Research article Special Issues

A GraphSAGE-based model with fingerprints only to predict drug-drug interactions


  • Received: 04 December 2023 Revised: 18 January 2024 Accepted: 21 January 2024 Published: 26 January 2024
  • Drugs are an effective way to treat various diseases. Some diseases are so complicated that the effect of a single drug for such diseases is limited, which has led to the emergence of combination drug therapy. The use multiple drugs to treat these diseases can improve the drug efficacy, but it can also bring adverse effects. Thus, it is essential to determine drug-drug interactions (DDIs). Recently, deep learning algorithms have become popular to design DDI prediction models. However, most deep learning-based models need several types of drug properties, inducing the application problems for drugs without these properties. In this study, a new deep learning-based model was designed to predict DDIs. For wide applications, drugs were first represented by commonly used properties, referred to as fingerprint features. Then, these features were perfectly fused with the drug interaction network by a type of graph convolutional network method, GraphSAGE, yielding high-level drug features. The inner product was adopted to score the strength of drug pairs. The model was evaluated by 10-fold cross-validation, resulting in an AUROC of 0.9704 and AUPR of 0.9727. Such performance was better than the previous model which directly used drug fingerprint features and was competitive compared with some other previous models that used more drug properties. Furthermore, the ablation tests indicated the importance of the main parts of the model, and we analyzed the strengths and limitations of a model for drugs with different degrees in the network. This model identified some novel DDIs that may bring expected benefits, such as the combination of PEA and cannabinol that may produce better effects. DDIs that may cause unexpected side effects have also been discovered, such as the combined use of WIN 55,212-2 and cannabinol. These DDIs can provide novel insights for treating complex diseases or avoiding adverse drug events.

    Citation: Bo Zhou, Bing Ran, Lei Chen. A GraphSAGE-based model with fingerprints only to predict drug-drug interactions[J]. Mathematical Biosciences and Engineering, 2024, 21(2): 2922-2942. doi: 10.3934/mbe.2024130

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  • Drugs are an effective way to treat various diseases. Some diseases are so complicated that the effect of a single drug for such diseases is limited, which has led to the emergence of combination drug therapy. The use multiple drugs to treat these diseases can improve the drug efficacy, but it can also bring adverse effects. Thus, it is essential to determine drug-drug interactions (DDIs). Recently, deep learning algorithms have become popular to design DDI prediction models. However, most deep learning-based models need several types of drug properties, inducing the application problems for drugs without these properties. In this study, a new deep learning-based model was designed to predict DDIs. For wide applications, drugs were first represented by commonly used properties, referred to as fingerprint features. Then, these features were perfectly fused with the drug interaction network by a type of graph convolutional network method, GraphSAGE, yielding high-level drug features. The inner product was adopted to score the strength of drug pairs. The model was evaluated by 10-fold cross-validation, resulting in an AUROC of 0.9704 and AUPR of 0.9727. Such performance was better than the previous model which directly used drug fingerprint features and was competitive compared with some other previous models that used more drug properties. Furthermore, the ablation tests indicated the importance of the main parts of the model, and we analyzed the strengths and limitations of a model for drugs with different degrees in the network. This model identified some novel DDIs that may bring expected benefits, such as the combination of PEA and cannabinol that may produce better effects. DDIs that may cause unexpected side effects have also been discovered, such as the combined use of WIN 55,212-2 and cannabinol. These DDIs can provide novel insights for treating complex diseases or avoiding adverse drug events.



    In this paper, Backward Difference Lattice Boltzmann (BD-LB) method is used to simulate the Burgers equation with variable coefficient in space and time. It is very difficult to construct its analytical solution directly for the partial differential equation (PDE) with variable coefficient in space and time, whereas finding its numerical solution is an effective way. The traditional macroscopic numerical simulation methods include finite difference (FD) method, finite volume (FV) method, and finite element (FE) method. Compared with these methods, lattice Boltzmann method (LBM) is a mesoscopic numerical approach with many advantages, such as clear physical background, simple algorithm, easy parallel computing, and easy to implement program and handle complex boundary conditions [1,2,3].

    The Boltzmann equation [4], a complex microintegral equation, is the fundamental equation of the gas kinetics theory. The right-end term is called the collision term, which is referred to as Ω(f). The existence of Ω(f) brings great difficulties to solve Eq (1.1).

    ft+ξfr+afξ=(ff1ff1)d2D|g|cosθdΩdξ1. (1.1)

    However, the collision operators used in the LBM are generally based on the much simpler Bhatnagar-Gross-Krook (BGK) collision operator. Chen and Qian et al. [5,6] put forward to single-relaxation-time lattice Boltzmann (SRT-LB) model. The model controls the speed of different particles near the equilibrium state by using the same time relaxation coefficient.

    fj(x+cjδt,t+δt)fj(x,t)=1τ[fj(x,t)feqj(x,t)], (1.2)

    where cj is the discrete lattice velocity, τ stands for the relaxation time, and feqj represents the distribution function of local equilibrium state.

    Qin et al. [7] adds a body force term in Eq (1.2) to simulate the incompressible Navier-Stokes flow and then investigate aqueous humor dynamics in human eye. Hu and Lan et al. [8,9] add the modified function hi to the Eq (1.2) to simulate the Gardner equation and KdV-Burgers equation with time-dependent variable coefficient respectively. Chai et al. [10,11] add the auxiliary distribution function Gj(x,t) and a source term Fj(x,t), proposing a multi-relaxed time lattice Boltzmann (MRT-LB) model to solve the Navier-Stokes equation and the convective diffusion equations. Gerasim V. Krivovichev [12] uses the Eq (1.2) formula to analyze the parameterized higher-order finite difference schemes of the linear advection equations. These schemes are based on a linear combinations of the spatial approximations of the convective term at the characteristic directions.

    We use a FD method to solve the PDE problems in computational mathematics, but the forward difference format is often conditionally stable and the backward difference format is unconditionally stable. In this paper, we do not add correction function and auxiliary function, but we adopt the backward difference format and then rewrite the Eq (1.2) as follow:

    fj(x,t)fj(xcjδt,tδt)=1τ[fj(xciδt,tδt)feq j(xcjδt,tδt)]. (1.3)

    The Burgers equation with variable coefficient is generally used to simulate the formation and decay of non-plane shock waves. The Burgers equation with the nonlinear term a(t) and the dispersion term b(t) can simulate the propagation of long shock waves in two shallow liquids [13].

    ut+a(t)uux+b(t)2ux2=0. (1.4)

    It is shown in the literature that when b(t) is equal to 1, the shock wave solution reverses its velocity and collapses after a(t) changes the critical point of its symbol. In literature, the soliton-type solutions are constructed by using B¨acklund transformation for the given form of a(t) and b(t). But these methods are very difficult for solving the variable coefficient partial differential equations, and we study the following Burgers equation with variable coefficient by using BD-LB method [14].

    ut+ux[a(x,t)u]+2x2[b(x,t)u]=0. (1.5)

    Equation (1.5) can represent a variety of physical models widely used in the fields of solid state materials, plasmas and fluid. u is amplitude function about time t and space x. a(x,t) and b(x,t) are both analytical functions about x and t. The subscripts represent partial derivatives.

    This paper is organized as follows: In Section 2, the LB method for the Burgers equation of variable coefficient is recovered by using both the Chapman-Enskog (CE) analysis and the direct Taylor expansion (DTE) method. In Section 3, we first test Example 3.1 and obtain some reasonable lattice parameters. Then we apply these parameters to other variable coefficient examples and compare the numerical result to the analytical solution. In Section 4, we discuss the obtained results and draw a conclusion.

    The Eq (1.3) is rewrote as follow:

    fj=feqj,n1+fneqj,n1. (2.1)

    We will use the following notations in this paper:

    fj:=fj(x,t);

    fj,n1:=fj(xcjδt,tδt);

    feqj,n1:=feqj(xcjδt,tδt);

    fneqj,n1:=(11τ)(fj,n1feqj,n1);

    δ is the Kronecker delta;

    is first-order spatial partial derivative.

    In Eq (2.1), feqj,n1 is the equilibrium distribution function, fneqj,n1 is the non-equilibrium distribution function. We define the macroscopic variable u as the sum of ifi.

    Judging from Eq (2.1), we can get the following equation:

    fjfeqj,n1=fj,n1feqj,n11τ(fj,n1feqj,n1). (2.2)

    According to the equation proposed by Eq (1.2),

    fjfj,n1=1τ(fjfeqj,n1), (2.3)

    and then

    feqj,n1=fj,n1+τ(fjfj,n1)=fj,n1[1+τfj,nfj,n1fj,n1]=fj,n1{1+τ[fj,nfj,n11]}. (2.4)

    Equation (2.4) can be written as follow:

    feqj=fj{1+τ[fj,n+1fj1]}, (2.5)

    we delimit fj,n+1=eDjfj, where Dj is the difference operator Dj=t+cj.

    The Eq (2.1) can be rewritten as follow:

    fj(x,t)=[1+τ(eDj1)]1feqj(x,t). (2.6)

    In the CE expansion, the LB equation is expanded by a dimensionless parameter ϵ, which is proportional the Knudsen number (Kn=λ/L), λ is the mean free path and L is the feature length. We choose the local equilibrium distribution function in the following form [2,10,15,16,17,18,20]:

    feqj(u)=wj,0r(u)+wj,1cjs(u)+wj,2(c2jc2S,2)t(u), (2.7)

    we adopt the weight family wj,a selection satisfy

    jwj,a=1,jwj,acj=0,jwj,ac2j=c2S,a, (2.8)

    here, cS,a is the lattice sound speed and the expansion of the function, r,s,t are expanded by CE as follows:

    r=n=0ϵnrn(u),s=n=0ϵnsn(u),t=n=0ϵntn(u). (2.9)

    Correspondingly, the expansion of the equilibrium state distribution function is as follow:

    feqj(u)=n=0ϵnf(eq,l)j(u), (2.10)

    that is to say, the equilibrium state distribution function varies when it is close to the scaling limit.

    We define the second moment of u in Eq (2.7) as follows:

    Rn(u)=jf(eq,n)j,Sn(u)=jf(eq,n)jcj,Tn(u)=jf(eq,n)jc2j, (2.11)

    here, j>0 breaks the conservation of u through the collision process because Rn(u)=rn(u) and the r0(u)=u. In this paper, we only consider Eq (1.5) with a conservation u, so we adopt rj=0 when j>0. Furthermore, we hypothesize that u is differentiable in the analysis.

    For ϵ, the temporal and spatial scale expansion as t=k=1ϵktk, =ϵ1, tk is expressed as k time scales, k is expressed as k space scales, respectively.

    Dj=k=1ϵkDj,k, (2.12)

    in which Dj,k:=tk+δk,1cj1. The solution of the Eq (2.6) can be written in the following form:

    fj(x,t)=k=0ϵkf(k)j(x,t), (2.13)

    where

    f(0)j(x,t)=f(eq,0)j,f(1)j(x,t)=τDj,1f(eq,0)j+f(eq,1)j,f(2)j(x,t)=τ[Dj,2(τ12)D2j,1]f(eq,0)jτDj,1f(eq,1)j+f(eq,2)j. (2.14)

    The Burgers equation has a second-order spatial derivative. It needs to sort the ϵ2 by using the formalized u, the results are summarized in Table 1 and we choose the following equation in order to make the results equivalent to the Burgers equation.

    J0=0,J1=a(x,t)u2/2,K0=b(x,t)u/(τ1/2). (2.15)
    Table 1.  Equations of all order with regarding to the parameter ϵ.
    ϵ order Equations of motion
    1 t1u+x1J0=0,
    2 t2u=x1{(τ12)x1K0}x1J1.

     | Show Table
    DownLoad: CSV

    We rewrite fneqj,n1 in Eq (2.1) into Eq (2.16) [19]:

    fneqj,n1=k=1(11τ)k{feqj(x(k+1)cjδt,t(k+1)δt)feqj(xkcjδt,tkδt)}. (2.16)

    Here, we assume feqj as follow, and the equilibrium state distribution function feqj is an analytic function.

    feqj=u[w(0)j+Kw(2)j]+u2Jw(1)j, (2.17)

    the moments of wj are shown in Eq (2.17) as in Table 2. Explicit forms for these weights are presented in Eqs (2.28) and (2.30).

    Table 2.  The moments of wj.
    Order w(0)j w(1)j w(2)j
    0 1 0 0
    1 0 1 0
    2 0 0 1

     | Show Table
    DownLoad: CSV

    In Eq (2.1), for j sum easy to calculate, on the left of the equation is u(x,t), and the equilibrium part to the right of Eq (2.1) changes to

    jfeqj,n1=jm=0(δt)mm!(t+(cjx))mfeqj(x,t)=uδt(ut+u[Ju]x)+12(δt)22[Ku]x2+O(3ux3,2u2xt,2ut2). (2.18)

    At the same time, for fneqj,n1 in Eq (2.16), one obtain

    jk=1(11τ)k{feqj(x(k+1)cjδt,t(k+1)δt)feqj(xkcjδt,tkδt)}=δtT1(ut+u[Ju]x)+12(δt)22[Ku]x2T2+O(3ux3,2u2xt,2ut2), (2.19)

    where T1=τ1,T2=2τ2τ1,T3=6τ36τ2+τ1 and τ>1/2 [20].

    Putting Eqs (2.18) and (2.19) into Eq (2.1), we can get

    ut=u[Ju]x+δt2!2[Ku]x2T2+1T1+1+O(3ux3,2u2xt,2ut2). (2.20)

    In order to recover the macroscopic Burgers equation, the parameters are defined as

    J=a(x,t)/2,K=b(x,t)/(τ1/2). (2.21)

    Compared with Eqs (2.15) and (2.21), the two analytical equations produce the same results for non-linear equations. These two methods are very different, but they recover the macroscopic Burgers equation.

    In numerical simulation, we assume the physical space X=δxx and T=δtt, and then put the X,T into Eq (2.20). Therefore, Eq (2.21) is as follows:

    J=a(x,t)δt/(2δx),K=b(x,t)δt/[(δx)2(τ1/2)]. (2.22)

    At the same time, the leading truncation error term at (δt)0 of Eq (2.20) is the fourth spatial derivative term whose coefficients involve K. This error term is

    (δx)2(T4+1)2(T2+1)4uX4. (2.23)

    For the D1Q5 lattice, the format of feqj is Eq (2.17). In order to remove the truncation error, the following δfeqj is added to feqj in the D1Q7 lattice,

    δfeqj=2(T1+1)δt(T2+1)(δx)2w(4)j. (2.24)

    For the D1Q7 lattice, the following feqj is employed:

    feqj=u[w(0)j+Kw(2)j2(T1+1)δt(T2+1)(δx)2w(4)j]+u2Jw(1)j. (2.25)

    We adopt the notation standardized in the LBM literature, where DdQq [6] refers to the d spatial dimensional model with q kinetic velocity. For the D1Q5 and D1Q7 models, a set of discrete weights having only the the unit n moment, is provided. These set of weights have the following properties:

    icpiw(n)i=δp,n. (2.26)

    In the case of D1Q5, when ci={0,±1,±2}, w(n)i can be obtained by inverting the matrix:

    (111110112201144011880111616). (2.27)

    The w(n)i is

    (w(0)iw(1)iw(2)iw(3)iw(4)i)=({1,0,0}{0,±23,112}{54,23,124}{0,16,±112}{14,16,124}). (2.28)

    In the case of D1Q7, when ci={0,±1,±2,±3}, w(n)i can be obtained by inverting the matrix:

    (11111110112233011449901188272701116168181011882432430111616729729). (2.29)

    The w(n)i is

    (w(0)iw(1)iw(2)iw(3)iw(4)iw(5)iw(6)i)=({1,0,0,0,0}{0,±34,320,±160}{4936,34,340,1180}{0,1348,±16,148}{718,1348,112,1144}{0,±148,160,±1240}{136,148,1120,1720}). (2.30)

    In this paper, the global relative error (GRE) is used to verify the effectiveness of the lattice Boltzmann model and the least-squares fitting is used to calculate the accuracy of the model.

    GRE=k=1|u(xk,t)u(xk,t)|k=1|u(xk,t)|,

    among these, u(x,t) and u(x,t) respectively represent the numerical solution and analytical solution.

    Example 3.1. For a(x,t)=C1,b(x,t)=C2, Eq (1.5) and the initial boundary conditions are taken as follows:

    ut+C1uux=C22ux2,0x1,0tT;u(x,0)=2C2πsin(πx)C3+cos(πx),0x1;u(0,t)=u(1,t)=0,0tT,

    analytical solution [21]

    u(x,t)=2C2πeπ2C2tsin(πx)C3+eπ2C2tcos(πx).

    We assume C1=1,C2=0.01,C3=2,T=1. Figures 13 depict the images when δx=0.05, δx=0.1, δx=0.2, and the left-hand plots in each figure show the minimum stability values of τ and δt about the D1Q5 and D1Q7 lattices. We compare them with the reference [19], and the fitting about the images is very good. The value of δt is shown in Table 3, δt displays the maximum time increments to maintain stability with τ=1.

    Figure 1.  the relaxation time τ (a) and the GRE (b) are presented for δx=0.05.
    Figure 2.  the relaxation time τ (a) and the GRE (b) are presented for δx=0.1.
    Figure 3.  the relaxation time τ (a) and the GRE (b) are presented for δx=0.2.
    Table 3.  The value of δt corresponding to δx.
    δx 0.05 0.10 0.20
    δt(D1Q5) 1.26×106 5.75×106 1.20×104
    δt(D1Q7) 9.66×107 4.07×106 7.24×105

     | Show Table
    DownLoad: CSV

    The plots on the right-hand in Figures 13 show the relationship of GRE and δt about D1Q5 and D1Q7 lattices. The crossed points show corresponding results between the D1Q5 and D1Q7 models. In Figure 4, the numerical results agree very well with the analytical solution for any t.

    Figure 4.  the D1Q5 scheme (a) and the D1Q7 scheme (b) are used to compare with the analytical solution for δx=0.2.

    Example 3.2. For a(x,t)=sech2(t),b(x,t)=C1sech2(t), Eq (1.5) becomes

    ut+ux[sech2(t)u]+2x2[C1sech2(t)u]=0,

    analytical solution [22]

    u(x,t)=C3C2±C1C2r21r+cosh(k(x)+c(t)+l)+C1C2sinh(k(x)+c(t)+l)r+cosh(k(x)+c(t)+l),

    where k(x)=C2x,c(t)=C3tanh(t), C2 and C3 are any constant, r21.

    Here, macro parameters are assumed to be r=2,l=0,C1=2,C2=C3=1 and lattice parameters are assumed δx=0.2,τ=1, and the value of δt is shown in Table 3. The calculation area is fixed at [10,10]. Figure 5 shows the comparison plot of numerical results and analytical solution at different moments. The space-time plots of the numerical result from t=0 to t=10 are shown in Figure 6.

    Figure 5.  Comparison with the analytical solution for each t using the basic D1Q5 scheme (a) and the proposed D1Q7 scheme (b) for δx=0.2.
    Figure 6.  The numerical results for D1Q5 (a) and D1Q7 (b) for propagation of the soliton from t=0 to t=10.

    Remark 3.1. The colormap default in the MATLAB color box represents the solution of the D1Q5 simulation and the colormap jet stands for the solution of the D1Q7 simulation.

    Then we do some numerical accuracy experiments. Several simulations are performed at different lattice resolutions δx={0.05,0.1,0.15,0.2}, and the value of δt is 1.0×105. Based on the GRE at t=1 and t=2, the slopes of the fitting lines are very close to 2 in Figure 7, which indicates all of three models have a second-order accuracy in space. When δt={1.0×105,5.0×105,1.0×104}, δx=0.2. Based on the GRE at x=1 and x=2, the slopes of the fitting lines are very close to 1 in Figure 8, which indicates all of three models have a first-order accuracy in time. The results are the same as Eq (2.20).

    Figure 7.  The numerical spatial accuracy diagram of D1Q5 (a) and D1Q7 (b) in Example 3.2.
    Figure 8.  The numerical time accuracy chart of D1Q5 (a) and D1Q7 (b) in Example 3.2.

    Example 3.3. For a(x,t)=2sech2(t)ln(cosh(x2)),b(x,t)=C1sech2(t). Equation (1.5) becomes

    ut+ux[2sech2(t)ln(cosh(x2))u]+2x2[C1sech2(t)u]=0,

    analytical solution [22]

    u(x,t)=C3C2±C1C2r21r+cosh(k(x)+c(t)+l)+C1C2sinh(k(x)+c(t)+l)r+cosh(k(x)+c(t)+l),

    where

    k(x)=C1C2a0[ln(cosh((C1C4a0x)C3C1a0C2))ln(cosh(C4C3a0C2))],
    c(t)=C3t0α(τ)dτ,α(t)=sech2(t).

    When the analytical solution is positive, and the parameters are a0=1,r=2, C1=2,C2=C3=1, C4=l=0, lattice parameters are assumed to be δx=0.2,τ=1, and the calculation area is fixed at [10,10].

    We present the comparison between detailed numerical results and analytical solution. Figure 9 shows the two-dimensional visual comparisons at some different times. The space-time evolution graph of the numerical results is shown in Figure 10. The numerical results show that the scheme has good long-time numeric simulation for the Burgers equation with variable coefficient in space and time. All of them clearly show that the numerical results agree with the analytical solutions well.

    Figure 9.  Comparison with analytical solution for each t using the D1Q5 (a) and D1Q7 (b) when δx=0.2.
    Figure 10.  The numerical results for D1Q5 (a) and D1Q7 (b) for propagation of the soliton from t=0 to t=10.

    The calculation cost is given in Table 4 so as to compare the calculation time of the proposed scheme on D1Q7 and the basic scheme on D1Q5. We find that D1Q7 costs less than D1Q5. This improvement is even more significant as spatial step δx is increased.

    Table 4.  The calculation cost is compared with D1Q5 and D1Q7 in various δx.
    δx D1Q5 D1Q7
    0.05 4534.35±5.38 675.75±2.15
    0.1 180.45±2.43 56.00±1.60
    0.2 19.30±1.05 8.00±0.00

     | Show Table
    DownLoad: CSV

    We derive the nonlinear Burgers equation with variable coefficient from the LB equation by using the CE analysis and the DTE methods. Based on comparative observations in Section 2, we suggest that when we derive LB models for macro equation, it is best to start with using the CE analysis of general equilibrium states. After obtaining some results on the equilibrium distribution function, we can apply DTE method to conduct error analysis to improve the stability and accuracy of the model.

    In this study, we derive the LB model of Burgers by using a SRT-LB model and compare the LB solutions of the model with the corresponding analysis, which verifies the accuracy of the model. What's more, the improvement of accuracy by increasing lattice speeds can be regarded as a compensation for the deteriorated precision due to increased τ. Increasing the spatial step size δx can reduce the computational cost. In the future, we prepare to use LB model to simulate more non-linear PDEs with variable coefficient.

    This study is supported by National Natural Science Foundation of China (Grant Nos. 11761005, 11861003), the Natural Science Foundation of Ningxia (2021AAC03206), Postgraduate Innovation Project of North Minzu University (YCX21156) and the First-Class Disciplines Foundation of Ningxia (Grant No. NXYLXK2017B09).

    All authors declare no conflicts of interest in this paper.



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