Loading [MathJax]/extensions/TeX/boldsymbol.js
Research article Topical Sections

Molecular dynamics study of homo-oligomeric ion channels: Structures of the surrounding lipids and dynamics of water movement

  • Molecular dynamics simulations were used to study the structural perturbations of lipids surrounding transmembrane ion channel forming helices/helical bundles and the movement of water within the pores of the ion-channels/bundles. Specifically, helical monomers to hexameric helical bundles embedded in palmitoyl-oleoyl-phosphatidyl-choline (POPC) lipid bilayer were studied. Two amphipathic α-helices with the sequence Ac-(LSLLLSL)3-NH2 (LS2), and Ac-(LSSLLSL)3-NH2 (LS3), which are known to form ion channels, were used. To investigate the surrounding lipid environment, we examined the hydrophobic mismatch, acyl chain order parameter profiles, lipid head-to-tail vector projection on the membrane surface, and the lipid headgroup vector projection. We find that the lipid structure is perturbed within approximately two lipid solvation shells from the protein bundle for each system (~15.0 Å). Beyond two lipid “solvation” shells bulk lipid bilayer properties were observed in all systems. To understand water flow, we enumerated each time a water molecule enters or exited the channel, which allowed us to calculate the number of water crossing events and their rates, and the residence time of water in the channel. We correlate the rate of water crossing with the structural properties of these ion channels and find that the movements of water are predominantly governed by the packing and pore diameter, rather than the topology of each peptide or the pore (hydrophobic or hydrophilic). We show that the crossing events of water fit quantitatively to a stochastic process and that water molecules are traveling diffusively through the pores. These lipid and water findings can be used for understanding the environment within and around ion channels. Furthermore, these findings can benefit various research areas such as rational design of novel therapeutics, in which the drug interacts with membranes and transmembrane proteins to enhance the efficacy or reduce off-target effects.

    Citation: Thuy Hien Nguyen, Catherine C. Moore, Preston B. Moore, Zhiwei Liu. Molecular dynamics study of homo-oligomeric ion channels: Structures of the surrounding lipids and dynamics of water movement[J]. AIMS Biophysics, 2018, 5(1): 50-76. doi: 10.3934/biophy.2018.1.50

    Related Papers:

    [1] Tingting Ma, Xinzhu Meng . Global analysis and Hopf-bifurcation in a cross-diffusion prey-predator system with fear effect and predator cannibalism. Mathematical Biosciences and Engineering, 2022, 19(6): 6040-6071. doi: 10.3934/mbe.2022282
    [2] Yongli Cai, Malay Banerjee, Yun Kang, Weiming Wang . Spatiotemporal complexity in a predator--prey model with weak Allee effects. Mathematical Biosciences and Engineering, 2014, 11(6): 1247-1274. doi: 10.3934/mbe.2014.11.1247
    [3] Yuxuan Zhang, Xinmiao Rong, Jimin Zhang . A diffusive predator-prey system with prey refuge and predator cannibalism. Mathematical Biosciences and Engineering, 2019, 16(3): 1445-1470. doi: 10.3934/mbe.2019070
    [4] Wanxiao Xu, Ping Jiang, Hongying Shu, Shanshan Tong . Modeling the fear effect in the predator-prey dynamics with an age structure in the predators. Mathematical Biosciences and Engineering, 2023, 20(7): 12625-12648. doi: 10.3934/mbe.2023562
    [5] Zuolin Shen, Junjie Wei . Hopf bifurcation analysis in a diffusive predator-prey system with delay and surplus killing effect. Mathematical Biosciences and Engineering, 2018, 15(3): 693-715. doi: 10.3934/mbe.2018031
    [6] Guangxun Sun, Binxiang Dai . Stability and bifurcation of a delayed diffusive predator-prey system with food-limited and nonlinear harvesting. Mathematical Biosciences and Engineering, 2020, 17(4): 3520-3552. doi: 10.3934/mbe.2020199
    [7] Yong Luo . Global existence and stability of the classical solution to a density-dependent prey-predator model with indirect prey-taxis. Mathematical Biosciences and Engineering, 2021, 18(5): 6672-6699. doi: 10.3934/mbe.2021331
    [8] Dongfu Tong, Yongli Cai, Bingxian Wang, Weiming Wang . Bifurcation structure of nonconstant positive steady states for a diffusive predator-prey model. Mathematical Biosciences and Engineering, 2019, 16(5): 3988-4006. doi: 10.3934/mbe.2019197
    [9] Ting Yu, Qinglong Wang, Shuqi Zhai . Exploration on dynamics in a ratio-dependent predator-prey bioeconomic model with time delay and additional food supply. Mathematical Biosciences and Engineering, 2023, 20(8): 15094-15119. doi: 10.3934/mbe.2023676
    [10] Jun Zhou . Bifurcation analysis of a diffusive plant-wrack model with tide effect on the wrack. Mathematical Biosciences and Engineering, 2016, 13(4): 857-885. doi: 10.3934/mbe.2016021
  • Molecular dynamics simulations were used to study the structural perturbations of lipids surrounding transmembrane ion channel forming helices/helical bundles and the movement of water within the pores of the ion-channels/bundles. Specifically, helical monomers to hexameric helical bundles embedded in palmitoyl-oleoyl-phosphatidyl-choline (POPC) lipid bilayer were studied. Two amphipathic α-helices with the sequence Ac-(LSLLLSL)3-NH2 (LS2), and Ac-(LSSLLSL)3-NH2 (LS3), which are known to form ion channels, were used. To investigate the surrounding lipid environment, we examined the hydrophobic mismatch, acyl chain order parameter profiles, lipid head-to-tail vector projection on the membrane surface, and the lipid headgroup vector projection. We find that the lipid structure is perturbed within approximately two lipid solvation shells from the protein bundle for each system (~15.0 Å). Beyond two lipid “solvation” shells bulk lipid bilayer properties were observed in all systems. To understand water flow, we enumerated each time a water molecule enters or exited the channel, which allowed us to calculate the number of water crossing events and their rates, and the residence time of water in the channel. We correlate the rate of water crossing with the structural properties of these ion channels and find that the movements of water are predominantly governed by the packing and pore diameter, rather than the topology of each peptide or the pore (hydrophobic or hydrophilic). We show that the crossing events of water fit quantitatively to a stochastic process and that water molecules are traveling diffusively through the pores. These lipid and water findings can be used for understanding the environment within and around ion channels. Furthermore, these findings can benefit various research areas such as rational design of novel therapeutics, in which the drug interacts with membranes and transmembrane proteins to enhance the efficacy or reduce off-target effects.


    The dynamic relationship between prey and predator has been one of the most interesting topics in the qualitative theory of population dynamics [1,2]. For example, Bazykin [2] considered a prey-predator system with the bounded functional response:

    $ {dudt=uku2uv1+mu,dvdt=avbv2+uv1+mu, $ (1.1)

    where $ u $ and $ v $ represent the densities of the prey and predator, respectively. Here, $ a $, $ b $, $ k $ and $ m $ are positive constants. The predator consumes the prey based on the prey-dependent functional response $ u/(1+mu) $, and the term $ -bv^2 $ represents the self-limitation for the predator. The above model takes account of not only the interspecies interactions between the prey species and predators, but also the density dependence of predator species.

    Recently, there is much evidence from biology and physiology to show that in many situations, especially when predators have to search, share or compete for food, the modeling and analysis of predator-prey systems for each specific case should be further developed according to the ratio-dependent theory [3,4,5], which can be roughly stated as that the per capita predator growth rate should depend on the ratio of prey to predators, but not just prey numbers, so it should be the ratio-dependent function response [6,7,8,9], i.e., a function of $ u/v $. On the basis of this consideration, substituting $ \frac{u/v}{1+mu/v} $ for $ \frac u{1+mu} $ in the above-mentioned model (1.1), we arrive at the following system:

    $ {dudt=uku2uvmu+v,dvdt=avbv2+uvmu+v. $ (1.2)

    If $ ma < 1 $, system (1.2) has a unique positive equilibrium $ \widetilde{\textbf{w}} = (\tilde{u}, \tilde{v})^\mathit{T} $, where

    $ ˜u=m(a+b˜v)k, ˜v=m(1ma)k+m2b. $ (1.3)

    Taking into account the inhomogeneous distribution of predator and prey populations in different spatial locations, in this study we will incorporate diffusion and cross-diffusion to system (1.2) and consider a more general system:

    $ {ut=d1Δu+d12Δv+uku2uvmu+v,  xΩ, t>0,vt=d21Δu+d2Δvavbv2+uvmu+v,  xΩ, t>0,uν=vν=0,xΩ, t>0,u(x,0)=u0(x)0, v(x,0)=v0(x)0,xΩ, $ (1.4)

    where $ \Omega $ is a fixed bounded domain with the sufficient smooth boundary $ \partial\Omega $ in $ \mathbb{R}^N $, $ \nu $ is the outward unit normal vector of the boundary $ \partial\Omega $, and $ \Delta $ is the Laplace operator. The positive constants $ d_{1} $ and $ d_{2} $ are the random diffusion coefficients of prey and predator, respectively, which describe the natural dispersive forces of random movement of individuals. Here, $ d_{12} $ and $ d_{21} $ are the cross-diffusion coefficients that express population fluxes of preys and predators due to the presence of other species [10,11,12]. The non-flux boundary condition indicates that the system is self-contained with no external energy exchange. For more related studies on dynamics of population models with diffusion and cross-diffusion, we refer the reader to [12,13,14,15,16,17] and references therein.

    One of our motivations of this study lies in a fact that, in the past few decades there have been continuous interests in the existence of positive steady states of diffusive predator-prey systems, and the majority of works have devoted to discovering the effect of diffusion on positive steady states [18,19,20,21,22,23,24]. However, little attention has been paid to the case that both diffusion and cross-diffusion are present in population systems [25,26,27,28]. Therefore, the main purpose of this paper is to make an attempt on the existence and non-existence of non-constant positive steady states of system (1.4). The corresponding steady-state equations of system (1.4) are:

    $ {d1Δud12Δv=uku2uvmu+v,d21Δud2Δv=avbv2+uvmu+v,uν=vν=0,xΩ,xΩ,xΩ. $ (1.5)

    This paper is organized as follows. In Section 2, we discuss a priori estimate of the positive steady state for system (1.4), that is, a priori upper and lower bounds for positive solutions of the elliptic system (1.5). In Section 3, we investigate the existence and non-existence of non-constant positive solutions of system (1.5). In particular, the impact of diffusion and cross-diffusion on the existence of non-constant positive steady states is explored.

    In this section, to establish a priori estimates of upper and lower bounds of positive solutions for system (1.5) in a straightforward manner, we need the following two lemmas.

    Lemma 2.1. [22,Maximun Principle] Assume that $ g\in C(\overline{\Omega} \times \mathbb{R}^1). $

    $ \rm{(i)} $ If $ \omega \in C^2(\Omega)\cap C^1(\overline{\Omega}) $ satisfies $ \Delta \omega(x)+g(x, \omega(x))\geq 0, \ x\in \Omega; \frac{\partial \omega}{\partial \nu}\leq 0, \ x\in \partial\Omega $; and $ \omega(x_{0}) = \max \limits_{\overline{\Omega}} \omega $, then $ g(x_{0}, \omega(x_{0}))\geq 0 $.

    $ \rm{(ii)} $ If $ \omega \in C^2(\Omega)\cap C^1(\overline{\Omega}) $ satisfies $ \Delta \omega(x)+g(x, \omega(x))\leq 0, \ x\in \Omega; \frac{\partial \omega}{\partial \nu}\geq 0, \ x\in \partial\Omega $; and $ \omega(x_{0}) = \min \limits_{\overline{\Omega}} \omega $, then $ g(x_{0}, \omega(x_{0}))\leq 0 $.

    Lemma 2.2. [27,Harnack's Inequality] Suppose that $ c(x)\in C(\overline{\Omega}) $. Let $ \omega(x)\in C^2(\Omega) \cap C^1(\overline{\Omega}) $ be a positive solution to

    $ \Delta \omega(x)+c(x)\omega(x) = 0, \; x\in\Omega; \; \; \frac{\partial\omega}{\partial\nu} = 0, \; x\in \partial\Omega. $

    Then there exists a positive constant $ C_{\ast} = C_{\ast}(\|c(x)\|_{\infty}, \, \Omega) $ such that

    $ \max \limits_{\overline{\Omega}} \omega \leq C_{\ast}\min \limits_{\overline{\Omega}} \omega. $

    In the following, the positive constants $ C_{1} $, $ C_{2} $, $ C_1^* $, $ C_2^* $, $ c_1 $ and $ c_2 $ will rely upon the domain $ \Omega $. However, while $ \Omega $ is fixed, we will not mention this dependence explicitly. Also, for convenience, we shall write $ \Lambda $ instead of the set of constants $ (a, b, m, k) $ in the sequel.

    Theorem 2.1. [Upper Bounds] Suppose that $ \frac{d_{12}}{d_{1}}\leq D $ and $ \frac{d_{21}}{d_{2}}\leq D $ for an arbitrary fixed number $ D $. Then there exist positive constants $ C_{i} = C_{i}(D, \Lambda) $, $ i = 1, 2 $, such that the positive solution $ (u, v) $ of system (1.5) satisfies

    $ max¯ΩuC1 and max¯ΩvC2. $ (2.1)

    Proof. Set $ \phi = d_{1}u+d_{12}v $ and $ \psi = d_{21}u+d_{2}v $. Then the problem (1.5) becomes

    $ {Δϕ=1kuvmu+vd1+d12vuϕ,Δψ=abv+umu+vd21uv+d2ψ,ϕν=ψν=0,xΩ,xΩ,xΩ. $ (2.2)

    Let $ x_{1}\in \overline{\Omega} $ be a point satisfying $ \phi(x_{1}) = \max \limits_{\overline{\Omega}} \phi $. Applying Lemma 2.1 to the first equation of (2.2), we can obtain $ u(x_{1})\leq \frac 1 k $ and $ v(x_{1})\leq \frac{m}{k}(1-ku(x_1))\leq \frac{m}{k} $. Hence, we obtain

    $ max¯Ωu1d1max¯Ωϕ=1d1(d1u(x1)+d12v(x1))1k+d12d1mkC1. $ (2.3)

    Analogously, let $ x_{2}\in \overline{\Omega} $ be a point satisfying $ \psi(x_{2}) = \max \limits_{\overline{\Omega}} \psi $. Using Lemma 2.1 to the second equation of (2.2), we have $ v(x_{2})\leq \frac{1-ma}{bm} $. Then we have

    $ max¯Ωv1d2max¯Ωψ=1d2(d21u(x2)+d2v(x2))d21C1d2+1mabmC2. $ (2.4)

    Before presenting the lower bound, we present the following lemma.

    Lemma 2.3. Let $ ma < 1 $, $ d_{1, n}, \, d_{2, n}, \, d_{12, n}, \, d_{21, n} $ be positive constants, $ n = 1, 2, ..., $ and $ (u_{n}, v_{n}) $ be the positive solution of system (1.5) with $ d_{1} = d_{1, n}, \, d_{12} = d_{12, n}, \, d_{21} = d_{21, n} $ and $ d_{2} = d_{2, n} $. Suppose that $ (d_{1, n}, d_{2, n}, d_{12, n}, d_{21, n})\rightarrow (d_{1}, d_{2}, d_{12}, d_{21}) $, and $ (u_{n}, v_{n})\rightarrow (u^{\ast}, v^{\ast}) $ uniformly on $ \overline{\Omega} $. If $ u^{\ast} $ and $ v^{\ast} $ are positive constants, then $ (u^{\ast}, v^{\ast}) $ satisfies

    $ 1-ku^{\ast}-\frac{v^{\ast}}{mu^{\ast}+v^{\ast}} = 0, \ \ -a-bv^{\ast}+\frac{u^{\ast}}{mu^{\ast}+v^{\ast}} = 0. $

    That is, $ (u^{\ast}, v^{\ast}) = (\tilde{u}, \tilde{v}) $, is the unique positive constant solution of system (1.5).

    Proof. It is clear that $ \int_{\Omega} u_{n}\left(1-ku_{n}-\frac{v_{n}}{mu_{n}+v_{n}}\right)dx = 0 $ holds for all $ n $. If $ 1-ku^{\ast}-\frac{v^{\ast}}{mu^{\ast}+v^{\ast}} > 0 $, then $ 1-ku_{n}-\frac{v_{n}}{mu_{n}+v_{n}} > 0 $ as $ n $ is getting large. However, $ u_{n} $ is positive, so this is impossible. Similarly, $ 1-ku^{\ast}-\frac{v^{\ast}}{mu^{\ast}+v^{\ast}} < 0 $ is impossible either. Hence, $ 1-ku^{\ast}-\frac{v^{\ast}}{mu^{\ast}+v^{\ast}} = 0 $. Using the same argument leads to $ -a-bv^{\ast}+\frac{u^{\ast}}{mu^{\ast}+v^{\ast}} = 0 $. Consequently, $ (u^{\ast}, v^{\ast}) = (\tilde{u}, \tilde{v}) $.

    Theorem 2.2. (Lower Bounds). Assume that $ d $ and $ D $ are fixed positive constants. Then there exist positive constants $ c_{i} = c_{i}(d, D, \Lambda) $, $ i = 1, 2 $, such that when $ d_{1}, d_{2}\geq d $ and $ \frac{d_{12}}{d_{1}} $, $ \frac{d_{21}}{d_{2}}\leq D $, the positive solution $ (u, v) $ of system (1.4) satisfies

    $ min¯Ωuc1 and min¯Ωvc2. $ (2.5)

    Proof. By a straightforward calculation, we have

    $ u(1kuvmu+v)d1u+d12v1d1(1+kmax¯Ωu+max¯Ω{1muv+1})=1d1(1+kmax¯Ωu+1min¯Ω(muv+1))1d1(2+kmax¯Ωu)1d1(2+kC1), $

    and

    $ v(abv+umu+v)d21u+d2v1d2(a+bmax¯Ωv+max¯Ω{1m+vu})=1d2(a+bmax¯Ωv+1min¯Ω(m+vu))1d2(a+bmax¯Ωv+1m)1d2(a+1m+bC2). $

    Then, Lemma 2.2 indicates that there exist positive constants $ C_{1}^{\ast} = C_{1}^{\ast}(d, \Lambda) $ and $ C_{2}^{\ast} = C_{2}^{\ast}(d, \Lambda) $ such that $ \max \limits_{\overline{\Omega}} \phi \leq C_{1}^{\ast}\min \limits_{\overline{\Omega}} \phi $ and $ \max \limits_{\overline{\Omega}} \psi \leq C_{2}^{\ast}\min \limits_{\overline{\Omega}} \psi $. Thus we obtain

    $ max¯Ωumin¯Ωumax¯Ωϕmin¯Ω(d1+d12vu)/min¯Ωϕmax¯Ω(d1+d12vu)=max¯Ωϕmin¯Ωϕmax¯Ω(d1+d12vu)min¯Ω(d1+d12vu)C1, $ (2.6)

    and

    $ max¯Ωvmin¯Ωvmax¯Ωψmin¯Ω(d21uv+d2)/min¯Ωψmax¯Ω(d21uv+d2)=max¯Ωψmin¯Ωψmax¯Ω(d21uv+d2)min¯Ω(d21uv+d2)C2. $ (2.7)

    Now, we estimate the positive lower bounds of $ u $ and $ v $. Suppose that (2.5) is not true. Then there exists a sequence $ \{d_{1, n}, d_{2, n}, d_{12, n}, d_{21, n}\}_{n = 1}^\infty $ with $ (d_{1, n}, d_{2, n}, d_{12, n}, d_{21, n})\in [d, \infty)\times [d, \infty)\times (0, \infty)\times (0, \infty) $ such that the positive solution $ (u_{n}, v_{n}) $ of system (1.5) satisfies

    $ {Δ(d1,nun+d12,nvn)=unku2nunvnmun+vn,Δ(d21,nun+d2,nvn)=avnbv2n+unvnmun+vn,νun=νvn=0,xΩ,xΩ,xΩ, $ (2.8)

    and

    $ min¯Ωun0 or min¯Ωvn0, as n. $ (2.9)

    For (2.8), it follows from the standard regularity theorem of elliptic equations [29] that there exists a subsequence of $ \{(u_{n}, v_{n})\}_{n = 1}^\infty $, still denoted by $ \{(u_{n}, v_{n})\}_{n = 1}^\infty $, and two nonnegative functions $ u, v\in C^2(\overline{\Omega}) $ satisfying $ (u_{n}, v_{n})\rightarrow (u, v) $ as $ n\rightarrow\infty $. Assume that $ (d_{1, n}, d_{2, n}, d_{12, n}, d_{21, n})\rightarrow (d_{1}, d_{2}, d_{12}, d_{21})\in [d, \infty)\times [d, \infty)\times (0, \infty)\times (0, \infty) $. In view of inequalities (2.6) and (2.7), it is easy to see that either $ \max \limits_{\overline{\Omega}} u_n\rightarrow0 $ or $ \max \limits_{\overline{\Omega}} v_n\rightarrow0 $, as $ n\rightarrow\infty $.

    So, there are three possible cases.

    If $ \max \limits_{\overline{\Omega}} u_n\rightarrow0 $ and $ \max \limits_{\overline{\Omega}} v_n\rightarrow0 $, then $ (u_{n}, v_{n})\rightarrow (0, 0) $ uniformly on $ \overline{\Omega} $, which is a contradiction to Lemma 2.3.

    If $ \max \limits_{\overline{\Omega}} u_n\rightarrow0 $ and $ \max \limits_{\overline{\Omega}} v_n\rightarrow \overline{v} $, where $ \overline{v} $ is a positive constant. By integrating the second equation of system (2.8) in $ \Omega $, it gives

    $ Ωvn(a+bvn)dx=0. $

    Then we get $ v_n\rightarrow-\frac{a}{b} $, which yields a contradiction to $ \overline{v} > 0 $. Hence, $ \min \limits_{\overline{\Omega}} u > 0 $.

    Similarly, if $ \max \limits_{\overline{\Omega}} u_n\rightarrow \overline{u} $ and $ \max \limits_{\overline{\Omega}} v_n\rightarrow 0 $, where $ \overline{u} $ is a positive constant. By integrating the first equation of system (2.8) in $ \Omega $, it gives

    $ Ωun(1kun)dx=0. $

    Then $ \overline{u} = \frac 1k $. It implies that $ (u_{n}, v_{n})\rightarrow \left(\frac 1 k, 0 \right) $ uniformly on $ \overline{\Omega} $. Apparently, this leads to a contradiction to Lemma 2.3. Consequently, $ \min \limits_{\overline{\Omega}} v > 0 $.

    In this section, we study the non-existence and existence of non-constant positive steady states of system (1.5).

    To consider the non-existence of non-constant solutions of system (1.5) in this subsection, for convenience, we denote

    $ f(u, v) = u-ku^2-\frac{uv}{mu+v}, \ \; \; g(u, v) = -av-bv^2+\frac{uv}{mu+v}. $

    Theorem 3.1. Let $ d_{2}^* $ be a fixed positive constant and satisfy $ \mu_{1}d_{2}^* > \frac{1}{m} $, where $ \mu_1 $ is the least positive eigenvalue of $ -\Delta $ on $ \Omega $ under the Neumann boundary condition. Then there exists a positive constant $ D_{1} = D_{1}(\Lambda, d_{2}^*) $ such that when $ d_{1}\geq D_{1} $ and $ d_2\geq d_2^* $, system (1.5) with $ d_{12} = d_{21} = 0 $ has no non-constant positive solution.

    Proof. Suppose that $ \textbf{w} = (u, v)^\mathit{T} $ is a positive solution of system (1.5). Let $ \tilde{\varphi} = \frac{1}{|\Omega|}\int_{\Omega}\varphi dx $ for any $ \varphi\in L^1(\Omega) $. Multiplying the first two equations of (1.5) by $ u-\tilde{u} $ and $ v-\tilde{v} $, respectively, and then integrating on $ \Omega $, by integration by parts we obtain

    $ Ωd1|u|2dx+Ωd2|v|2dx=Ωf(u,v)(u˜u)dx+Ωg(u,v)(v˜v)dx=Ω[f(u,v)f(˜u,˜v)](u˜u)dx+Ω[g(u,v)g(˜u,˜v)](v˜v)dx=Ω{1k(u+˜u)v˜v(mu+v)(m˜u+˜v)}(u˜u)2+Ω{mu˜u(mu+v)(m˜u+˜v)+v˜v(mu+v)(m˜u+˜v)}(u˜u)(v˜v)dx+Ω{ab(v+˜v)+mu˜u(mu+v)(m˜u+˜v)}(v˜v)2dxΩ{(u˜u)2+mu˜u+v˜v(mu+v)(m˜u+˜v)|u˜u||v˜v|+1m(v˜v)2}dx. $ (3.1)

    In view of Theorems 2.1 and 2.2, using Young's inequality to (3.1) we find

    $ Ωd1|u|2dx+Ωd2|v|2dxΩ{(u˜u)2+2L|u˜u||v˜v|+1m(v˜v)2}dxΩ{(1+Lε)(u˜u)2+(1m+εL)(v˜v)2}dx $ (3.2)

    for some positive constant $ L $, where $ \varepsilon $ is the arbitrary small positive constant arising from Young's inequality.

    Using Poincaré's inequality, we see that $ \mu_1\int_{\Omega}(u-\tilde{u})^2dx\leq \int_{\Omega}|\nabla u|^2dx $ and $ \mu_1\int_{\Omega}(v-\tilde{v})^2dx\leq \int_{\Omega}|\nabla v|^2dx $, where $ \mu_1 $ is the least positive eigenvalue of $ -\Delta $ on $ \Omega $ under the Neumann boundary condition. It follows from inequality (3.2) that

    $ μ1Ω[d1(u˜u)2+d2(v˜v)2]dxΩ{(1+Lε)(u˜u)2+(1m+εL)(v˜v)2}dx. $

    Choosing a sufficiently small $ \varepsilon_0 > 0 $ such that $ d_{2}^*\mu_{1}\geq \frac 1m+\varepsilon_0L $, and taking $ D_{1}\triangleq \frac 1{\mu_1}(1+\frac L{\varepsilon_0}) $, we arrive at the desired result $ (u, v) = (\tilde{u}, \tilde{v}) $.

    From the discussion in the preceding subsection, we know that when the cross-diffusion terms are absent, there might be no non-constant positive solutions for system (1.5). In this subsection, we shall discuss the existence of non-constant positive solutions of system (1.5) with respect to cross-diffusion coefficients $ d_{21} $ and $ d_{12} $ as the other parameters $ d_1 $ and $ d_2 $ are fixed by means of the Leray-Schauder degree theory.

    To facilitate the discussion, we rewrite system (1.5) as

    $ \begin{eqnarray} \left\{ \begin{array}{l} -\Delta \mathit{\boldsymbol{\Phi}}(\textbf{w}) = \textbf{F}(\textbf{w}), \\ \frac{\partial \textbf{w}}{\partial \nu} = \textbf{0}, \end{array}\; \; \begin{array}{l} x\in \Omega, \\[1.5mm] x\in \partial\Omega, \end{array} \right. \end{eqnarray} $ (3.3)

    where $ \textbf{w} = (u, v)^\mathit{T} $, $ \mathit{\boldsymbol{\Phi}}(\textbf{w}) = (\phi, \psi)^\mathit{T} $, and $ \textbf{F}(\textbf{w}) = (f(u, v), \, g(u, v))^\mathit{T} $.

    Let $ \textbf{X} = \left\{\textbf{w}\in \left[C^1(\overline{\Omega})\right]^2 \Big|\frac{\partial\textbf{w}}{\partial\nu} = \textbf{0} \ \text{on}\ \partial\Omega \right\} $, and define

    $ \begin{eqnarray*} &&\textbf{X}^+ = \{\textbf{w}\in \textbf{X} |\textbf{w} \gt 0 \; \; on\; \; \overline{\Omega}\}, \\ &&\mathcal{B}(c) = \{\textbf{w}\in \textbf{X}|c^{-1} \lt u, v \lt c \; \; on\; \; \overline{\Omega}\}, \end{eqnarray*} $

    where $ c $ is a positive constant that is guaranteed to exist by Theorems 2.1 and 2.2.

    Assume that

    $ \begin{eqnarray} d_1d_2-d_{12}d_{21}\neq0. \end{eqnarray} $ (3.4)

    Since $ \mathit{\boldsymbol{\Phi}}_{\textbf{w}}(\textbf{w}) = \left(\begin{matrix} d_{1} & d_{12}\\ d_{21} & d_{2} \end{matrix}\right) $, the determinant $ \det \mathit{\boldsymbol{\Phi}}_{\textbf{w}}(\textbf{w}) $ is nonzero for all non-negative $ \textbf{w} $, $ \mathit{\boldsymbol{\Phi}}_{\textbf{w}}^{-1}(\textbf{w}) $ exists and $ \det\mathit{\boldsymbol{\Phi}}_{\textbf{w}}^{-1}(\textbf{w}) $ is of the same sign as $ \det\mathit{\boldsymbol{\Phi}}_{\textbf{w}}(\textbf{w}) $. Thus, $ \textbf{w} $ is a positive solution of system (3.3) if and only if

    $ \begin{eqnarray} \textbf{G}(\textbf{w})\triangleq \textbf{w}-(\textbf{I}-\Delta)^{-1} \big\{\mathit{\boldsymbol{\Phi}}_{\textbf{w}}^{-1}(\textbf{w})[\textbf{F}(\textbf{w}) +\nabla\textbf{w}\mathit{\boldsymbol{\Phi}}_{\textbf{ww}}(\textbf{w})\nabla\textbf{w}] +\textbf{w}\big\} = 0, \; \; \textbf{w}\in\textbf{X}^+, \end{eqnarray} $ (3.5)

    where $ (\textbf{I}-\Delta)^{-1} $ is the inverse of $ \textbf{I}-\Delta $ in $ \textbf{X} $ under the homogeneous Neumann boundary condition. Since $ \textbf{G}(\cdot) $ is a compact perturbation of the identity operator, for any $ \mathcal{B} = \mathcal{B}(c) $, the Leray-Schauder degree of $ \deg(\textbf{G}(\cdot), \textbf{0}, \mathcal{B}) $ is well-defined if $ \textbf{G}(\textbf{w})\neq 0 $ on $ \partial\mathcal{B} $.

    Note that

    $ \begin{eqnarray*} D_{\textbf{w}}\textbf{G}(\widetilde{\textbf{w}}) = \textbf{I}-(\textbf{I}-\Delta)^{-1} \big\{\mathit{\boldsymbol{\Phi}}_{\textbf{w}}^{-1}(\widetilde{\textbf{w}})\textbf{F}_{\textbf{w}}(\widetilde{\textbf{w}}) +\textbf{I}\big\}, \end{eqnarray*} $

    where

    $ \mathit{\boldsymbol{\Phi}}_{\textbf{w}}(\widetilde{\textbf{w}}) = \left(\begin{matrix} d_{1} & d_{12}\\ d_{21} & d_{2} \end{matrix}\right), \; \; \ \textbf{F}_{\textbf{w}}(\widetilde{\textbf{w}}) = \left(\begin{matrix} -\frac{m^2\tilde{u}^2}{(m\tilde{u}+\tilde{v})^2} & -\frac{m\tilde{u}^2}{(m\tilde{u}+\tilde{v})^2}\\ \frac{\tilde{v}^2}{(m\tilde{u}+\tilde{v})^2} & -\frac{\tilde{u}\tilde{v}}{(m\tilde{u}+\tilde{v})^2}-b\tilde{v} \end{matrix}\right). $

    We recall that if $ D_{\textbf{w}}\textbf{G}(\widetilde{\textbf{w}}) $ is invertible, the index of $ \textbf{G} $ at $ \widetilde{\textbf{w}} $ is defined by

    $ {\rm index}(\textbf{G}(\cdot), \widetilde{\textbf{w}}) = (-1)^{\gamma}, $

    where $ \gamma $ is the total number of eigenvalues of $ D_{\textbf{w}}\textbf{G}(\widetilde{\textbf{w}}) $ with negative real parts (counting multiplicities), then the degree $ \deg(\textbf{G}(\cdot), \textbf{0}, \mathcal{B}) $ is equal to the sum of the indices over all isolated solutions when $ \textbf{G} = \textbf{0} $ in $ \mathcal{B}(c) $, provided that $ \textbf{G}\neq \textbf{0} $ on $ \partial\mathcal{B} $.

    Let $ 0 = \mu_{0} < \mu_{1} < \mu_{2} < \cdots $ be the eigenvalues of the operator $ -\Delta $ on $ \Omega $ under the homogeneous Neumann boundary condition and $ E(\mu_{i}) $ be the eigenspaces with respect to $ \mu_{i} $. Let $ \{\phi_{ij}; j = 1, 2, \cdots, \dim E(\mu_{i})\} $ be a set of orthonormal basis of $ E(\mu_{i}) $ and $ \textbf{X}_{ij} = \{\textbf{c}\phi_{ij}|\textbf{c}\in \mathbb{R}^2\} $.

    Denote

    $ \textbf{X} = \left\{\textbf{w}\in \left[C^1(\overline{\Omega})\right]^2 \Big|\frac{\partial\textbf{w}}{\partial\nu} = \textbf{0} \ \text{on}\ \partial\Omega \right\} \ \, \text{and}\ \, \textbf{X}_{ij} = \{\textbf{c}\phi_{ij}\ |\, \textbf{c}\in \mathbb{R}^2\}. $

    Then

    $ \begin{eqnarray*} \textbf{X} = \bigoplus\limits_{i = 1}^{\infty}\textbf{X}_{i} \; \; \; and\; \; \; \textbf{X}_{i} = \bigoplus \limits_{j = 1}^{\dim E(\mu_{i})}\textbf{X}_{ij}. \end{eqnarray*} $

    We refer to the decomposition above in the following discussions of the eigenvalues of $ D_{\textbf{w}}\textbf{G}(\widetilde{\textbf{w}}) $. We know that $ \textbf{X}_{ij} $ is invariant under $ D_{\textbf{w}}\textbf{G}(\widetilde{\textbf{w}}) $ for each $ i\in \mathbb{N} $ and each $ j\in [1, \dim E(\mu_{i})]\cap\mathbb{N} $, i.e., $ D_{\textbf{w}}\textbf{G}(\widetilde{\textbf{w}})\textbf{w}\in \textbf{X}_{ij} $ for any $ \textbf{w}\in \textbf{X}_{ij} $. Thus, $ \lambda $ is an eigenvalue of $ D_{\textbf{w}}\textbf{G}(\widetilde{\textbf{w}}) $ on $ \textbf{X}_{ij} $ if and only if it is an eigenvalue of the matrix

    $ \begin{eqnarray*} \textbf{I}-\frac{1}{1+\mu_{i}}\left[\mathit{\boldsymbol{\Phi}}_{\textbf{w}}^{-1}(\widetilde{\textbf{w}}) \textbf{F}_{\textbf{w}}(\widetilde{\textbf{w}})+\textbf{I}\right] = \frac{1}{1+\mu_{i}}\left[\mu_{i}\textbf{I}-\mathit{\boldsymbol{\Phi}}_{\textbf{w}}^{-1}(\widetilde{\textbf{w}}) \textbf{F}_{\textbf{w}}(\widetilde{\textbf{w}})\right]. \end{eqnarray*} $

    Hence, $ D_{\textbf{w}}\textbf{G}(\widetilde{\textbf{w}}) $ is invertible if and only if the matrix

    $ \begin{eqnarray*} \textbf{I}-\frac{1}{1+\mu_{i}}\left[\mathit{\boldsymbol{\Phi}}_{\textbf{w}}^{-1}(\widetilde{\textbf{w}}) \textbf{F}_{\textbf{w}}(\widetilde{\textbf{w}})+\textbf{I}\right] \end{eqnarray*} $

    is non-singular for any $ i\geq 0 $.

    Denote

    $ \begin{eqnarray} H(\mu) = H(\widetilde{\textbf{w}}; \mu)\triangleq \det \left\{\mu\textbf{I}-\mathit{\boldsymbol{\Phi}}_{\textbf{w}}^{-1}(\widetilde{\textbf{w}}) \textbf{F}_{\textbf{w}}(\widetilde{\textbf{w}})\right\}. \end{eqnarray} $ (3.6)

    We observe that if $ H(\mu_{i})\neq 0 $, then for each $ 1\leq j\leq \dim E(\mu_{i}) $, the number of negative eigenvalues of $ D_{\textbf{w}}\textbf{G}(\widetilde{\textbf{w}}) $ on $ \textbf{X}_{ij} $ is odd if and only if $ H(\mu_{i}) < 0 $.

    We summarize our result as follows.

    Theorem 3.2. Assume that the matrix $ \mu_{i}\bf{I}-\mathit{\boldsymbol{\Phi}}_{\bf{w}}^{-1}(\widetilde{\bf{w}}) \bf{F}_{\bf{w}}(\widetilde{\bf{w}}) $ is non-singular for an arbitrary $ i\geq 0 $. Then there holds

    $ \begin{eqnarray*} {\rm index}(\bf{G}(\cdot), \widetilde{\bf{w}}) = (-1)^{\sigma}, \end{eqnarray*} $

    where $ \sigma = \sum \limits_{i\geq 0, \, H(\mu_{i}) < 0}\dim E(\mu_{i}) $.

    According to the above Theorem, we need to consider the sign of $ H(\mu_{i}) $ in order to calculate the index of $ (\textbf{G}(\cdot), \widetilde{\textbf{w}}) $. In addition, by (3.6) we have

    $ H(\mu) = \det \left\{\mathit{\boldsymbol{\Phi}}_{\textbf{w}}^{-1}(\widetilde{\textbf{w}})\right\} \det \left\{\mu\mathit{\boldsymbol{\Phi}}_{\textbf{w}}(\widetilde{\textbf{w}})-\textbf{F}_{\textbf{w}}(\widetilde{\textbf{w}})\right\}. $

    Hence, we need to consider the signs of $ \det\big\{\mathit{\boldsymbol{\Phi}}_{\textbf{w}}^{-1}(\widetilde{\textbf{w}})\big\} $ and $ \det \big\{\mu\mathit{\boldsymbol{\Phi}}_{\textbf{w}}(\widetilde{\textbf{w}})-\textbf{F}_{\textbf{w}}(\widetilde{\textbf{w}})\big\}, $ respectively.

    By a direct calculation we get

    $ \begin{equation} \begin{split} \det\left\{\mathit{\boldsymbol{\Phi}}_{\textbf{w}}^{-1}(\widetilde{\textbf{w}})\right\} & = \frac 1 A, \\ \det \left\{\mu\mathit{\boldsymbol{\Phi}}_{\textbf{w}}(\widetilde{\textbf{w}})-\textbf{F}_{\textbf{w}}(\widetilde{\textbf{w}})\right\} & = A\mu^2-B\mu+C\triangleq\mathcal{A}(\mu), \end{split} \end{equation} $ (3.7)

    where

    $ \begin{eqnarray*} &&A = d_{1}d_{2}-d_{12}d_{21}, \\ &&B = -\frac{m^2\tilde{u}^2}{(m\tilde{u}+\tilde{v})^2}d_2-\left(\frac{\tilde{u}\tilde{v}}{(m\tilde{u}+\tilde{v})^2}+b\tilde{v}\right)d_1 -\frac{\tilde{v}^2}{(m\tilde{u}+\tilde{v})^2}d_{12}+\frac{m\tilde{u}^2}{(m\tilde{u}+\tilde{v})^2}d_{21}, \\ &&C = \det\big\{\textbf{F}_{\textbf{w}}(\widetilde{\textbf{w}})\big\} = \frac{ma\tilde{u}\tilde{v}}{(m\tilde{u}+\tilde{v})^2}+kb\tilde{u}\tilde{v} \gt 0. \end{eqnarray*} $

    Let $ \tilde{\mu}_{1} $ and $ \tilde{\mu}_{2} $ be the two roots of $ \mathcal{A}(\mu) = 0 $ with $ Re\{\tilde{\mu}_{1}\}\leq Re\{\tilde{\mu}_{2}\} $. Then $ \tilde{\mu}_{1}\tilde{\mu}_{2} = \frac C A $, which is of the same sign as $ A $.

    Next, we discuss the dependence of $ \mathcal{A}(\mu) $ on $ d_{12} $ and $ d_{21} $, respectively. Due to the following limits:

    $ \begin{eqnarray*} &&\lim\limits_{d_{12}\rightarrow\infty}\frac{A}{d_{12}} = -d_{21} \lt 0, \ \; \; \lim\limits_{d_{21}\rightarrow\infty}\frac{A}{d_{21}} = -d_{12} \lt 0, \\ &&\lim\limits_{d_{12}\rightarrow\infty}\frac{B}{d_{12}} = -\frac{\tilde{v}^2}{(m\tilde{u}+\tilde{v})^2} \lt 0, \; \; \ \lim\limits_{d_{21}\rightarrow\infty}\frac{B}{d_{21}} = \frac{m\tilde{u}^2}{(m\tilde{u}+\tilde{v})^2} \gt 0, \end{eqnarray*} $

    it follows from (3.7) that

    $ \begin{eqnarray} \begin{split} & \lim\limits_{d_{12}\rightarrow\infty}\frac{\mathcal{A}(\mu)}{d_{12}} = \mu\left[-d_{21}\mu+ \frac{\tilde{v}^2}{(m\tilde{u}+\tilde{v})^2}\right], \\ & \lim\limits_{d_{21}\rightarrow\infty}\frac{\mathcal{A}(\mu)}{d_{21}} = \mu\left[-d_{12}\mu- \frac{m\tilde{u}^2}{(m\tilde{u}+\tilde{v})^2}\right]. \end{split} \end{eqnarray} $ (3.8)

    Note that the above two limits hold only when $ d_{12} $ (or $ d_{21} $) is chosen to be large enough. This is certainly possible if $ A = d_1d_2-d_{12}d_{21} < 0 $ for the fixed $ d_1 $ and $ d_2 $. In this case, we have

    $ \det\big\{\mathit{\boldsymbol{\Phi}}_{\textbf{w}}^{-1}(\widetilde{\textbf{w}})\big\} = \frac 1 A \lt 0\ \, \text{and}\ \, \tilde{\mu}_{1}\tilde{\mu}_{2} = \frac C A \lt 0. $

    Based on the above analyses, in summary we obtain the following results.

    Theorem 3.3. Assume $ A < 0 $. Then there exists a positive constant $ d_{12}^* $ such that when $ d_{12}\geq d_{12}^* $, both of the two roots $ \tilde{\mu}_{1}(d_{12}) $ and $ \tilde{\mu}_{2}(d_{12}) $ of $ \mathcal{A}(\mu) = 0 $ are real and satisfy

    $ \begin{equation} \begin{split} & \lim\limits_{d_{12}\rightarrow\infty}\tilde{\mu}_1(d_{12}) = 0, \\ & \lim\limits_{d_{12}\rightarrow\infty}\tilde{\mu}_2(d_{12}) = \frac{\tilde{v}^2}{d_{21}(m\tilde{u}+\tilde{v})^2}\triangleq \bar{\mu} \gt 0, \end{split} \end{equation} $ (3.9)

    where

    $ \begin{equation} \left\{ \begin{aligned} &\tilde{\mu}_{1}(d_{12}) \lt 0 \lt \tilde{\mu}_{2}(d_{12}), \\ &\mathcal{A}(\mu; d_{12}) \lt 0 \ \mathit{\text{when}}\ \mu\in(-\infty, \tilde{\mu}_{1}(d_{12}))\cup(\tilde{\mu}_{2}(d_{12}), \infty), \\ &\mathcal{A}(\mu; d_{12}) \gt 0 \ \mathit{\text{when}}\ \mu\in(\tilde{\mu}_{1}(d_{12}), \tilde{\mu}_{2}(d_{12})). \end{aligned} \right. \end{equation} $ (3.10)

    Theorem 3.4. Assume $ A < 0 $. Then there exists a positive constant $ d_{21}^* $ such that when $ d_{21}\geq d_{21}^* $, both of the two roots $ \tilde{\mu}_{1}(d_{21}) $ and $ \tilde{\mu}_{2}(d_{21}) $ of $ \mathcal{A}(\mu) = 0 $ are real and satisfy

    $ \begin{equation} \begin{split} & \lim\limits_{d_{21}\rightarrow\infty}\tilde{\mu}_1(d_{21}) = -\frac{m\tilde{u}^2}{d_{12}(m\tilde{u}+\tilde{v})^2}\triangleq \bar{\bar{\mu}} \lt 0, \\ &\lim\limits_{d_{21}\rightarrow\infty}\tilde{\mu}_2(d_{21}) = 0, \end{split} \end{equation} $ (3.11)

    where

    $ \begin{equation} \left\{ \begin{aligned} &\tilde{\mu}_{1}(d_{21}) \lt 0 \lt \tilde{\mu}_{2}(d_{21}), \\ &\mathcal{A}(\mu; d_{21}) \lt 0 \; \; when\; \; \mu\in(-\infty, \tilde{\mu}_{1}(d_{21}))\cup(\tilde{\mu}_{2}(d_{21}), \infty), \\ &\mathcal{A}(\mu; d_{21}) \gt 0 \; \; when\; \; \mu\in(\tilde{\mu}_{1}(d_{21}), \tilde{\mu}_{2}(d_{21})). \end{aligned} \right. \end{equation} $ (3.12)

    The following theorem is regarding the existence of non-constant positive solutions to system (3.3) with respect to the cross-diffusion coefficient $ d_{12} $, while all of other parameters are fixed.

    Theorem 3.5. Suppose that $ A < 0 $, and the parameters $ \Lambda $, $ d_{1} $, $ d_{2} $ and $ d_{21} $ are fixed. Let $ \bar{\mu} $ be given by (3.9). If $ \bar{\mu}\in (\mu_{n}, \mu_{n+1}) $ for some $ n\geq 1 $, and the sum $ \sigma_{n} = \sum\limits_{i = 1}^{n}\dim E(\mu_{i}) $ is odd, then there exists a positive constant $ d_{12}^* $ such that when $ d_{12}\geq d_{12}^* $, system (3.3) has at least one non-constant positive solution.

    Proof. According to Theorem 3.3, there exists a positive constant $ d_{12}^* $ such that, if $ d_{12}\geq d_{12}^* $, then (3.10) holds and

    $ \begin{eqnarray} \tilde{\mu}_{1}(d_{12}) \lt 0 = \mu_{0} \lt \tilde{\mu}_{2}(d_{12}), \; \; \ \tilde{\mu}_{2}(d_{12})\in(\mu_{n}, \mu_{n+1}). \end{eqnarray} $ (3.13)

    It suffices to prove that for all $ d_{12}\geq d_{12}^* $, system (3.3) has at least one non-constant positive solution. By the way of contradiction, we assume that this is not true for some $ d_{12}\ (\geq d_{12}^*) $. By applying the homotopy invariance of the topological degree, we can see a contradiction explicitly.

    For any fixed $ d_{12} $ that satisfies $ d_{12}\geq d_{12}^* $, we take $ \hat{d}_2\geq d_{2}^* $ and $ \hat{d}_1\geq D_1 $, where $ \mu_{1}d_{2}^* > \frac{1}{m} $ and $ D_{1} = D_{1}(\Lambda, d_{2}^*) $ are given by Theorem 3.1. For $ t\in [0, 1] $, we define

    $ \mathit{\boldsymbol{\Phi}}(t; \textbf{w}) = \left(\begin{matrix} [td_{1}+(1-t)\hat{d}_1]u+td_{12}v\\ td_{21}u+[td_{2}+(1-t)\hat{d}_2]v \end{matrix}\right). $

    Consider the system

    $ \begin{eqnarray} \left\{ \begin{array}{l} -\Delta \mathit{\boldsymbol{\Phi}}(t; \textbf{w}) = \textbf{F}(\textbf{w}), \\ \frac{\partial \textbf{w}}{\partial \nu} = \textbf{0}, \end{array}\; \; \begin{array}{l} x\in \Omega, \\[1.5mm] x\in \partial\Omega. \end{array} \right. \end{eqnarray} $ (3.14)

    Then $ \textbf{w} $ is a non-constant positive solution of system (3.3) if and only if it is a positive solution of system (3.14) when $ t = 1 $. It is obvious that $ \widetilde{\textbf{w}} $ is the unique positive constant solution of system (3.14) for any $ t\in[0, 1] $. From (3.5), we know that for any $ t\in[0, 1] $, $ \textbf{w} $ is a positive solution of system (3.3) if and only if

    $ \begin{eqnarray*} \textbf{G}(t; \textbf{w}) &\triangleq& \textbf{w}-(\textbf{I}-\Delta)^{-1} \big\{\mathit{\boldsymbol{\Phi}}_{\textbf{w}}^{-1}(t; \textbf{w})[\textbf{F}(\textbf{w}) +\nabla\textbf{w}\mathit{\boldsymbol{\Phi}}_{\textbf{ww}}(t; \textbf{w})\nabla\textbf{w}] +\textbf{w}\big\} \\ & = &0, \ \text{for}\ \textbf{w}\in\textbf{X}^+. \end{eqnarray*} $

    Analogous to (3.6), we set

    $ H(t, \mu) = \det \left\{\mathit{\boldsymbol{\Phi}}_{\textbf{w}}^{-1}(t, \widetilde{\textbf{w}})\right\} \det \big\{\mu\mathit{\boldsymbol{\Phi}}_{\textbf{w}}(t, \widetilde{\textbf{w}})-\textbf{F}_{\textbf{w}}(\widetilde{\textbf{w}})\big\}. $

    It is easy to see that $ \textbf{G}(1; \textbf{w}) = \textbf{G}(\textbf{w}) $ and

    $ \begin{eqnarray*} D_{\textbf{w}}\textbf{G}(t; \widetilde{\textbf{w}}) = \textbf{I}-(\textbf{I}-\Delta)^{-1} \big\{\mathit{\boldsymbol{\Phi}}_{\textbf{w}}^{-1}(t; \widetilde{\textbf{w}})\textbf{F}_{\textbf{w}}(\widetilde{\textbf{w}}) +\textbf{I}\big\}. \end{eqnarray*} $

    In particular, we obtain

    $ \begin{eqnarray*} &&D_{\textbf{w}}\textbf{G}(0; \widetilde{\textbf{w}}) = \textbf{I}-(\textbf{I}-\Delta)^{-1} \big\{\hat{\mathit{\boldsymbol{\Phi}}}_{\textbf{w}}^{-1}(\widetilde{\textbf{w}})\textbf{F}_{\textbf{w}}(\widetilde{\textbf{w}}) +\textbf{I}\big\}, \\ &&D_{\textbf{w}}\textbf{G}(1; \widetilde{\textbf{w}}) = \textbf{I}-(\textbf{I}-\Delta)^{-1} \big\{\mathit{\boldsymbol{\Phi}}_{\textbf{w}}^{-1}(\widetilde{\textbf{w}})\textbf{F}_{\textbf{w}}(\widetilde{\textbf{w}}) +\textbf{I}\big\} = D_{\textbf{w}}\textbf{G}(\widetilde{\textbf{w}}), \end{eqnarray*} $

    where

    $ \hat{\mathit{\boldsymbol{\Phi}}}_{\textbf{w}}(\widetilde{\textbf{w}}) = \left(\begin{matrix} \hat{d}_{1} & 0\\ 0 & \hat{d}_{2} \end{matrix}\right)\ \, \text{and}\ \, \mathit{\boldsymbol{\Phi}}_{\textbf{w}}(\widetilde{\textbf{w}}) = \left(\begin{matrix} d_{1} & d_{12}\\ d_{21} & d_{2} \end{matrix}\right). $

    More specifically, when $ t = 1 $, from (3.6) and (3.7) we have

    $ \begin{eqnarray} H(1, \mu) & = & H(\mu) \\ & = & \det \big\{\mathit{\boldsymbol{\Phi}}_{\textbf{w}}^{-1}(\widetilde{\textbf{w}})\big\} \det \big\{\mu\mathit{\boldsymbol{\Phi}}_{\textbf{w}}(\widetilde{\textbf{w}})-\textbf{F}_{\textbf{w}}(\widetilde{\textbf{w}})\big\}\\ & = & \det \big\{\mathit{\boldsymbol{\Phi}}_{\textbf{w}}^{-1}(\widetilde{\textbf{w}})\big\}\mathcal{A}(\mu), \end{eqnarray} $ (3.15)

    where

    $ \det \big\{\mathit{\boldsymbol{\Phi}}_{\textbf{w}}^{-1}(\widetilde{\textbf{w}})\big\} = \left( {\det \big\{\mathit{\boldsymbol{\Phi}}_{\textbf{w}}(\widetilde{\textbf{w}})\big\}}\right)^{-1} = \frac 1A \lt 0, $

    and $ \mathcal{A}(\mu) $ is defined by (3.7).

    According to (3.10), (3.13) and (3.15), we deduce that

    $ \begin{equation*} \left\{ \begin{aligned} &H(1, \mu_{0}) = H(0) \lt 0, \\ &H(1, \mu_{i}) \lt 0 \; \; \; \; \; \; \; \; \; \; \; \; \; \text{when}\; \; 1\leq i\leq n, \\ &H(1, \mu_{i+1}) \gt 0 \; \; \; \; \; \; \; \; \; \; \text{when}\; \; i\geq n+1. \end{aligned} \right. \end{equation*} $

    Therefore, zero is not an eigenvalue of the matrix $ \mu_{i}\textbf{I}-\mathit{\boldsymbol{\Phi}}_{\textbf{w}}^{-1}(\widetilde{\textbf{w}}) \textbf{F}_{\textbf{w}}(\widetilde{\textbf{w}}) $ for any $ i\geq 0 $, and

    $ \sum\limits_{i\geq 0, H(1, \mu_{i}) \lt 0}\dim E(\mu_{i}) = \sum \limits_{i = 1}^{n}\dim E(\mu_{i}) = \sigma_{n} $

    is odd. It follows from Theorem 3.2 that

    $ \begin{equation} {\rm index}(\mathit{\boldsymbol{\Phi}}(1; \cdot), \widetilde{\textbf{w}}) = (-1)^{\gamma} = (-1)^{\sigma_{n}} = -1. \end{equation} $ (3.16)

    When $ t = 0 $, we have

    $ \begin{eqnarray} \begin{split} H(0, \mu)& = \det \big\{\hat{\mathit{\boldsymbol{\Phi}}}_{\textbf{w}}^{-1}(\widetilde{\textbf{w}})\big\} \det \big\{\mu\hat{\mathit{\boldsymbol{\Phi}}}_{\textbf{w}}(\widetilde{\textbf{w}})-\textbf{F}_{\textbf{w}}(\widetilde{\textbf{w}})\big\}\\ & = \det \big\{\hat{\mathit{\boldsymbol{\Phi}}}_{\textbf{w}}^{-1}(\widetilde{\textbf{w}})\big\}\mathcal{\hat{A}}(\mu), \end{split} \end{eqnarray} $ (3.17)

    where

    $ \det \big\{\hat{\mathit{\boldsymbol{\Phi}}}_{\textbf{w}}^{-1}(\widetilde{\textbf{w}})\big\} = \left(\hat{d}_1\hat{d}_2\right)^{-1} \gt 0, $

    and we use $ \mathcal{\hat{A}}(\mu) $ to represent $ \mathcal{A}(\mu) $ given in (3.7) under the restriction of $ d_{12} = d_{21} = 0 $. In this case, by Theorem 3.1 it implies that $ \textbf{G}(0; \textbf{w}) $ = 0 only has the positive constant solution $ \widetilde{\textbf{w}} $ in $ \textbf{X}^+ $. By a direct calculation we have

    $ \mathcal{\hat{A}}(\mu) = \hat{d}_1\hat{d}_2\mu^2+\left[\left(\frac{\tilde{u}\tilde{v}}{(m\tilde{u}+\tilde{v})^2} +b\tilde{v}\right)\hat{d}_1+ \frac{m^2\tilde{u}^2}{(m\tilde{u}+\tilde{v})^2}\hat{d}_2\right]\mu+\frac{ma\tilde{u}\tilde{v}}{(m\tilde{u}+\tilde{v})^2}+kb\tilde{u}\tilde{v}, $

    and so $ H(0, \mu_{i}) > 0 $ for all $ i\geq 0 $.

    Discussing in the same manner, we can prove that

    $ \begin{equation} {\rm index}(\mathit{\boldsymbol{\Phi}}(0; \cdot), \widetilde{\textbf{w}}) = (-1)^{0} = 1. \end{equation} $ (3.18)

    According to Theorems 2.1 and 2.2, there exists a positive constant $ c $ such that the positive solution of system (3.14) satisfies $ c^{-1} < u $ and $ v < c $ for any $ t\in[0, 1] $. Thus for any $ t\in [0, 1] $, there holds $ \mathit{\boldsymbol{\Phi}}(t; \textbf{w})\neq 0 $ on $ \partial\mathcal{B}(M) $. By the homotopy invariance of the topological degree, we have

    $ \begin{equation} {\rm deg}(\mathit{\boldsymbol{\Phi}}(1; \cdot), 0, \mathcal{B}(M)) = {\rm deg}(\mathit{\boldsymbol{\Phi}}(0; \cdot), 0, \mathcal{B}(M)). \end{equation} $ (3.19)

    Note that both $ \mathit{\boldsymbol{\Phi}}(1; \textbf{w}) = 0 $ and $ \mathit{\boldsymbol{\Phi}}(0; \textbf{w}) = 0 $ have the unique positive solution $ \widetilde{\textbf{w}} $ in $ \mathcal{B}(M) $. From (3.16) and (3.18) we get

    $ \begin{equation} \left\{ \begin{aligned} &{\rm deg}(\mathit{\boldsymbol{\Phi}}(0; \cdot), 0, \mathcal{B}(M)) = {\rm index}(\mathit{\boldsymbol{\Phi}}(0; \cdot), \widetilde{\textbf{w}}) = 1, \\ &{\rm deg}(\mathit{\boldsymbol{\Phi}}(1; \cdot), 0, \mathcal{B}(M)) = {\rm index}(\mathit{\boldsymbol{\Phi}}(1; \cdot), \widetilde{\textbf{w}}) = -1, \end{aligned} \right. \end{equation} $ (3.20)

    which yields a contradiction with (3.19).

    Processing in an analogous way as we just did in the proof of Theorem 3.5, we can obtain the following result regarding the existence of non-constant positive steady states for system (3.3) with respect to the cross-diffusion coefficient $ d_{21} $. So, we omit the proof.

    Theorem 3.6. Suppose that $ A < 0 $, and the parameters $ \Lambda $, $ d_{1} $, $ d_{2} $ and $ d_{12} $ are fixed. Let $ \bar{\bar{\mu}} $ be given by (3.11). If $ \bar{\bar{\mu}}\in (\mu_{n}, \mu_{n+1}) $ for some $ n\geq 1 $, and the sum $ \sigma_{n} = \sum\limits_{i = 1}^{n}\dim E(\mu_{i}) $ is odd, then there exists a positive constant $ d_{21}^* $ such that when $ d_{21}\geq d_{21}^* $, system (3.3) has at least one non-constant positive solution.

    This work is supported by NSF of China (No. 41575004) and UTRGV Faculty Research Council Award 1100127.

    The authors declare that there is no conflict of interests regarding the publication of this paper.

    [1] Lee AG (2004) How lipids affect the activities of integral membrane proteins. BBA-Biomembranes 1666: 62–87. doi: 10.1016/j.bbamem.2004.05.012
    [2] Pohorille A, Schweighofer K, Wilson MA (2006) The origin and early evolution of membrane channels. Astrobiology 5: 1–17. doi: 10.1017/S1473550406002886
    [3] Hille B (2001) Ion Channels of Excitable Membranes, 3 Eds., Sinauer.
    [4] Kaczorowski GJ, Mcmanus OB, Priest BT, et al. (2008) Ion channels as drug targets: The next GPCRs. J Gen Physiol 131: 399–405. doi: 10.1085/jgp.200709946
    [5] Ackerman MJ, Clapham DE (1997) Ion channels-basic science and clinical disease. N Engl J Med 336: 1575–1586. doi: 10.1056/NEJM199705293362207
    [6] Lear JD, Wasserman ZR, Degrado WF (1988) Synthetic amphiphilic peptide models for protein ion channels. Science 240: 1177–1181. doi: 10.1126/science.2453923
    [7] Kienker PK, Degrado WF, Lear JD (1994) A helical-dipole model describes the single-channel current rectification of an uncharged peptide ion channel. Proc Natl Acad Sci USA 91: 4859–4863. doi: 10.1073/pnas.91.11.4859
    [8] Petrache HI, Zuckerman DM, Sachs JN, et al. (2002) Hydrophobic matching mechanism investigated by molecular dynamics simulations. Langmuir 18: 1340–1351. doi: 10.1021/la011338p
    [9] Nguyen THT, Liu Z, Moore PB (2013) Molecular dynamics simulations of homo-oligomeric bundles embedded within a lipid bilayer. Biophys J 105: 1569–1580. doi: 10.1016/j.bpj.2013.07.053
    [10] Howard KP, Lear JD, Degrado WF (2002) Sequence determinants of the energetics of folding of a transmembrane four-helix-bundle protein. Proc Natl Acad Sci USA 99: 8568–8572. doi: 10.1073/pnas.132266099
    [11] Arseneault M, Dumont M, Otis F, et al. (2012) Characterization of channel-forming peptide nanostructures. Biophys Chem 162: 6–13. doi: 10.1016/j.bpc.2011.12.001
    [12] Fischer WB (2005) Viral Membrane Proteins: Structure, Function, and Drug Design, In: Protein Rev, Kluwer Academic/Plenum Publishers.
    [13] Wang J, Kim S, Kovacs F, et al. (2001) Structure of the transmembrane region of the M2 protein H+ channel. Protein Sci 10: 2241–2250.
    [14] Stouffer AL, Acharya R, Salom D, et al. (2008) Structural basis for the function and inhibition of an influenza virus proton channel. Nature 451: 596–599. doi: 10.1038/nature06528
    [15] Schnell JR, Chou JJ (2008) Structure and mechanism of the M2 proton channel of influenza A virus. Nature 451: 591–595. doi: 10.1038/nature06531
    [16] Kovacs FA, Cross TA (1997) Transmembrane four-helix bundle of influenza A M2 protein channel: Structural implications from helix tilt and orientation. Biophys J 73: 2511–2517. doi: 10.1016/S0006-3495(97)78279-1
    [17] Acharya R, Carnevale V, Fiorin G, et al. (2010) Structure and mechanism of proton transport through the transmembrane tetrameric M2 protein bundle of the influenza A virus. Proc Natl Acad Sci USA 107: 15075–15080. doi: 10.1073/pnas.1007071107
    [18] Moore PB, Zhong Q, Husslein T, et al. (1998) Simulation of the HIV-1 Vpu transmembrane domain as a pentameric bundle. FEBS Lett 431: 143–148. doi: 10.1016/S0014-5793(98)00714-5
    [19] Woolley GA, Wallace BA (1992) Model ion channels: Gramicidin and alamethicin. J Membrane Biol 129: 109–136.
    [20] Opella SJ, Marassi FM, Gesell JJ, et al. (1999) Structures of the M2 channel-lining segments from nicotinic acetylcholine and NMDA receptors by NMR spectroscopy. Nat Struct Mol Biol 6: 374–379. doi: 10.1038/7610
    [21] Akerfeldt KS, Kienker PK, Lear JD (1996) Structure and conduction mechanisms of minimalist ion channels. Compr Supramol Chem 10: 659–686.
    [22] Gratkowski H, Lear JD, Degrado WF (2001) Polar side chains drive the association of model transmembrane peptides. Proc Natl Acad Sci USA 98: 880–885. doi: 10.1073/pnas.98.3.880
    [23] Randa HS, Forrest LR, Voth GA, et al. (1999) Molecular dynamics of synthetic leucine-serine ion channels in a phospholipid membrane. Biophys J 77: 2400–2410. doi: 10.1016/S0006-3495(99)77077-3
    [24] Oiki S, Danho W, Madison V, et al. (1988) M2 δ, a candidate for the structure lining the ionic channel of the nicotinic cholinergic receptor. Proc Natl Acad Sci USA 85: 8703–8707. doi: 10.1073/pnas.85.22.8703
    [25] Carruthers A, Melchior DL (1986) How bilayer lipids affect membrane protein activity. Trends Biochem Sci 11: 331–335. doi: 10.1016/0968-0004(86)90292-6
    [26] Palsdottir H, Hunte C (2004) Lipids in membrane protein structures. BBA-Biomembranes 1666: 2–18. doi: 10.1016/j.bbamem.2004.06.012
    [27] Lindahl E, Sansom MSP (2008) Membrane proteins: Molecular dynamics simulations. Curr Opin Struct Biol 18: 425–431. doi: 10.1016/j.sbi.2008.02.003
    [28] Phillips JC, Braun R, Wang W, et al. (2005) Scalable molecular dynamics with NAMD. J Comput Chem 26: 1781–1802. doi: 10.1002/jcc.20289
    [29] Duan Y, Wu C, Chowdhury S, et al. (2003) A point-charge force field for molecular mechanics simulations of proteins based on condensed-phase quantum mechanical calculations. J Comput Chem 24: 1999–2012. doi: 10.1002/jcc.10349
    [30] Hu Z, Jiang J (2010) Assessment of biomolecular force fields for molecular dynamics simulations in a protein crystal. J Comput Chem 31: 371–380.
    [31] Darden T, York D, Pedersen L (1993) Particle mesh Ewald: An N.log(N) method for Ewald sums in large systems. J Chem Phys 98: 10089–10092.
    [32] Feller SE, Yin D, Pastor RW, et al. (1997) Molecular dynamics simulation of unsaturated lipid bilayers at low hydration: Parameterization and comparison with diffraction studies. Biophys J 73: 2269–2279. doi: 10.1016/S0006-3495(97)78259-6
    [33] Jojart B, Martinek TA (2007) Performance of the general amber force field in modeling aqueous POPC membrane bilayers. J Comput Chem 28: 2051–2058. doi: 10.1002/jcc.20748
    [34] Taylor J, Whiteford NE, Bradley G, et al. (2009) Validation of all-atom phosphatidylcholine lipid force fields in the tensionless NPT ensemble. BBA-Biomembranes 1788: 638–649. doi: 10.1016/j.bbamem.2008.10.013
    [35] Rosso L, Gould IR (2007) Structure and dynamics of phospholipid bilayers using recently developed general all-atom force fields. J Comput Chem 29: 24–37.
    [36] Kučerka N, Tristram-Nagle S, Nagle JF (2006) Structure of fully hydrated fluid phase lipid bilayers with monounsaturated chains. J Membrane Biol 208: 193–202. doi: 10.1007/s00232-005-7006-8
    [37] Nielsen SO, Ensing B, Ortiz V, et al. (2005) Lipid bilayer perturbations around a transmembrane nanotube: A coarse grain molecular dynamics study. Biophys J 88: 3822–3828. doi: 10.1529/biophysj.104.057703
    [38] De Planque MR, Killian JA (2003) Protein-lipid interactions studied with designed transmembrane peptides: Role of hydrophobic matching and interfacial anchoring (Review). Mol Membr Biol 20: 271–284. doi: 10.1080/09687680310001605352
    [39] Nyholm TKM, Oezdirekcan S, Killian JA (2007) How protein transmembrane segments sense the lipid environment. Biochemistry 46: 1457–1465. doi: 10.1021/bi061941c
    [40] Sonne J, Jensen MO, Hansen FY, et al. (2007) Reparameterization of all-atom dipalmitoylphosphatidylcholine lipid parameters enables simulation of fluid bilayers at zero tension. Biophys J 92: 4157–4167. doi: 10.1529/biophysj.106.087130
    [41] Venturoli M, Smit B, Sperotto MM (2005) Simulation studies of protein-induced bilayer deformations, and lipid-induced protein tilting, on a mesoscopic model for lipid bilayers with embedded proteins. Biophys J 88: 1778–1798. doi: 10.1529/biophysj.104.050849
    [42] Chung LA, Lear JD, Degrado WF (1992) Fluorescence studies of the secondary structure and orientation of a model ion channel peptide in phospholipid vesicles. Biochemistry 31: 6608–6616. doi: 10.1021/bi00143a035
    [43] Choma C, Gratkowski H, Lear JD, et al. (2000) Asparagine-mediated self-association of a model transmembrane helix. Nat Struct Mol Biol 7: 161–166. doi: 10.1038/72440
    [44] Douliez JP, Leonard A, Dufourc EJ (1995) Restatement of order parameters in biomembranes: Calculation of C-C bond order parameters from C-D quadrupolar splittings. Biophys J 68: 1727–1739. doi: 10.1016/S0006-3495(95)80350-4
    [45] Douliez JP, Ferrarini A, Dufourc EJ (1998) On the relationship between C-C and C-D order parameters and its use for studying the conformation of lipid acyl chains in biomembranes. J Chem Phys 109: 2513–2518. doi: 10.1063/1.476823
    [46] Seelig J, Niederberger W (1974) Two pictures of a lipid bilayer. A comparison between deuterium label and spin-label experiments. Biochemistry 13: 1585–1588.
    [47] Seelig J, Waespesarcevic N (1978) Molecular order in cis and trans unsaturated phospholipid bilayers. Biochemistry 17: 3310–3315. doi: 10.1021/bi00609a021
    [48] Smart OS, Breed J, Smith GR, et al. (1997) A novel method for structure-based prediction of ion channel conductance properties. Biophys J 72: 1109–1126. doi: 10.1016/S0006-3495(97)78760-5
    [49] Smart OS, Neduvelil JG, Wang X, et al. (1996) HOLE: A program for the analysis of the pore dimensions of ion channel structural models. J Mol Graphics 14: 354–360. doi: 10.1016/S0263-7855(97)00009-X
    [50] Jorgensen WL, Chandrasekhar J, Madura JD, et al. (1983) Comparison of simple potential functions for simulating liquid water. J Chem Phys 79: 926–935. doi: 10.1063/1.445869
    [51] Zhong Q, Jiang Q, Moore PB, et al. (1998) Molecular dynamics simulation of a synthetic ion channel. Biophys J 74: 3–10. doi: 10.1016/S0006-3495(98)77761-6
    [52] Larsen RJ, Marx ML (2017) An Introduction to Mathematical Statistics and Its Applications, Pearson, 742.
    [53] Heijmans RDH, Pollock DSG, Satorra A (2000) Innovations in Multivariate Statistical Analysis, Springer US, 298.
    [54] Canal L (2005) A normal approximation for the chi-square distribution. Comput Stat Data An 48: 803–808. doi: 10.1016/j.csda.2004.04.001
  • This article has been cited by:

    1. Yuxuan Zhao, Lingzhi Nie, Hongli Yang, Kai Song, Huilin Hou, Tailored fabrication of TiO2/In2O3 hybrid mesoporous nanofibers towards enhanced photocatalytic performance, 2021, 629, 09277757, 127455, 10.1016/j.colsurfa.2021.127455
    2. Tongtong Chen, Jixun Chu, Hopf bifurcation for a predator-prey model with age structure and ratio-dependent response function incorporating a prey refuge, 2023, 28, 1531-3492, 408, 10.3934/dcdsb.2022082
    3. Debjit Pal, Dipak Kesh, Debasis Mukherjee, Study of a diffusive intraguild predation model, 2025, 2, 2997-6006, 10.20935/AcadEnvSci7508
  • Reader Comments
  • © 2018 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(5938) PDF downloads(1005) Cited by(2)

Figures and Tables

Figures(13)  /  Tables(4)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog