Universal Properties of Concentration Sensing in Large Ligand-Receptor Networks

Cells estimate concentrations of chemical ligands in their environment using a limited set of receptors. Recent work has shown that the temporal sequence of binding and unbinding events on just a single receptor can be used to estimate the concentrations of multiple ligands. Here, for a network of many ligands and many receptors, we show that such temporal sequences can be used to estimate the concentration of a few times as many ligand species as there are receptors. Crucially, we show that the spectrum of the inverse covariance matrix of these estimates has several universal properties, which we trace to properties of Vandermonde matrices. We argue that this can be used by cells in realistic biochemical decoding networks.

Figure 1

(a) Eigenvalue spectrum of $-\log P^{\prime \prime }$ plotted for different total observation time $T$ for a network of four receptors and 40 ligands. The eigenvalues are averaged over ten random realizations of the unbinding rates. All concentrations and binding rates are set to 1. The unbinding rates are chosen from a log-normal distribution with mean parameter 1 and the standard deviation parameter 0.1. In these simulations, we have set the minimum eigenvalue (the inverse of the prior variance) as ${10}^{-10}$. In reality, eigenvalues much smaller than 1 (red dotted line) are dominated by the prior and are not physically relevant. The largest eigenvalue corresponds to measuring the total concentration of all ligands. Other eigenvalues group together in subsets of the number of receptors (here $N_R=4$). These subsets are nearly equally spaced on the log axis. (b) Averaged (over 1000 random realizations of the unbinding rates) eigenvalue spectrum of $-\log P^{\prime \prime }$ vs the number of receptors, $N_R$. Here $N_L=40$ and $T={10}^4$. The concentrations and the unbinding rates are as above. As the number of receptors changes, the size of the split subsets follows. (c) Averaged (over 1000 random realizations of the unbinding rates) eigenvalue spectrum of $-\log P^{\prime \prime }$ vs the standard deviation of the unbinding rate distribution, which is log normal with the mean parameter 1. We simulated a network of four receptors and 40 ligands for $T={10}^4$. The concentrations and binding rates were chosen as earlier. A network with wider range of unbinding rates estimates concentrations better (larger eigenvalues).

Figure 2

Asymptotic properties of the inference are reflected in the negative logarithm of the determinant of the Fisher information matrix per ligand species, $\mathcal{F}$. We plot $\mathcal{F}$ vs. $N_L/N_R$, connecting the data points with (a) constant $N_L$ and (b) constant $N_R$. Parameters of the simulations are as in Fig. 1, with the log variance of the log-normal distribution of the unbinding rate $\mathrm{\Delta }r=0.1$, and $T=100$. Asymptotically, $\mathcal{F}$ scales as ${(N_L)}^{\alpha }$, $\alpha \approx 3.7$, and as $1/{(N_R)}^{\beta }$, $\beta \approx 1$. Insets in both panels show how $\alpha$ and $\beta$ change with $T$. Error bars represent ($\pm$) 1 standard error obtained from the linear fits.

Figure 3

The eigenvalue structure observed for the Hessian for the ligand-receptor networks has its origin in the Vandermode matrix, through the expansion of the type $(VR{)}^T(VR)$ (see text). The first four columns show spectra of $R^{T}R$, where $R$ is a random matrix, $V^{T}V$ [18], $(VR{)}^T(VR)$, and a sum of different $(VR{)}^T(VR)$ matrices. $R$ and $V$ are $40\times 40$ square matrices and five such matrices are added for the spectra in the fourth column. Such matrices are ill conditioned due to the $V^{T}V$ factor, and their eigenvalue spectra are dominated by this factor. Notice that the spectrum of the $V^{T}V$ factor determines the spectrum of $(VR{)}^T(VR)$, and adding many such matrices results in level splitting. A related argument is illustrated in the last two columns. We suggest that the eigenvalues of $\log P_i^{\prime \prime }$ result from the exponential part in the numerator of Eq. (4). Matrix $M_{40\times 40}$ in the fifth column has the same structure $(M_i{)}_{\alpha \beta }={\sum }_{m=1}^N(x_{\alpha }{x}_{\beta }{)}^{{\tau }_m}$, resulting in exponentially spaced eigenvalues. For these simulations, we chose $\tau$ uniformly at random in [0 1], $x$ uniformly at random in [0.9, 1.1], and $N=20$. Adding many such matrices together again results in level splitting (last column).