
Microalgae have a large high-quality protein content that can be used as human protein supplements. Dunaliella salina and Spirulina platensis have been identified as rich sources of natural bioactive compounds. We aimed to examine the antioxidant properties of bioactive peptides using the enzymatic digestion of the ribulose-1,5-bisphosphate carboxylase/oxygenase (RuBisCO) enzyme derived from D. salina and S. platensis microalgae. This was an in vitro and in silico study. Cell walls of D. salina and S. platensis were lysed, proteins were isolated, and RuBisCO fraction was concentrated. Then, the protein was enzymatically digested using pepsin, trypsin, and chymotrypsin. Finally, antioxidant activity was assessed at different stages (pre- and post-digestion). The 1,1-diphenyl-2-picrylhydrazyl (DPPH) assay was used to assess the antioxidant potential of hydrolyzates both before and after digestion. Findings indicated that digestion over time, particularly by chymotrypsin, produced bioactive fragments with enhanced antioxidant properties. For the 0–35 protein fraction (which likely includes RuBisCO), the antioxidant potential of peptides derived from S. platensis was significantly greater than that from D. salina. We showed that chymotrypsin may be an appropriate enzyme to yield the highest peptide concentrations from the protein extracts of these microalgae with the highest antioxidant activity. Moreover, the results of digesting the RuBisCO sequence with digestive enzymes showed that antioxidant properties increased with the production of hidden bioactive peptides. This finding may lead to the application of RuBisCO protein and its derivative peptides in the food and pharmaceutical industries of S. platensis and D. salina.
Citation: Seyedeh Fahimeh Razavi, Leila Zarandi Miandoab, Elaheh Zadeh Hosseingholi, Nader Chaparzadeh. Radical scavenging capacity of RuBisCO bioactive peptides derived from Dunaliella salina and Spirulina platensis: An in silico and in vitro study[J]. AIMS Molecular Science, 2025, 12(1): 49-66. doi: 10.3934/molsci.2025004
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Microalgae have a large high-quality protein content that can be used as human protein supplements. Dunaliella salina and Spirulina platensis have been identified as rich sources of natural bioactive compounds. We aimed to examine the antioxidant properties of bioactive peptides using the enzymatic digestion of the ribulose-1,5-bisphosphate carboxylase/oxygenase (RuBisCO) enzyme derived from D. salina and S. platensis microalgae. This was an in vitro and in silico study. Cell walls of D. salina and S. platensis were lysed, proteins were isolated, and RuBisCO fraction was concentrated. Then, the protein was enzymatically digested using pepsin, trypsin, and chymotrypsin. Finally, antioxidant activity was assessed at different stages (pre- and post-digestion). The 1,1-diphenyl-2-picrylhydrazyl (DPPH) assay was used to assess the antioxidant potential of hydrolyzates both before and after digestion. Findings indicated that digestion over time, particularly by chymotrypsin, produced bioactive fragments with enhanced antioxidant properties. For the 0–35 protein fraction (which likely includes RuBisCO), the antioxidant potential of peptides derived from S. platensis was significantly greater than that from D. salina. We showed that chymotrypsin may be an appropriate enzyme to yield the highest peptide concentrations from the protein extracts of these microalgae with the highest antioxidant activity. Moreover, the results of digesting the RuBisCO sequence with digestive enzymes showed that antioxidant properties increased with the production of hidden bioactive peptides. This finding may lead to the application of RuBisCO protein and its derivative peptides in the food and pharmaceutical industries of S. platensis and D. salina.
Ion channels are protein channels in the cell membrane that regulate the balance and transmission of ions inside and outside the cell. Ion channels are involved in critical physiological processes such as cellular potentiation, signaling, cellular excitability, and metabolic regulation by selectively allowing specific types of ions to pass through the cell membrane in order to maintain the difference between the ionic concentrations inside and outside the cell. In this way, ion channels control a wide range of biological functions of living organisms [1,2,3,4]. Currently, the two most relevant properties of ion channels, permeation and selectivity, are characterized by the I-V relations (current-voltage relations) for each ionic species [1,5]. The ionic transport through the ion channel is governed by fundamental physical laws of electrodiffusion. The macroscopic properties of ionic flows also depend on external driving forces, namely boundary concentrations and boundary potentials [6].
There are two related major topics in the study of ion channel problems: structures of ion channels and ionic flow properties. The physical structure of ion channels is defined by the channel shape and the spatial distribution of permanent and polarization charge. Usually, the shape of a typical ion channel is approximated as a cylindrical-like domain with a varying cross-section area. Within a large class of ion channels, amino acid side chains are distributed mainly over a "short" and "narrow" portion of the channel [7], with acidic side chains contributing permanent negative charges and basic side chains contributing permanent positive charges. For an open channel with a given structure, the main interest is to understand its electrodiffusion property. One major challenge in the study of ionic flow properties is the nonlinear interaction among different physical parameters involved in the diffusion process of charged particles. On the other hand, all present experimental measurements about ionic flow are of the input-output type [1]; that is, the internal dynamics within the channel cannot be measured with current technology. Thus, it is extremely hard to extract coherent properties or to formulate specific characteristic quantities from experimental measurements. Therefore, a suitable mathematical analysis with a physically sound model plays critical and unique roles for a possible comprehensive understanding of ion channel problems.
Poisson-Nernst-Planck (PNP) systems are basic primitive models for ionic motion through ion channels, in which the channel is considered as a line segment with a position-dependent conductivity related to the cross-sectional area. Recently, there have been some successes in mathematical analysis of PNP models for ionic flows through membrane channels. Those [6,7,8,9,10,11,12,13,14,15,16,17] that were studied in the framework of geometric singular perturbation analysis implied certain interesting threshold phenomena of ionic flows for relatively simple setups [7,18].
Considering the structural characteristics of the channel, a basic continuum model for ionic flows is the PNP system, which treats the aqueous medium as a dielectric continuum (see [19,20,21,22,23,24,25,26]). PNP systems, under some further conditions, can be derived as reduced models from molecular dynamic models, Boltzmann equations, and variational principles [27,28,29,30,31]. The PNP type models have different levels of resolutions: The classical PNP system is the simplest one, which treats ions as point-charges and ignores ion-to-ion interactions [32,33,34,35,36,37,38,39,40]. The PNP-HS model takes into consideration volume exclusion by treating ions as hard-spheres with the charges located at their centers (see [8,12,41,42] and references therein). More sophisticated models, such as coupling PNP and Navier-Stokes equations for aqueous solutions (see, e.g., [43,44,45,46,47,48,49]), and a PNP-Cahn-Hilliard model which includes steric effects [50,51] have been developed. The sophisticated models are able to model the physical problem more precisely, however it is very difficult to analytically and even numerically study their dynamics. Here, we incorporate key features while keeping the equations amenable to both analysis and numerical simulations of ionic flows.
Since the channel is narrow, and one can effectively view it as a one-dimensional channel [0,l], where l is the length of the channel together with the baths linked by the channel. A quasi-one-dimensional steady-state PNP model for a mixture of n ion species though a single channel reads [52]
1A(X)ddX(εr(X)ε0A(X)dΦdX)=−e(n∑s=1zsCs(X)+Q(X)),dJkdX=0,−Jk=1kBTDk(X)A(X)Ck(X)dμkdX,k=1,2,…,n | (1.1) |
where e is the elementary charge, kB is the Boltzmann constant, T is the absolute temperature, Φ(X) is the electric potential, Q(X) is the permanent charge density of the channel, ε0(X) is the local dielectric coefficient, εr(X) is the relative dielectric coefficient, and A(X) represents the area of the cross-section over the point X and, for the jth ion species, Cj(X) is the concentration, zj is the valence, μj is the electrochemical potential, Jj is the flux density, and Dj(X) is the diffusion coefficient.
For system (1.1), the following boundary conditions are posed [9] for k=1,2,⋯,n
Φ(0)=V, Ck(0)=Lk>0;Φ(1)=0, Ck(1)=Rk>0,k=1,2,⋯,n. | (1.2) |
We point out that, for our analysis in current work, we are only interested in solutions of systems (1.1) and (1.2) with positive concentrations. For both time evolution of the three-dimensional PNP system and the quasi-one-dimensional PNP system with two ion species, the positivity-preserving properties for the ion concentration Ck has been rigorously discussed in [53] (see Proposition 2.1 for details). For the steady-state quasi-one-dimensional classical PNP system with n≥3 different types of ion species, the authors in [15] show that there is a unique solution of the system with Ck>0. At the PDE level, we do not have a result when the system involves more than two different types of ion species. However, for the set-up in this work, it is not difficult to verify from [15] that the concentration Ck is positive for ν sufficiently small. Interested readers can also refer to [51] for more discussions of the positivity-preserving properties of the concentrations from the point of view of a numerical approach. We also remark that our analysis is for the steady-state PNP model, and so the method in the current work cannot be directly applied to time-dependent PNP models and higher dimensional PNP models, but it provides important insights into related problems and could be a starting point for a more complicated model.
For a solution to the boundary value problem (BVP) (1.1) and (1.2), the current I is
I=n∑s=1zsJs. | (1.3) |
For fixed boundary concentrations, the Jk's depend only on V, and Eq (1.3) defines the so-called I-V relations.
The electrochemical potential μi(X) for the ith ion species consists of the ideal component μidi(X) and the excess component μexi(X): μi(X)=μidi(X)+μexi(X), where
μidi(X)=zieΦ(X)+kBTlnCi(X)C0 | (1.4) |
where C0 is some characteristic number density. The classical PNP system takes into consideration of the ideal component μidi(X) only, which reflects the collision between ion particles and water (medium) molecules. Ion species in the classical PNP system are treated as point charges, and ion-to-ion interactions are ignored. The classical PNP system is reasonable in dilute cases. Meanwhile, this is a major weakness of the classical PNP model since many important properties of ionic flows do depend on finite ion sizes, particularly for those cations with the same valences but different finite ion sizes, such as Na+ (sodium) and K+ (potassium). This is closely related to the selectivity property of ion channels. To study the effects on ionic flows from ion sizes, one needs to consider excess components in the electrochemical potential, and one choice is to include hard-sphere (HS) potentials. PNP models with ion size effects have been investigated computationally with great successes (see [24,28,30,44], etc.), and have been mathematically analyzed (see, for example, [12,41,42,54,55]).
In this work, we take Bikerman's local hard-sphere (LHS) model [56] to approximate μexi(X)
1kBTμBiki(X)=−ln(1−n∑j=1νjCj(X))=νn∑j=1λjCj(X)+o(ν), | (1.5) |
where νj represents the volume of the jth ion species. Particularly, we take νn=ν so that λn=1 in our following discussion.
We point out that since Ci is the number density of ith ion species, then, ∑nj=1vjCj<1. In this sense, Bikerman's LHS takes into consideration nonzero ion sizes.
For definiteness, we will take the following settings:
(A1). Consider three ion species (n=3) with z1=z2=−z3=1.
(A2). Q(X)=0 over the whole interval [0,1].
(A3). μi(X) consists of both the ideal component μidi(X) and the LHS potential μBiki(X) in (1.5).
(A4). εr(X)=εr and Di(X)=Di.
Under the assumptions (A1)–(A4), system (1.1) reads
1A(X)ddX(εrε0A(X)dΦdX)=−e3∑j=1zjCj(X),dJidX=0,−Ji=1kBTDiA(X)Ci(X)dμidX,i=1,2,3. | (1.6) |
The boundary conditions become, for i=1,2,3,
Φ(0)=V, Ci(0)=Li>0;Φ(1)=0, Ci(1)=Ri>0. | (1.7) |
We first make a dimensionless rescaling following [57]. Set C0=max{Li,Ri:i=1,2} and let
ε2=εrε0kBTe2l2C0, x=Xl, h(x)=A(X)l2, Di=lC0Di;ϕ(x)=ekBTΦ(X), ci(x)=Ci(X)C0, Ji=JiDi; V=ekBTV, Li=LiC0, Ri=RiC0. | (1.8) |
Correspondingly, for k=1,2,3, the BVP (1.6) and (1.7) becomes
ε2h(x)ddx(h(x)ddxϕ)=−(c1+c2−c3),dc1dx+c1dϕdx+c1(x)kBTddxμBik1(x)=−J1h(x),dc2dx+c2dϕdx+c2(x)kBTddxμBik2(x)=−J2h(x),dc3dx−c3dϕdx+c3(x)kBTddxμBik3(x)=−J3h(x),dJkdx=0, | (1.9) |
with the boundary conditions
ϕ(0)=V, ck(0)=Lk>0; ϕ(1)=0, ck(1)=Rk>0. | (1.10) |
We take h(x)=1 over [0,1] in our analysis since eventually it appears in the form of ∫101h(s)ds, a positive constant, which will not affect the behavior of ionic flows.
We take advantage of the work done in [8], and further apply regular perturbation analysis to study the effects on the I-V relations from finite ion sizes under further restrictions. To be specific, we expand the I-V relation I(V,ν) along ν=0 for fixed boundary concentrations and obtain
I(V;ν)=I0(V)+νI1(V)+o(ν), |
where, under our assumptions,
I0(V)=D1J10+D2J20−D3J30,I1(V)=D1J11+D2J21−D3J31. | (2.1) |
Here, from [8],
J10=f0f1(L1−R1e−V),J20=f0f1(L2−R2e−V),J30=f0(lnL3−ln(R3eV)),J11=−[L3J10+J20(A1+Tc0Tm0(1)−1)]−1{−(1+F2)J10L232(J10+J20)(A2+Tc0Tm0(1)−1)+F1(Tc0Tm1−Tc1Tm0)(Tm0)2lnA(1)+J10(Tc0Tm1−Tc1Tm0)L32Tm0(J10+J20)2(A1+Tc0Tm0(1)−1)+(λ2−λ1)F21(1+Tc02J30)(A−Tc0Tm0(1)−1)+M1(1)−M3(1)},J21=−[L3J10+J20(A1+Tc0Tm0(1)−1)]−1{−(1+F2)J20L232(J10+J20)(A2+Tc0Tm0(1)−1)−F1(Tc0Tm1−Tc1Tm0)(Tm0)2lnA(1)+J20(Tc0Tm1−Tc1Tm0)L32Tm0(J10+J20)2(A1+Tc0Tm0(1)−1)−(λ2−λ1)F21(1+Tc02J30)(A−Tc0Tm0(1)−1)+M2(1)+M4(1)},J31=Tm1−Tc12, | (2.2) |
where
f0=L3−R3lnL3−lnR3, f1=lnL3−ln(R3e−V)L3−R3e−V, Tc0=J10+J20−J30,Tm0=J10+J20+J30, F1=J20L1−J10L2J10+J20, F2=λ1J10+λ2J20J10+J20, A(x)=1−Tm02L3x,Tm1=−L3[(λ2−λ1)F1Tc0J30(A1−Tc0Tm0(1)−1)+L3(1+F2)(A2(1)−1)],Tc1=Tm0lnA(1)[Tc0Tm0(Tm1Tm0−(λ2−λ1)F1Tc02J30−L3(1+F2)2)(A−1(1)−1)−L3(1+F2)Tc02Tm0(A(1)−1)+(λ2−λ1)F1Tc02J30(A−Tc0Tm0(1)−1)]+Tm1Tc0Tm0, |
and
M1(x)=(Tm1Tm0−(λ2−λ1)F1Tc02J30−L3(1+F2)2)[F1Tc0Tm0(A−1(x)−1)−J10L3J10+J20(ATc0Tm0(x)−1)],M2(x)=(−Tm1Tm0+(λ2−λ1)F1Tc02J30+L3(1+F2)2)[F1Tc0Tm0(A−1(x)−1)+J20L3J10+J20(ATc0Tm0(x)−1)],M3(x)=F1L3Tm0[(λ2−λ1)Tc0J10J10+J20(1+Tc02J30)+(1+F2)(Tm0+Tc02)](A(x)−1),M4(x)=F1L3Tm0[(λ2−λ1)Tc0J20J10+J20(1−Tc02J30)+(1+F2)(Tm0+Tc02)](A(x)−1). |
Our main focus in this section is to examine the finite ion size effects on the I-V relations. To be specific, we are interested in the leading term I1(V), which contains the finite ion size effects. More precisely, with respect to the membrane potential V, we will characterize
(i) the monotonicity of I1(V);
(ii) the sign of I1(V);
(iii) the magnitude of I(V) equivalent to I0(V)I1(V).
Our analysis is further based on the following assumptions
L3=L1+L2, R3=R1+R2 and L1L2=R1R2 | (3.1) |
where L3=L1+L2 and R3=R1+R2 are the so-called electroneutrality boundary conditions.
To get started, we rewrite Jk1 for k=1,2,3, under assumption (3.1), as follows:
J11=J10N3R3e−V−L3,J21=J20N3R3e−V−L3,J31=(1+F2)(L3+R3)2(L3−R3−q2Tc02), | (3.2) |
where
q2=1+2(L3−R3)(R3+L3)lnA(1),N3=(1+F2)(R3−L3eV)2eV(V−lnA(1))((R3+L3)(V−lnA(1))−2f0V). |
It then follows that
I1(V)=DzJ11+D2J21−D3J31=D1J10+D2J20R3e−V−L3N3−D3(1+F2)(L3+R3)2(L3−R3−q2Tc02). | (3.3) |
For Jk0, I0, Jk1, and I1, we observe the following:
Lemma 3.1. The following scaling law holds.
(i) Jk0 and I0 scale linearly in the boundary concentrations; that is, for any real number s>0,
Jk0(V;sL1,sL2,sL3,sR1,sR2,sR3)=sJk0(V;L1,L2,L3,R1,R2,R3) |
and
I0(V;sL1,sL2,sL3,sR1,sR2,sR3)=sI0(V;L1,L2,L3,R1,R2,R3). |
(ii) Jk1 and I1 scale quadratically in the boundary concentrations; that is, for any s>0,
Jk1(V;sL1,sL2,sL3,sR1,sR2,sR3)=s2Jk1(V;L1,L2,L3,R1,R2,R3) |
and
I1(V;sL1,sL2,sL3,sR1,sR2,sR3)=s2I1(V;L1,L2,L3,R1,R2,R3). |
We remark that the scaling law actually provides an efficient way to adjust the effects on ionic flows from the finite ion sizes by controlling the boundary concentrations.
We study the monotonicity of the leading term I1(V) in (3.3) under the assumption (3.1), which is critical for one to further consider the sign of I1(V).
To get started, we establish the following result.
Lemma 3.2. Assume L3<R3. One has q2>0.
Proof. We rewrite q2 as
q2=1+2(L3−R3)(L3+R3)lnA(1)=(L3+R3)lnA(1)+2(L3−R3)(L3+R3)lnA(1). |
Note that lnA(1)=lnR3L3>0. One has that the sign of q2 is the same as that of (L3+R3)lnA(1)+2(L3−R3). For convenience, we define
ˆq2(L3,R3)=(L3+R3)lnA(1)+2(L3−R3)=R3ˉq2(L3,R3), |
where
ˉq2(L3,R3)=−(L3R3+1)lnL3R3+2(L3R3−1). |
Upon introducing x=L3R3, ˉq2(L3,R3) can be written as
ˉq2(x)=−(x+1)lnx+2(x−1). |
Direct calculation gives ˉq′2(x)=−lnx−1x+1 and ˉq″2(x)=1−xx2>0 for 0<x<1, which implies that ˉq′2(x) is increasing in x. Note also that ˉq′2(1)=0. One has ˉq′2(x)<0 for 0<x<1, which indicates that ˉq2(x) is decreasing for 0<x<1. Together with ˉq2(1)=0, one has ˉq2(x)>0 for 0<x<1. Hence, ˆq2(x)>0 for 0<x<1. Therefore, q2>0 for L3<R3.
We have the following result:
Theorem 3.3. Under the assumption (3.1), for V>0, I1(V) is increasing in the potential V.
Proof. Under assumption (3.1), from (3.2), direct calculations yield
J′11(V)=−f0(L3,R3)(1+F2)(L1eV−R1)(2f0(L3,R3)−L3−R3)2(L3eV−R3)eV,J′21(V)=−f0(L3,R3)(1+F2)(L2eV−R2)(2f0(L3,R3)−L3−R3)2(L3eV−R3)eV,J′31(V)=−q2f0(L3,R3)(L3+R3)(1+F2)2. |
Note that, for V>0,
f0(L3,R3)>0, L1eV−R1L3eV−R3>0, 2f0(L3,R3)−L3−R3<0 and 1+F2>0. |
Together with Lemma 3.2, one has J′11(V)>0, J′21(V)>0, and J′31(V)<0. By the definition of I1(V)=D1J11(V)+D2J21(V)−D3J31(V), we have I′1(V)>0, and thus I1(V) is increasing in the potential V.
Based on the analysis in Section 3.1, we now study the sign of the leading term I1(V), which provides important information on the finite ion size effects of the I-V relations.
Lemma 3.4. Under assumption (3.1), one has
(i) If (D1−D3)(L1−R1)+(D2−D3)(L2−R2)>0, then, I1>0;
(ii) If (D1−D3)(L1−R1)+(D2−D3)(L2−R2)<0, then, there exists a unique zero Vzc1s of I1(V)=0, such that I1<0 for (0,Vzc1s) and I1>0 for (Vzc1s,+∞).
Proof. Evaluating I1(V) in (3.3) at V=0, one has
I1(0)=K[(D1−D3)(L1−R1)+(D2−D3)(L2−R2)], |
where
K=(1+λ1)(L1−R1)+(1+λ2)(L2−R2)2(L3−R3)(L3+R3). |
It is easy to verify that, under the assumption L1L2=R1R2, K>0. Hence, the sign of I1(0) is determined by the factor (D1−D3)(L1−R1)+(D2−D3)(L2−R2). More precisely,
● if (D1−D3)(L1−R1)+(D2−D3)(L2−R2)>0, then I1(0)>0. Together with the fact that I1(V) is increasing, one has I1(V)>0 for all V>0;
● if (D1−D3)(L1−R1)+(D2−D3)(L2−R2)<0, then I1(V)=0 has a unique zero, say Vzc1s such that I1(V)<0 for V∈(0,Vzc1s) while I1(V)>0 for V∈(Vzc1s,∞) since again I1(V) is increasing in the potential V.
This completes the proof.
One of our main results then follows directly.
Theorem 3.5. Under assumption (3.1) and V>0, one has, for small ν>0,
(i) if (D1−D3)(L1−R1)+(D2−D3)(L2−R2)>0, then the finite ion size always enhances the current I(V);
(ii) if (D1−D3)(L1−R1)+(D2−D3)(L2−R2)<0, then the finite ion size enhances the current I(V) if V∈(Vzc1,∞), while it reduces the current I(V) if V∈(0,Vzc1).
We consider the finite ion size effects on the magnitude of the I-V relation I(V), that is |I(V)|, which is equivalent to analyzing I0(V)I1(V). For the zeroth order term I0(V), one has
Lemma 3.6. Under assumption (3.1), one has I0(V) is increasing in the potential V; furthermore,
(i) If (D1−D3)(L1−R1)+(D2−D3)(L2−R2)>0, then I0(V)>0;
(ii)If (D1−D3)(L1−R1)+(D2−D3)(L2−R2)<0, then there exists a unique zero Vzc0s of I0(V)=0, such that I0<0 for (0,Vzc0s) and I0>0 for (Vzc0s,+∞).
Proof. To study the monotonicity of I0(V), we will focus on the monotonicity of the individual fluxes Jk0 for k=1,2,3 first. From (2.2), direct calculations yield
dJ10dV(V)=zD1R1R3e−V(L3−R3e−V)2f0(L3,R3)g0(V;L3/R3,L1/R1), |
where, with x=V, a=L3/R3>0, and b=L1/R1>0,
g0(x;a,b)=(a−b)(lna+x)+(b−e−x)(a−e−x)ex. |
Since g′0(x)=(a−e−x)(b+e−x)ex, g0(x;a,b) has only one critical point x∗=−lna. It follows from g0(x∗;a,b)=0 and limx→∞g0(x;a,b)=∞ that g0(x;a,b)>0 for x≠x∗. Note that the sign of dJ10dV is the same as that of g0(x;a,b). One has that J10 is increasing in the potential V. Similarly, one can show that J20(V) is also increasing in the potential V. As for J30(V), one has dJ30dV(V;0)=−zD3f0(L3,R3)<0 for all V, from which J30(V) is decreasing in the potential V. It then directly follows from the definition of I0=D1J10+D2J20−D3J30 that the zeroth order term I0(V) is increasing in the potential V. Also, from (2.2), we have
I0(V)=f0(L3,R3)[f1(V;L3,R3)(D1L1+D2L2−(D1R1+D2R2)e−V)−D3(lnL3−ln(R3eV))]. |
Note that I0(0)=(D1−D3)(L1−R1)+(D2−D3)(L2−R2). We have
● If (D1−D3)(L1−R1)+(D2−D3)(L2−R2)>0, then I0(V)>0;
● If (D1−D3)(L1−R1)+(D2−D3)(L2−R2)<0, then there exists a unique zero Vzc0 of I0(V)=0, such that I0<0 for (0,Vzc0) and I0>0 for (Vzc0,+∞).
This completes the proof. It follows that:
Theorem 3.7. Suppose L1L2=R1R2. For V>0, one has
(i) For (D1−D3)(L1−R1)+(D2−D3)(L2−R2)>0, one has I0(V)I1(V)>0;
(ii) For (D1−D3)(L1−R1)+(D2−D3)(L2−R2)<0, one has
(a) If Vzc0<Vzc1, then
(a1) I0I1<0 if V∈(Vzc0,Vzc1);
(a2) I0I1>0 if V∈(0,Vzc0)∪(Vzc1,+∞).
That is, for V∈(Vzc0,Vzc1), ion sizes ν reduces |I(ν)|, and for V∈(0,Vzc0)∪(Vzc1,+∞), ion sizes ν strengthens |I(ν)|.
(b) If Vzc1<Vzc0, then
(b1) I0I1<0 if V∈(Vzc1,Vzc0);
(b2) I0I1>0 if V∈(0,Vzc1)∪(Vzc0,+∞).
That is, for V∈(Vzc1,Vzc0), ion sizes ν reduces |I(ν)|, and for V∈(0,Vzc1)∪(Vzc0,+∞), ion sizes ν strengthens |I(ν)|.
To end this section, we demonstrate that the ionic flow properties of interest are very sensitive to the sign of (D1−D3)(L1−R1)+(D2−D3)(L2−R2), the interplay between the boundary concentrations and the diffusion coefficients. For any experiment to be conducted, once the ion species are fixed, the values for the diffusion coefficients D1,D2, and D3 will be fixed, and one can then adjust/control the boundary concentrations to observe different dynamics of ionic flows through membrane channels, which further depends on the boundary potential.
In this part, numerical simulations are performed to provide more intuitive illustrations of some analytical results. To be specific, we numerically identify the critical potentials Vzc0 and Vzc1, which characterize the effects on ionic flows from finite ion sizes. To further illustrate the finite ion size effects on ionic flows, we also numerically obtain the zeroth order I-V relations (corresponding to the classical PNP system which ignores the finite ion sizes) and the I-V relations with small positive ν, respectively. Corresponding critical potentials for each setup are identified, from which one is able to observe the effects on ionic flows from the finite ion sizes.
We will conduct two experiments:
(I) For the first one, we perform numerical simulations to the system under the assumption L1L2=R1R2, from which one can better understand our analytical results obtained in Section 3 (see Figure 1);
(II) For the second one, we perform numerical simulations to the system without the restriction L1L2=R1R2, which provides more general insights into the study of ionic flows properties of interest (see Figure 2).
To get started, we rewrite system (1.9) and (1.10) as a system of first-order ordinary differential equations. Upon introducing u=εddxϕ, one has
ε˙ϕ=u,ε˙u=−(c1+c2−c3),ε˙C=−u(I+N)−1ZC−ε(I+N)−1J,˙J=0 | (4.1) |
where
C=(c1c2c3),J=(J1J2J3),Z=diag(1,1,−1),N=(c1∇cμBik1c2∇cμBik2c3∇cμBik3)=(c1∂c1μBik1c1∂c2μBik1c1∂c3μBik1c2∂c1μBik2c2∂c2μBik2c2∂c3μBik2c3∂c1μBik3c3∂c2μBik3c3∂c3μBik3). |
The boundary conditions are
ϕ(0)=V, ck(0)=Lk>0; ϕ(1)=0, ck(1)=Rk>0. | (4.2) |
In our simulations of system (4.1) and (4.2), we take ε=0.08, λ1=1.32, λ2=1.85 and ν=0.01 for both experiments, while taking L1=12, L2=8, R1=24, R2=16 for the first one with L1L2=R1R2, and L1=12, L2=8, R1=20, R2=12 for the second one. For each experiment, we take D1=1.032, D2=1.334, D3=1.45 and D1=1.032, D2=1.334, D3=0.98, respectively.
Our numerical simulations show that
(A) Under the restriction L1L2=R1R2, both the zeroth order term I0(V) and the leading term I1(V) that contains the finite ion size effects, are increasing in the boundary potential V (see Figure 1). For V>0,
(A1) if the quantity (D1−D3)(L1−R1)+(D2−D3)(L2−R2) is positive, then both I0(V) and I1(V) are positive, and it follows that I0(V)I1(V)>0 for V>0;
(A2) if the quantity (D1−D3)(L1−R1)+(D2−D3)(L2−R2) is negative, then both I0(V)=0 and I1(V)=0 have unique zeros, Vzc0s and Vzc1s. That is, I0(Vzc0s)=0 and I1(Vzc1s)=0. For the setup in our numerical simulations, Vzc0s<Vzc1s, and hence, I0(V)I1(V)>0 for V∈(0,Vzc0s)∪(Vzc1s,∞), and I0(V)I1(V)<0 for V∈(Vzc0s,Vzc1s).
This is consistent with our analytical results stated in Theorem 3.3, Lemma 3.4, Lemma 3.6, and Theorem 3.7.
(B) Without the restriction L1L2=R1R2, for V>0, similar numerical results are obtained (see Figure 2). This indicates the qualitative properties of ionic flows through membrane channels are stable.
To end this section, we demonstrate that the purpose of the numerical simulations is just to provide more intuitive illustration of our analytical results, and it is simply based on "bvp4c" from [MATLAB]. There is no numerical convergence analysis involved, which is also beyond the goal of this work. Those who are interested in the numerical approach to similar problems may refer to the following works [51,58].
In this work, we examine the qualitative properties of ionic flows through membrane channels via PNP systems with two cations and finite ion sizes effects modeled through Bikerman's local hard-sphere potential. The main goal is to understand the problem from the mathematical point of view, which could provide some insights into related studies of ion channel problems. Viewing the ion sizes as small parameters, we expand the I-V relation along the characteristic ion size ν=0 as
I(V)=I0(V)+νI1(V)+0(ν) |
where I1(V) is the leading term that contains the finite ion size effects, and is our main focus in the study. The sign and the monotonicity of I1(V) is rigorously analyzed, from which one is able to examine the finite ion size effects on ionic flows. Critical potentials Vzc0s and Vzc1s balancing ion size effects on the I-V relations are identified, and their critical roles played in the study of ionic flow properties are analyzed. Nonlinear interplays between system parameters are characterized particularly, the ionic flow properties are sensitive to the interplay between the diffusion coefficients and the boundary concentrations of the two cations through (D1−D3)(L1−R1)+(D2−D3)(L2−R2), which provides some efficient ways to adjust/control the finite ion size effects on the I-V relations. Interesting scaling laws are observed, which is another way to adjust/control the effects on the I-V relations from the finite ion sizes. Numerical simulations are then performed to provide more intuitive illustrations of the analytical results and better understanding of the internal dynamics of ionic flows, while current technology cannot detect it in experiments.
We comment that the setup in this work is relatively simple (e.g., no permanent charges considered), and the study is the first step for analysis on more realistic and complicated models. On the other hand, the simple model studied in this work allows one to obtain a more explicit expression of the I-V relations in terms of system parameters, from which we are able to extract concrete information of the effects from small finite ion sizes. Furthermore, the detailed discussion may have great potential for the studies of synthetic ion channels.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
Y. Wang thanks the Math Department at New Mexico Tech for the hospitality during his one-year visit to the department when the main part of this work is completed under the instruction of M. Zhang. The authors are partially supported by Simons Foundation (No. 628308) and NSF of China (No. 12172199).
The authors declare there is no conflict of interest. Mingji Zhang is a guest editor for Mathematical Biosciences and Engineering and was not involved in the editorial review or the decision to publish this article.
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