the first strain with high-level resistance to ceftriaxone[END_REF], was found to be resistant to Ceftriaxone and other cephalosporins. Despite this resistance, all algorithms predicted Ceftriaxone as a treatment option, which aligns with the empirical practice of using Ceftriaxone as the first-line treatment. However, the mentioned paper reported that H041 was indeed resistant to this antibiotic, making the prediction incorrect.

On the other hand, all algorithms, except for GRMF, predicted Piperacillin/tazobactam, which is in line with the susceptibility of H041 to this drug combination as indicated in Table 8 of the mentioned paper. Therefore, this prediction is correct. GRDMF specifically predicted Ampicillin among the top 5 (or top 7 in other algorithms), and according to the mentioned paper, H041 was found to be susceptible to this drug, further validating the prediction.

GRMF predicted Penicillin G as a treatment option for H041, but as mentioned in the previous paper, H041 was found to be resistant to this antibiotic. Therefore, this prediction is also incorrect.

All algorithms predicted Vancomycin and Amoxicillin/Clavulanate as potential treatments for H041. However, since these drugs were not tested in the aforementioned paper, it cannot be definitively stated whether they would be effective against H041. While Vancomycin was occasionally used as a drug of choice for treating such infections in the 70s and 80s [START_REF] Brorson | Vancomycin-sensitive strains of Neisseria gonorrhoeae. A problem for the diagnostic laboratory[END_REF], there has been limited research on its use for treating multi-drug resistant gonorrhea since then. Similarly, Clavulanate has been sparingly used in the January 6, 2024 13/22 past [START_REF] Munday Pe | Treatment of gonorrhea with clavulanate-potentiated amoxicillin (Augmentin®)[END_REF][START_REF] Lim | Two dose Augmentin treatment of acute gonorrhoea in men[END_REF], but there is no recent research available to support its efficacy. GRDMF predicts gentamicin, which has been found to be effective for uncomplicated gonorrhea in previous studies [START_REF] Hathorn | The effectiveness of gentamicin in the treatment of Neisseria gonorrhoeae: a systematic review[END_REF]. In vitro studies indicate that this drug has the potential to treat multi-drug resistant gonorrhea [START_REF] Li | In vitro efficacy of gentamicin alone and in combination with ceftriaxone, ertapenem, and azithromycin against multidrug-resistant Neisseria gonorrhoeae[END_REF][START_REF] Bala | Gentamicin susceptibility among a sample of multidrug-resistant Neisseria gonorrhoeae isolates in India[END_REF][START_REF] Liu | Gentamicin susceptibility of Neisseria gonorrhoeae isolates from 7 provinces in China[END_REF]. However, further clinical trials are required to confirm its effectiveness in this context. The NIRTX011 sequence was obtained from the Chennai Bioproject 15 .From Table 9, it can be observed that Amoxycillin/Clavunate is predicted by GRDMF, GRBMC, and GRMC. This drug has been demonstrated to be effective in patients with multi-drug resistant TB [START_REF] Mishra | First successful use of low dose amoxicillin-clavulanic acid in management of drug resistant tuberculosis[END_REF] and is also included in the WHO guidelines 16 . Similarly, Levofloxacin, which is predicted by GRMC and GRBMC, is recommended as a treatment option for multi-drug resistant TB according to the WHO guidelines. It is also used to treat individuals who have been exposed to such patients.

All the methods predicted Vancomycin. However, we could not find definitive research supporting its usage in clinical settings for multi-drug resistant TB. Nonetheless, in vitro studies indicate its potential effectiveness against multi-drug resistant tuberculosis [START_REF] Soetaert | Increased vancomycin susceptibility in mycobacteria: a new approach to identify synergistic activity against multidrug-resistant mycobacteria[END_REF][START_REF] Srivastava | Potency of vancomycin against Mycobacterium tuberculosis in the hollow fiber system model[END_REF]. Similarly, Ceftriaxone (and Cefazolin, a related cephalosporin) has been found to be effective against multi-drug resistant TB in in vitro studies [START_REF] Srivastava | Repurposing cefazolin-avibactam for the treatment of drug resistant Mycobacterium tuberculosis[END_REF][START_REF] Nguyen | Exploring Cell Wall Targets to Overcome Mycobacterium tuberculosis (Mtb): Ceftriaxone (CRO) Inhibits LdtMt2, a Major Peptidoglycan (PG) Synthase[END_REF]. With the exception of GRMF, three of the other algorithms predicted doxycycline. In vitro studies have demonstrated that this drug, either alone [START_REF] Zhang | In vitro activity of tetracycline analogs against multidrug-resistant and extensive drug resistance clinical isolates of Mycobacterium tuberculosis[END_REF] or in combination with amikacin [START_REF] Gonzalo | Combination of amikacin and doxycycline against multidrug-resistant and extensively drug-resistant tuberculosis[END_REF], has potential for use in multi-drug resistant tuberculosis. GRMF predicted Piperacillin and Tazobactam, Cefoperazone and Sulbactam. Although known as broad-spectrum antibiotics, we could not find any research supporting their treatment for multi-drug resistant tuberculosis for Piperacillin. In-vitro studies have shown the limited efficacy of sulbactam when administered with meropenem [START_REF] Zhang | In vitro activity of β-lactams in combination with β-lactamase inhibitors against multidrug-resistant Mycobacterium tuberculosis isolates[END_REF]. We could not find any literature on the effectiveness of Tazobactam and Cefoperazone on multi-drug resistant TB.

Discussion

We have compiled a database called the Drug-Bacteria Association Database (DBA), which contains information about bacterial strains and the drugs effective against them. This database also includes similarity information associated with bacteria and drugs (refer to Methods for details). Various matrix completion techniques, including graph-regularized methods, have been applied to this database. The information on drug-bacteria associations and their associated metadata, such as similarity information, is stored in three matrices: the drug-bacteria association January 6, 2024 15/22 matrix (Y), the bacteria similarity matrix (Sb), and the drug similarity matrix (Sd). Eight matrix completion methods are explored and evaluated. The matrix completion techniques that do not incorporate similarity information utilize the partially observed association matrix as their input data, aiming to recover the underlying low-rank structure by imputing the missing entries. These methods also incorporate a masking operator that indicates the positions of test and train indices. Conversely, the techniques incorporating graph regularization not only utilize the association matrix but also leverage the associated metadata, namely the similarity information. The drug similarity information is obtained from the chemical structures of drugs, while the similarity information for bacteria is derived by calculating the genome distance among bacterial strains. This metadata, along with the association matrix, is provided as input to the graph-regularized algorithms. Table 4, Table 5, and Table 6 clearly demonstrates that the graph-regularized methods, which consider the similarity information among bacteria and drugs, exhibit superior performance in association prediction (as indicated by improved AUC and AUPR values) compared to the non-graph-regularized methods.

We have also focused on several multidrug-resistant bacterial strains, namely Klebsiella pneumoniae strain U25, Neisseria gonorrhoeae strain H041, and Mycobacterium tuberculosis strain NIRTX011. Using the graph-regularized method, we have generated drug recommendations (top 6) for these strains, and the outcomes are presented in Table 7, Table 8, and Table 9. It is noteworthy that the predictions made by GRMC, GRDMF, and GRBMC consistently align with each other, while GRMF exhibits inconsistencies. This discrepancy is particularly evident in the case of the NIRTX011 strain. Based on these findings, we advise against using GRMF for antibiotic repurposing and suggest relying on the recommendations from the other graph-regularized methods instead. The data set and code can be freely accessed on GitHub at the following URL: 17 .

Conclusion

This study represents the first effort to curate a dedicated dataset specifically tailored for drug-bacteria associations. The primary aim in creating this dataset is to streamline the repurposing of antibiotics for treating bacterial infections in humans. Beyond documenting the associations between antibiotics and bacteria, the dataset includes essential genomic similarity data for bacteria and structural similarity data for drugs.

To tackle this problem, we have primarily resorted to matrix completion techniques. The evaluation encompassed three distinct and significant test scenarios. The first scenario focused on uncovering new applications for existing drugs in the realm of known bacterial infections. The second scenario involved identifying potential targets for newly developed antibiotics. Finally, the third scenario entailed repurposing existing drugs to combat emerging bacterial infections.

While the treatment options for uncomplicated bacterial infections are well-established, our dataset holds potential for training algorithms specifically designed to address the challenges posed by drug-resistant strains of known bacteria. Computational tools for addressing the acute problem of antimicrobial resistance are being proposed in recent times18 [START_REF] Singh | Multi-scale temporal convolutional networks and continual learning based in silico discovery of alternative antibiotics to combat multi-drug resistance[END_REF][START_REF] Tarín-Pelló | Computer-aided drug repurposing to tackle antibiotic resistance based on topological data analysis[END_REF] To demonstrate the practical relevance of our approach, we conducted three in-depth case studies centered around multi-drug-resistant strains of Klebsiella pneumoniae, Neisseria gonorrhea, and Mycobacterium tuberculosis. Notably, an analysis of existing

  1 > 0 and µ 2 > 0 are penalization weights, tr(•) corresponds to the trace operator, L d = D d -S d and L b = D b -S b represent the graph Laplacians [38] for S d (row/drug) and S b (column/bacteria) similarity matrices, respectively. D d , D b are diagonal matrices comprising entries D ii d = j S ij d and D ii b = j S ij b

Fig 1 .

 1 Fig 1. Schematic diagram illustrating the DBA framework.

Table 1 .= 2 ;

 12 Parameters used in several algorithms for k 1 = 25; k 2 = 5; GRMC p = 7; λ = 0.03; µ 1 = 0.02; µ 2 = 0.005; ν 1 = 0.4; ν 2 = 0.3; GRMF p = 7; λ l = 0.05; λ d = 0.03; λ t = 0.03; GRDMF µ 1 = 0.1; µ 2 = 0.05, α = 1; k 1 = 25; k 2 = 5; p = 2; GRBMC p = 2; λ = 0.8; µ 1 = 0.1; µ 2 = 0.1; Table 2. Parameters used in several algorithms for CV2 Algorithms Hyperparameters for CV2 MC δ = 15.15; MF k = 20;α = 0.05; BMC σ = 0.3; DMF α = 0.7; k 1 = 30; k 2 = 20; GRMC p = 2; λ = 0.1; µ 1 = 0.01; µ 2 = 1; ν 1 = 0.4; ν 2 = 0.1; GRMF p = 8; λ l = 1.5; λ d = 0.1; λ t = 0.5; GRDMF µ 1 = 0.06; µ 2 = 1.6, α = 20; k 1 = 25; k 2 = 20; p = 2; GRBMC p = 2; λ = 0.5; µ 1 = 3; µ 2 = 0.02; Table 3. Parameters used in several algorithms for CV3 Algorithms Hyperparameters for CV3 MC δ = 5.05; MF k = 5; α = 0.5; BMC σ = 0.5; DMF α = 1.5; k 1 = 20; k 2 = 5; GRMC p = 2; λ = 0.01; µ 1 = 1; µ 2 = 0.1; ν 1 = 0.4; ν 2 = 0.1 GRMF p = 7; λ l = 0.0413; λ d = 0.02; λ t = 0.03; GRDMF µ 1 = 0.4; µ 2 = 0.7, α = 1; k 1 = 25; k 2 = 5; p = 2; GRBMC p = 2; λ = 0.8; µ 1 = 0.08; µ 2 = 1;

Table 4 .

 4 Performance comparison of association prediction for all techniques under CV1

	Metric MC	MF	BMC DMF GRMC GRMF GRDMF GRBMC
	AU C	0.5318 0.9239 0.6883 0.9168 0.9313	0.9558	0.9244	0.9501
	AU P R	0.2484 0.8270 0.4122 0.7961 0.8108	0.8928	0.8232	0.8775

Table 5 .

 5 Performance comparison of association prediction for all techniques under CV2.

	Metric MC	MF	BMC DMF GRMC GRMF GRDMF GRBMC
	AU C	0.4703 0.5138 0.5046 0.6049 0.7340	0.7096	0.7320	0.7301
	AU P R	0.2306 0.2233 0.2079 0.3383 0.4315	0.3922	0.4216	0.4320

Table 6 .

 6 Performance comparison of association prediction for all techniques under CV3

	Metric MC	MF	BMC DMF GRMC GRMF GRDMF GRBMC
	AU C	0.5206 0.5018 0.4872 0.5837 0.7969	0.7904	0.7115	0.8029
	AU P R	0.2521 0.3080 0.2293 0.2856 0.5523	0.5480	0.4344	0.5825

Table 7 .

 7 Predicting drugs for Klebsiella pneumoniae strain U25

	Algorithms Used Recommended drugs
		Tazobactam
		Piperacillin
	GRMF	Vancomycin Amikacin
		Colistin
		Ceftriaxone
		Piperacillin
		Tazobactam
	GRDMF	Colistin Amikacin
		Ertapenem
		Meropenem
		Tazobactam
		Piperacillin
	GRBMC	Amikacin Colistin
		Meropenem
		Vancomycin
		Piperacillin
		Tazobactam
		Colistin
		Amikacin
	GRMC	Meropenem Sulbactam
	*Green -Effective, White -Unknown.

Table 8 .

 8 Predicting drugs for Neisseria gonorrhoeae strain H041

	Algorithms Used Recommended drugs
		Ceftriaxone
		Clavulanate
	GRMF	Amoxicillin Vancomycin
		PenicillinG
		Ampicillin
		Ceftriaxone
		Vancomycin
		Piperacillin
		Tazobactam
	GRDMF	Ampicillin Gentamicin
		Ceftriaxone
		Tazobactam
		Piperacillin
		Vancomycin
	GRBMC	Clavulanate Amoxicillin
		Ceftriaxone
		Tazobactam
		Piperacillin
		Vancomycin
	GRMC	Clavulanate Amoxicillin

*Green -Effective, Red -Ineffective, Blue -Experimental

  15 https://www.ncbi.nlm.nih.gov/bioproject/492975 16 https://apps.who.int/iris/bitstream/handle/10665/250125/9789241549639-eng.pdf

	January 6, 2024	14/22

Table 9 .

 9 Predicting drugs for Mycobacterium tuberculosis strain NIRTX011

	Algorithms Used Recommended drugs
		Vancomycin
		Ceftriaxone
	GRMF	Tazobactam Piperacillin
		Cefoperazone
		Sulbactam
		Ceftriaxone
		Vancomycin
	GRDMF	Doxycycline Clavulanate
		Amoxicillin
		Tazobactam
		Ceftriaxone
		Vancomycin
	GRBMC	Amoxicillin Clavulanate
		Doxycycline
		Levofloxacin
		Ceftriaxone
		Vancomycin
	GRMC	Doxycycline Clavulanate
		Amoxicillin
		Levofloxacin

*Green -Effective, White -Unknown, Blue -Experimental

https://www.aiimsjodhpur.edu.in/quick%20docs/Antibiotic%20Policy%20AIIMS%20JDH%202018. pdf

https://go.drugbank.com/categories/DBCAT000104

https://www.genome.jp/tools/simcomp/simcomp_help.html

https://www.dsmz.de/services/online-tools/genome-to-genome-distance-calculator-ggdc

https://github.com/sayantika21175/Thesis_2023/blob/main/Antibiotic%20guidelines% 20for%20adults%202018-%20CMC%20vellore.pdfJanuary 6, 2024

January 6, 2024

January 6, 2024 5/22

https://github.com/sayantika21175/Thesis_2023

https://www.sciencedirect.com/science/article/abs/pii/S0010482524000118 January 6, 2024

Acknowledgments

The authors thank Inria Saclay for the funding of this research, through the Inria International Associate Team program COMPASS.

medical literature underscored the promising performance of our benchmark algorithms when trained on our meticulously curated dataset.