Balancing specificity, sensitivity, and speed of ligand discrimination by zero-order ultraspecificity

Specific interactions between receptors and their target ligands in the presence of nontarget ligands are crucial for biological processes such as T cell ligand discrimination. To discriminate between the target and nontarget ligands, cells have to increase specificity to the target ligands by amplifying the small differences in affinity among ligands. In addition, sensitivity to the ligand concentration and quick discrimination are also important to detect low amounts of target ligands and facilitate fast cellular decision making after ligand recognition. In this work we propose a mechanism for nonlinear specificity amplification (ultraspecificity) based on zero-order saturating reactions, which was originally proposed to explain nonlinear sensitivity amplification (ultrasensitivity) to the ligand concentration. In contrast to the previously proposed proofreading mechanisms that amplify the specificity by a multistep reaction, our model can produce an optimal balance of specificity, sensitivity, and quick discrimination. Furthermore, we show that a model for insensitivity to a large number of nontarget ligands can be naturally derived from a model with the zero-order ultraspecificity. The zero-order ultraspecificity, therefore, may provide an alternative way to understand ligand discrimination from the viewpoint of nonlinear properties in biochemical reactions.

Figure 1

(a) Schematic diagram of ligand binding to receptors. Target and nontarget ligands have different unbinding rates $k_{-1}$ and $k_{-1}^{\prime }$. The unbinding rate of the target ligand is smaller than that of the nontarget ligand as $k_{-1}<k_{-1}^{\prime }$. (b) Receptor phosphorylation and dephosphorylation reactions in a ligand discrimination system. Receptor phosphorylation is induced by the ligand whereas receptor dephosphorylation is catalyzed by phosphatase enzymes.

Figure 2

Schematic diagram of the models analyzed in this paper: (a) the kinetic proofreading model with multistep proofreading model, (b) the zero-order ultrasensitivity model, (c) the kinetic proofreading model with zero-order ultraspecificity, (d) the generalized kinetic proofreading model, and (e) the kinetic proofreading model with concentration compensation. The $\text{KPM}$ model and the $\text{ZU}$ model are the models published in [9, 10, 11]. The $\text{KPZU}$ model, the $\text{GKP}$ model, and the $\text{KPC}$ model are the original models proposed in this paper. To compare these models, we modify the published models based on the minimum assumptions described in Fig. 1. See Eqs. (1), (10, 11, 12), and (30) for the detail of the models.

Figure 3

Steady-state response and the time series of the $\text{KPM}$ model for various numbers of proofreading steps $n$ ($0\le n\le 3$). The stationary fraction of the phosphorylated receptor ${\overline{R}}_{\mathrm{p}}^{*}$ is shown as a function of (a) the unbinding rate $k_{-1}$ with $L_{\mathrm{T}}=5$ and $k_1=0.11$ and (b) the total ligand concentration $L_{\mathrm{T}}$ with $k_{-1}=1$ and $k_1=0.02$. The curves are analytically obtained from the approximated equation (9) and the points are obtained from the numerical simulation. (c) Time series of the fraction of phosphorylated receptor $R_{\mathrm{p}}^{*}\left(t\right)$ with $L_{\mathrm{T}}=5, k_{-1}=0.1$, and $k_1=10$. In (a)–(c) the values of the other parameters are $R_{\mathrm{T}}=100, w=1$, and $k_4=0.01$. The results of the numerical simulations are obtained from Eqs. (2, 3, 4, 5, 6). We use the same initial values $R(t=0)=R_{\mathrm{T}}, R_{\mathrm{p}}(t=0)=0, C_i(t=0)=0$ (for $i=0,1,\cdots ,n$), and $L(t=0)=L_{\mathrm{T}}$.

Figure 4

Steady-state response and the time series of the $\text{GKP}$ model for (a)–(c) different values of the unsaturation level $K$ when the number of the proofreading step is $n=1$ and (d)–(f) different values of the proofreading steps $n$ when $K=10$. The $\text{GKP}$ model with small $K$ and small $n$ corresponds to the $\text{KPZU}$ model, and the $\text{GKP}$ model with large $K$ and large $n$ corresponds to the $\text{KPM}$ model. (a) and (d) Stationary fraction of the phosphorylated receptor ${\overline{R}}_{\mathrm{p}}^{*}$ as a function of the unbinding rate $k_{-1}$. The semilogarithmic plot is also shown in the inset of (d). (b) and (e) Stationary fraction of the phosphorylated receptor ${\overline{R}}_{\mathrm{p}}^{*}$ as a function of the total ligand concentration $L_{\mathrm{T}}$. (c) and (f) Time series of the fraction of phosphorylated receptor $R_{\mathrm{p}}^{*}\left(t\right)$ obtained from the numerical simulation. The curves in (a), (b), (d), and (e) are analytically obtained from the approximated equations (22) and (23) and the points are obtained from the numerical simulation. In (a)–(c) the values of the unsaturation level are $K=1$, 10, and 100. The parameters are (a) $L_{\mathrm{T}}=3$ and (c) $L_{\mathrm{T}}=5$. The values of the other parameters are $R_{\mathrm{T}}=100, P_{\mathrm{T}}=1, w=1, k_{-2}=10$, and $k_3=1$. In (d)–(f) the number of the proofreading steps is $0\le n\le 4$. The parameters are (d) $L_{\mathrm{T}}=4$ and (f) $L_{\mathrm{T}}=20$. The values of the other parameters are $R_{\mathrm{T}}=100, P_{\mathrm{T}}=1, w=1, k_{-2}=10$, and $k_3=1$. Here $k_1$ and $k_2$ are obtained as $k_1=({\overline{k}}_{-1}+w)/K_1$ and $k_2=(k_{-2}+k_3)/K_2$, where $K=K_1=K_2$. We use (a) and (d) ${\overline{k}}_{-1}=10$, (b), (c), and (e) ${\overline{k}}_{-1}=k_{-1}=1$, and (f) ${\overline{k}}_{-1}=k_{-1}=0.1$. We use the same initial values $R(t=0)=R_{\mathrm{T}}, R_{\mathrm{p}}(t=0)=0, C_i(t=0)=0$ (for $i=0,1,...,n$), $P(t=0)=P_{\mathrm{T}}, D(t=0)=0$, and $L(t=0)=L_{\mathrm{T}}$.

Figure 5

Specificity, sensitivity, and speed of the $\text{GKP}$ model as functions of the number of proofreading steps $n$ and the unsaturation level $K$ are plotted as color maps: (a) specificity function $\lambda (n,K)$ with $L_{\mathrm{T}}=3$, (b) sensitivity function $\mu (n,K)$ with $k_{-1}=1$, and (c) speed function $\nu (n,K)$ with $L_{\mathrm{T}}=3$ and $k_{-1}=1$. (a) $\lambda (n,K)$ and (b) $\mu (n,K)$ are calculated from ${\overline{R}}_{\mathrm{p}}^{*}\left(k_{-1}\right)$ and (c) ${\overline{R}}_{\mathrm{p}}^{*}\left(k_{-1}\right)$ and $\nu (n,K)$ are obtained from the numerical simulations of Eqs. (13, 14, 15, 16, 17, 18, 19). We use the initial states indicated in Fig. 4. The other parameters are $R_{\mathrm{T}}=100, P_{\mathrm{T}}=1, w=1, k_{-2}=10$, and $k_3=1$. In (a)–(c) $k_1$ and $k_2$ are obtained as $k_1=({\overline{k}}_{-1}+w)/K_1$ and $k_2=(k_{-2}+k_3)/K_2$ where $K=K_1=K_2$. Here (a) ${\overline{k}}_{-1}=10$ and (b) and (c) ${\overline{k}}_{-1}=1$.

Figure 6

Specificity $\lambda (n,K)$ (red circles), sensitivity $\mu (n,K)$ (blue triangles), and speed $\nu (n,K)$ (green diamonds) as functions of the number of proofreading steps $n$ for different unsaturation levels $K=1$ (solid curves) and $K=1000$ (dashed curves), which correspond to the zero-order regime and the first order regime, respectively. The data plotted are the same as in Fig. 5.

Figure 7

Steady-state fraction of the phosphorylated receptor ${\overline{R}}_{\mathrm{p}}^{*}$ of (a) the $\text{GKP}$ model with $n=1$ (the $\text{KPZU}$ model) and (b) and (c) the $\text{KPC}$ model with $n=10$ as a function of the unbinding rate $k_{-1}$ and the total ligand concentration (a) $L_{\mathrm{T}}$ or (b) and (c) $X_{\mathrm{T}}$. Here ${\overline{R}}_{\mathrm{p}}^{*}$ is shown by colors and is obtained from the numerical simulation of (a) Eqs. (13, 14, 15, 16, 17, 18, 19) and (b) and (c) Eqs. (31)–(36). The initial states are as in Fig. 4 and Figs. 8, respectively. In (c) the data are the same as those of (b), but the plotted region is changed from $0\le X_{\mathrm{T}}\le 10$ to $0\le X_{\mathrm{T}}\le 100$. The parameters are $R_{\mathrm{T}}=100, P_{\mathrm{T}}=1, k_1=10, k_2=10$, and $k_{-2}=10$. The other parameters are (a) $w=2$ and $k_3=1$ and (b) and (c) $w=10$ and $k_3=0.1$.

Figure 8

Steady-state response and the time series of the $\text{KPC}$ model for various numbers of proofreading steps $n$ ($0\le n\le 4$). The stationary fraction of the phosphorylated receptor ${\overline{R}}_{\mathrm{p}}^{*}$ is shown as a function of (a) the unbinding rate $k_{-1}$ with $X_{\mathrm{T}}=2, k_1=10, w=10, k_2=10.1$, and $k_3=0.1$ and (b) the total ligand concentration $L_{\mathrm{T}}$ with $k_1=11, k_{-1}=1, w=10, k_2=10.1$, and $k_3=0.1$. The curves are analytically obtained from the approximated equation (41) and the points are obtained from the numerical simulation. The semilogarithmic plot is also shown in the inset of (a). (c) Time series of the fraction of phosphorylated receptor $R_{\mathrm{p}}^{*}\left(t\right)$ with $L_{\mathrm{T}}=1, k_1=10, k_{-1}=0.1, w=1, k_2=1$, and $k_3=1$. In (a)–(c) the values of the other parameters are $R_{\mathrm{T}}=100$ and $k_{-2}=10$. The results of the numerical simulations are obtained from Eqs. (31, 32, 33, 34, 35, 36). For the numerical simulations, we use the same initial values $R(t=0)=R_{\mathrm{T}}, R_{\mathrm{p}}(t=0)=0, C_i(t=0)=0$ (for $i=0,1,...,n$), $D(t=0)=0$, and $X(t=0)=X_{\mathrm{T}}$.

Figure 9

Schematic diagram of the $\text{KPZU}$ model and the $\text{KPC}$ model. (a) In the $\text{KPZU}$ model, ligand $\mathrm{L}$ is interpreted as a complex of a ligand (pMHC), an adaptor molecule (CD4 or CD8), and a kinase (Lck). (b) In the $\text{KPC}$ model, molecule $\mathrm{X}$ is interpreted as a complex of a ligand (pMHC), an adaptor molecule (CD4 or CD8), a kinase (Lck), and a phosphatase (CD45).