Allosteric proteins as logarithmic sensors

Abstract

Many sensory systems, from vision and hearing in animals to signal transduction in cells, respond to fold changes in signal relative to background. Responding to fold change requires that the system senses signal on a logarithmic scale, responding identically to a change in signal level from 1 to 3, or from 10 to 30. It is an ongoing search in the field to understand the ways in which a logarithmic sensor can be implemented at the molecular level. In this work, we present evidence that logarithmic sensing can be implemented with a single protein, by means of allosteric regulation. Specifically, we find that mathematical models show that allosteric proteins can respond to stimuli on a logarithmic scale. Next, we present evidence from measurements in the literature that some allosteric proteins do operate in a parameter regime that permits logarithmic sensing. Finally, we present examples suggesting that allosteric proteins are indeed used in this capacity: allosteric proteins play a prominent role in systems where fold-change detection has been proposed. This finding suggests a role as logarithmic sensors for the many allosteric proteins across diverse biological processes.

Keywords: allosteric regulation; fold-change detection; logarithmic sensing.

Fig. 1.

Sensory systems have conflicting goals. (A) A sensitive system detects small changes in signal but has a narrow response range. (B) A broad-ranged system responds to a large range of signal but is not sensitive to small changes. (C) A tunable sensor is both sensitive to small changes in signal and is capable of adjusting its response curve logarithmically across a broad range. Proposed molecular circuits for fold-change detection. (D) An incoherent feedforward loop is a common motif in gene regulatory systems, where an input activates an output, and at the same time a repressor of the output. (E) A nonlinear feedback loop has also been proposed as a mechanism for fold-change detection. (F) A logarithmic-feedback circuit, built from a logarithmic sensor coupled to linear feedback. In this study, we ask how a logarithmic sensor might be implemented at the molecular level.

Fig. 2.

An MWC protein can act as a logarithmic sensor. (A) The MWC model describes a protein that can switch between an active and inactive conformation at a rate determined by the allosteric constant $e^{{\epsilon }_0}$. The active state has a ligand-binding affinity $K_A$ and the inactive state has an affinity $K_I$. The white and blue triangles represent binding sites unoccupied and occupied by ligand, respectively. (B) Within a certain range, activity of the MWC protein, $a\left(c,{\epsilon }_0\right)$, depends logarithmically on the ligand concentration. The blue line indicates the ideal logarithmic sensor, whose activity directly corresponds to the logarithm of ligand concentration. The gray range indicates the range where activity of the MWC protein coincides with that of the ideal logarithmic sensor with a certain tolerable error. In this illustration, we set the error to be at most 10% (corresponding to $\tau =6$ in Eq. S9). (C) The sensitivity function $S\left(c,{\epsilon }_0\right)$ is related to the derivative of the activity function, $a\left(c,{\epsilon }_0\right)$. The sensitivity function allows us to define a range (in gray) where the sensitivity is above a certain threshold. In this illustration, the threshold is set to $\frac{N}{8}$, corresponding approximately to $\tau =6$. In both B and C, we use $N=4$, $K_A={10}^{-3}\mu M$, $K_I={10}^2\mu M$, and ${\mathit{\epsilon }}_0=13$. (D) The activity curve of an MWC protein can be tuned on a logarithmic scale, by modulating the allosteric parameter ${\epsilon }_0$.

Fig. S1.

Effects of ${\epsilon }_0$ on the sensitivity function. (A) Activation curves for the MWC model in Eq. 1. The parameters used here are $K_A={10}^{-2},K_I={10}^2,N=4,\text{and}\hspace{0.17em}{\epsilon }_0\in \left[\mathrm{15,25}\right]$. (B) Sensitivity functions corresponding to the MWC activation curves in A.

Fig. S2.

Saturation effects in the sensitivity function. (A) Activation curves for the MWC model in Eq. 1, across a full range of ligand concentration, c. The parameters used here are $K_A={10}^{-2},K_I={10}^2,N=4,\hspace{0.17em}\text{and}\hspace{0.17em}{\epsilon }_0\in \left[\mathrm{0,60}\right]$. (B) $S\left(c,{\epsilon }_0\right)$ as c nears lower saturation. The solid black curves are $S\left(c,{\epsilon }_0\right)$ for the same parameters as in A, the dashed blue line is $S_{lower}\left(c,{\epsilon }_0\right)$ from Eq. S15, and the dotted black line is the scaling function $\frac{N}{4}\frac{c/K_A}{1+c/K_A}$. (C) $S\left(c,{\epsilon }_0\right)$ as c nears upper saturation. The solid black curves are plots of $S\left(c,{\epsilon }_0\right)$ for the same parameters as in A, the dashed red line is from Eq. S16, and the dotted black line is the scaling function $\frac{N}{4}\frac{1}{1+c/K_I}$.

Fig. S3.

Logarithmic tuning in the KNF model. Here, we show the capacity of the KNF model to be logarithmically tuned. This plot uses $K_D={10}^2,K_{ab}=1,\hspace{0.17em}\text{and}\hspace{0.17em}{K}_{bb}\in \left[{10}^0,{10}^2\right]$. For these parameters, we observe approximately three orders-of-magnitude in logarithmic shifting before the response curve begins to change shape. Much like the MWC and GPCR models, the KNF model can potentially act as a logarithmic sensor over a broad range of signal.

Fig. 3.

The regulatory circuit of the GPCRs can act as a logarithmic sensor. (A) Upon activation by ligand (c), the receptor (R) changes conformation and activates a G protein (T), which then break into an α and a $\beta \gamma$ subunit. The α subunit is responsible for downstream signaling, after which, it recombines with a $\beta \gamma$ subunit and recover the pool of G proteins. (B) Activity of the GPCR system, (i.e., the concentration of ${\widehat{\alpha }}_{GTP}$) is logarithmically tuned by $\frac{k_6}{k_4}$, the effective allosteric constant in the system. The logarithmic tuning breaks down when $k_4$ is much slower than $k_6$. In this plot, $k_1=1,k_2=10,k_3=10,k_5=50,k_6=.01,$ and $k_4\mathrm{\in }\left[{10}^{-2},{10}^2\right]$.

Fig. 4.

Logarithmic-feedback circuit. (A) A logarithmic sensor can produce fold-change detection when coupled with negative feedback. In our model, the logarithmic sensor is an allosteric protein and the feedback comes from downstream modulation of an allosteric effector. (B) In fold-change detection a step increase in signal from 25 to 50, or from 50 to 100, will produce identical outputs. (C) An illustration of how the logarithmic-feedback circuit can produce fold-change detection. In the upper row, a logarithmic sensor experiences a twofold change in signal from 25 to 50. This stimulus produces a change in the sensor’s activity (orange arrow). The change in activity turns on downstream feedback which allosterically tunes the activation curve on a logarithmic scale (blue arrow), returning the sensor’s activity to its basal level. In the lower row, the same sensor now experiences another twofold change in signal, from 50 to 100. Despite the different in signal magnitude, this twofold change produces a change in activity that is identical to the previous one (dashed lines). Feedback will eventually takes effect and the system will return again to its basal level of activity.

Fig. S4.

Fold-Change detection with an MWC Tar/Tsr Sensor. In this simulation, $K_A={10}^{-2},K_I={10}^2,N=4,m=10,\hspace{0.17em}\text{and}\hspace{0.17em}{a}_0=\frac{1}{3}$. The blue line indicates the activity of the Tar/Tsr receptor. The orange line indicates ligand concentration, varied by threefold at each step increase.

Fig. S5.

Fold-Change detection with a GPCR sensor. In this simulation, $k_1=0.01,k_2=15,k_3=10,k_5=20,k_6=0.05,\beta =1,m=0.15,\hspace{0.17em}\text{and}\hspace{0.17em}{a}_0=\frac{1}{3}$. The blue line indicates the activity of the GPCR system. The orange line indicates ligand concentration, varied by threefold at each step increase.

Fig. 5.

Biophysical measurements show that allosteric proteins are logarithmically tunable. In A–F, activity of an allosteric protein is plotted against ligand concentration. Within each plot, each activity curve corresponds to a different level of allosteric modulation. The arrow indicates modulation of the concentration of allosteric effectors. Data points (black circles) were extracted from the original studies using Web Plot Digitizer, except in A and B, where the original data were available. The data were fit with Hill equations using a nonlinear least-square fit in Matlab. The range of logarithmic tuning is defined as the ratio of $\frac{K_A}{K_I}$, which we estimated from the published measurements with empirical $K_D$ values from the Hill equation and is depicted in the blue regions. These regions are meant to be a visual aid to highlight the effects of allosteric regulation and are not analytical. (A) PFK1 is a key enzyme in glycolysis and is allosterically regulated by ADP and ATP. In this study, ADP was varied from 0 to 2 mM (23). (B) Hemoglobin is the primary oxygen transport protein in vertebrates. Hemoglobin is allosterically regulated by blood pH. In this study, pH was varied from 6.6 to 7.8 (26). (C) Cyclic nucleotide-gate ion channels are allosterically modulated by calmodulin. In this study, the ion channels were treated with 0 and 0.5 μM calmodulin (27). (D) The Tar receptor in the Escherichia coli chemotaxis pathway is allosterically regulated by methylation level. In this study, the methylation level was varied through receptor mutants (28). (E) Muscarinic acetylcholine receptors are GPCRs responsible for signaling often found in neurons. The receptors are allosterically regulated by benzyl quinolone carboxylic acid (BQCA). In this study, BQCA was varied from 0 to 10 μM (29). (F) EGFRs are allosterically regulated by receptor density (30). In this study, receptor density was varied by overexpression from $2\mathrm{\times }{10}^4$ to $1.2\times {10}^6$ receptors per cell. (G) A table summary of more allosteric proteins, whose measured activity shows logarithmically tuning. When measurements were performed in vivo, the systems were either Chinese hamster ovary (CHO) cells, human embryonic kidney (HEK) cells, Xenopus oocytes, or E. coli. Data for CNG ion channels in phototransduction from ref. ; data for ${\text{M}}_2$ mAChR, ${\mathrm{A}}_1$-AR, and GLP-1 GPCRs from ref. ; and data for the lac repressor from ref. .

Fig. 6.

Allostery and feedback in glycolysis and oxygen regulation. (A) PFK1 (PFK) is a key enzyme in glycolysis. PFK1 both catalyzes downstream production of ATP and is allosterically regulated by ATP itself. This system of interactions resembles a logarithmic-feedback circuit, which has the capacity to produce fold-change detection. (B) Oxygen transport in vertebrates is mediated by hemoglobin and feedback via multiple effectors (including carbon dioxide level). This system of interactions forms a logarithmic-feedback circuit, which can produce fold-change detection.