## Abstract

Chemical reactions are responsible for information processing in living cells, and their accuracy and speed have been discussed from a thermodynamic viewpoint [1–5]. The recent development in stochastic thermodynamics enables evaluating the thermodynamic cost of information processing [6–8]. However, because experimental estimation of the thermodynamic cost based on stochastic thermodynamics requires a sufficient number of samples [9], it is only estimated in simple living systems such as RNA folding [10] and F_{1}-ATPase [11]. Therefore, it is challenging to estimate the thermodynamic cost of information processing by chemical reactions in living cells. Here, we evaluated the thermodynamic cost and its efficiency of information processing in living systems at the singlecell level for the first time by establishing an informationgeometric method to estimate them with a relatively small number of samples. We evaluated the thermodynamic cost of the extracellular signal-regulated kinase (ERK) phosphorylation from the time series of the fluorescence imaging data by calculating the intrinsic speed in information geometry. We also evaluated a thermodynamic efficiency based on the thermodynamic speed limit [8, 12, 13], and thus this paper reports the first experimental test of thermodynamic uncertainty relations in living systems. Our evaluation revealed the change of the efficiency under the conditions of different cell densities and its robustness to the upstream pathway perturbation. Because our approach is widely applicable to other signal transduction pathways such as the G-protein coupled receptor pathways for sensation [14], it would clarify efficient mechanisms of information processing in such a living system.

The Gibbs free energy change mainly drives information transmission in living cells, and its mechanism follows thermodynamic laws. For example, systems biology reveals a deep connection between information transmission accuracy and the Gibbs free energy change in a cell [1, 2]. In recent developments of stochastic thermodynamics [15] and chemical thermodynamics [16, 17], a similar connection occurs more deeply. The trade-off relations among speed, thermodynamic cost, and accuracy in a living cell have been proposed [3, 4], and its general thermodynamic bound of information processing in signal transduction has been discussed in terms of Maxwell’s demon [5, 18–20]. Moreover, as thermodynamic generalizations of uncertainty relations, thermodynamic trade-off relations have been intensively studied in terms of thermodynamic uncertainty relations [21, 22] and thermodynamic speed limits [8, 23–29]. These trade-off relations are mainly based on mathematical properties of the Fisher information [8, 12, 30], which gives a metric of information geometry [31]. Historically, information geometry has been considered as a possible choice of differential geometry for equilibrium thermodynamics [7, 32]. In recent years, information geometry meets nonequilibrium thermodynamics such as stochastic thermodynamics [8, 12, 33] and chemical thermodynamics [13], and it provides a unified framework to derive these trade-off relations in living systems [8, 12, 13]. While theoretical progress of thermodynamic trade-off relations has been intensively made for biological applications, the thermodynamic uncertainty relation has been experimentally tested only in artificial systems [34, 35]. An experimental test of thermodynamic trade-off relations in living systems has not been reported, and the thermodynamic speed limit has not been tested even in artificial systems.

This paper experimentally evaluated the thermodynamic cost by quantifying the Fisher information of time from the fluorescence imaging of the ERK phosphorylation in normal rat kidney epithelial (NRK-52E) cells. In information geometry, the Fisher information of time is regarded as the square of the intrinsic speed. This Fisher information is calculated from the time evolution of the phosphorylated ERK fraction, which can be experimentally measured by the Förster resonance energy transfer (FRET) signal. We also evaluated other information-geometric quantities for activation and inactivation processes from this Fisher information, and these information-geometric quantities illustrate the thermodynamic cost of the ERK phosphorylation, which is related to the Gibbs free energy change and the fluctuation-response ratio. We focused on the thermodynamic speed limit based on these information-geometric quantities and found that the efficiency ranges from 0.4 to 0.9 for the ERK phosphorylation in living NRK-52E cells. We quantified the thermodynamic variability for cell density changes and the perturbation of the upstream Raf pathway. While the cell density increases the thermodynamic cost and reduces efficiency, the Raf inhibitor addition reduces the thermodynamic cost, but the efficiency is robust to the Raf inhibitor addition.

The information-geometric method proposed in this paper is generally applicable to switching dynamics between the active state and the inactive state. We consider the following chemical reaction
where *A*_{on} is an active state, *A*_{off} is an inactive state, *s*′ is stimulus, and *k*_{on}(*s*′) and *k*_{off} are rate constants. Because *k*_{on}(*s*′) can depend on stimulus *s*′, it causes a pulsatile response of *A*_{on} if this stimulus *s*′ is excitable. This paper focuses on the ERK phosphorylation in the Ras-Raf-MEK-ERK pathway that shows a pulsatile response [36–38]. In Fig. 1**a**, we show the Ras-Raf-MEK-ERK pathway that relays extracellular stimuli such as growth factors from the plasma membrane to targets in the cytoplasm and nucleus. This three-tiered Raf-MEK-ERK mitogen-activated protein kinase (MAPK) cascade plays an essential role in various cellular processes, including cell proliferation, differentiation, and tumorigenesis [39]. Upon growth factor stimulation, the receptor tyrosine kinase (RTK) activates the Ras small GTPase at the plasma membrane, which recruits and activates the Raf. The activated Raf induces activation and phosphorylation of the MEK. The upstream kinase MEK phosphorylates the ERK to increase kinase activation of the ERK. The phosphorylated ERK is finally dephosphorylated by phosphatases, thereby shutting down the ERK activation. The phosphorylated MEK catalyzes a phosphate transfer from the adenosine triphosphate (ATP) to the ERK, and the Gibbs free energy difference of ATP hydrolysis thermodynamically drives this phosphorylation of the ERK [39, 40]. Here, *A*_{on} corresponds to a phosphorylated state of the ERK, *A*_{off} corresponds to a nonphosphorylated state of the ERK, and *s*′ corresponds to the stimulus by upstream proteins in the Ras-Raf-MEK-ERK pathway, respectively. We try to estimate the thermodynamic cost of the ERK phosphorylation in living cells using the time series of the phosphorylated ERK fraction, which can be experimentally measured by the FRET technique at the singlecell level [37].

We discuss the thermodynamic cost estimation from the time series of the phosphorylated ERK fraction *P*_{1} = [*A*_{on}]/([*A*_{on}] + [*A*_{off}]), where [*A*_{on}] and [*A*_{off}] are concentrations corresponding to *A*_{on} and *A*_{off}, respectively (see also Supplementary Note). In general, it is hard to estimate the Gibbs free energy change in living cells because it needs prior knowledge about the equilibrium concentration [16]. However, this equilibrium concentration is not estimated well from the time series of *P*_{1}. We propose a novel informationgeometric method to estimate the thermodynamic cost from *P*_{1}, which does not require prior knowledge about the equilibrium concentration. Because the total concentration [*A*_{tot}] = [*A*_{off}] + [*A*_{on}] is conserved, the nonphosphorylated and phosphorylated ERK fractions *P*_{0} = 1 – *P*_{1} = [*A*_{off}]/[*A*_{tot}] and *P*_{1} = [*A*_{on}]/[*A*_{tot}] can be regarded as the probability distribution, and a Riemannian manifold can be introduced as the set of probability distributions in information geometry [31].

This method focuses on an intrinsic speed on this manifold related to the thermodynamic cost, such as the Gibbs free energy change and the fluctuation-response ratio of the ERK phosphorylation [8, 12, 13]. The square of the intrinsic speed is given by the Fisher information of time

Because *ds*^{2}/*dt*^{2} only consists of the concentration fraction *P _{i}* and its change speed

*dP*/

_{i}*dt*, we can estimate it from the time series of

*P*

_{1}.

The square of the intrinsic speed can quantify the thermodynamic cost of ERK phosphorylation (see also Supplementary Note). Under near-equilibrium condition, we obtain *ds*^{2}/*dt*^{2} ≃ −(*dσ*/*dt*)/(2*R*[*A*_{tot}]), where *R* is the gas constant, and σ is the entropy production that is the minus sign of the Gibbs free energy change over the temperature [13]. Then, the Fisher information of time is proportional to the entropy production rate change under near-equilibrium condition. Even for a system far from equilibrium, the square of the intrinsic speed gives the fluctuation-response ratio of the ERK phosphorylation. Let *f _{i}* be an observable of the ERK states, where

*i*= 0(

*i*=1) means the nonphosphorylated (phosphorylated) state. The square of the intrinsic speed

*ds*

^{2}/

*dt*

^{2}is equal to the fluctuation-response ratio where is the mean value, |

*d*〈

_{t}*f*〉

_{t}| = |

*d*〈

*f*〉

_{t}/

*dt*| implies the time-response of the observable, and implies the observable fluctuation. This relation is a consequence of the Crame’r-Rao inequality for an efficient estimator, where any observable of a binary state can be regarded as an efficient estimator.

As shown in Fig. 1**b**, a pulsatile response consists of activation and inactivation processes, and we introduce informationgeometric quantities of these processes as measures of the thermodynamic cost and the speed limit’s efficiency (see also Supplementary Note). During the activation (inactivation) process, the phosphorylated ERK fraction *P*_{1} is monotonically increasing (decreasing) in time. The activation (inactivation) starts at time *t*_{0} = *t*_{ini}(*t*_{0} = *t*_{peak}) and ends at time *t*_{1} = *t*_{peak} (*t*_{1} = *t*_{fin}). We here introduce three informationgeometric quantities [7, 8], the action
the length
and the mean velocity
during the process from time *t*_{0} to *t*_{1}. The action quantifies the thermodynamic cost of the process because the action is approximately proportional to the entropy production rate at time *t*_{0} under near-equilibrium condition. Even for a system far from equilibrium, the action can be interpreted as the thermodynamic cost in terms of the fluctuation-response ratio in Eq. (3) and becomes large if the observable change speed |*d _{t}*〈

*f*〉

_{t}| is relatively larger than its fluctuation during the process. The length is given by twice the Bhattacharyya angle, which is a measure of a difference between two concentration fractions at time

*t*

_{0}and time

*t*

_{1}. The mean velocity quantifies the speed of the concentration fraction change during the process. In information geometry, is regarded as the arc length of a circle in coordinate, and

*ds*/

*dt*is the intrinsic speed on this circle (see also Fig. 1

**c**). From the Cauchy-Schwarz inequality, we obtain the thermodynamic speed limit [8] which is a trade-off relation between the thermodynamic cost and the transition time

*t*

_{1}–

*t*

_{0}during the process. To quantify how much the thermodynamic cost converts into the concentration fraction change speed, we can consider the speed limit’s efficiency [8]

The efficiency *η* satisfies 0 ≤ *η* ≤ 1, and *η* =1 (*η* = 0) implies that this conversion is most efficient (inefficient). The efficiency becomes higher if the intrinsic speed is close to constant because *η* =1 if and only if the intrinsic speed *ds*/*dt* is constant regardless of time, (*d*/*dt*)|*ds*^{2}/*dt*^{2}| = 0.

We experimentally measured the phosphorylated ERK fraction in living NRK-52E cells with the FRET-based ERK biosensor, the EKAREV-NLS (Fig. 2**a**) [41]. By comparing the fluorescence ratio of cells with the phosphorylated ERK fraction obtained from the western blotting (Fig. 2**b**), we quantified *P*_{1} and the square of the intrinsic speed *ds*^{2}/*dt*^{2} for the ERK activation (*p* = a) and inactivation (*p* = i) processes under the condition of different cell densities: 2.0 × 10^{3} cells/cm^{2} (low, *d* = L), 2.0 × 10^{4} cells/cm^{2} (medium, *d* = M), and 2.0 × 10^{5} cells/cm^{2} (high, *d* = H), where the indices *p* ∈ {a, i} and *d* ∈ {L, M, H} regard the process and cell density, respectively. The pulses of the ERK activation were observed under these conditions (Fig. 2**c** and Supplementary Video 1), and the behavior of *ds*^{2}/*dt*^{2} characterizes these conditions *p* and *d* (Fig. 2**d**). It reveals thermodynamic differences of the intracellular ERK activation under these conditions.

The information-geometric quantities also differentiate these conditions (Fig. 3 and Extended Data Table 2). Firstly, we discuss the histogram of the action. The mean value of the action becomes larger as cell density increases for both *p* = a and *p* = i. It reflects the fact that the speed under the condition of the high density appeared faster than the low density. The difference of between the high (medium) and low densities is at least twice . In comparison with the activation and inactivation processes, is approximately twice as much as , and the distributions for the activation have longer tails than the inactivation. Because the action is a measure of the thermodynamic cost, these results suggest that the activation’s thermodynamic cost is larger than the inactivation’s one and becomes larger as cell density increases. Secondly, we discuss the histogram of the length. While the length does not distinguish the activation process from the inactivation process, the mean value of the length differentiates the cell densities. It reflects the fact that the peak of the spike becomes higher as the cell density increases. Finally, we discuss the histogram of the mean velocity. The mean value of the mean velocity for the activation is larger than the inactivation . It shows that the speed of the activation is faster than the inactivation.

From these information-geometric quantities , we obtained the speed limit’s efficiency (Fig. 3 right column). The mean value of the speed limit’s efficiency is almost under these conditions, and the distribution of shifted to lower than those of and (see also Extended Data Table 2). The efficiencies range between and , and the distributions’ shapes are biased to the higher efficiency. We can detect that the activation process is less efficient than the inactivation process . We also find that the efficiency becomes worse in the higher cell density . These results indicate that the process is accelerated under the higher cell density, and the activation process is also more accelerated than the inactivation process by the thermodynamic cost. This idea is supported by the histogram of |*d*/*dt*(*ds*^{2}/*dt*^{2})| (Extended Data Fig. 1), which becomes zero when *η* = 1. The mean values of |*d*/*dt*(*ds*^{2}/*dt*^{2})| for the activation and the high cell density are larger than that for the inactivation and the low cell density, respectively (Extended Data Table. 2). The scatter plot (Extended Data Fig. 2, see also Extended Data Table 4) suggests no correlation between and , while and seem to have a correlation. Because cell density modulates the ERK activation’s excitability through both changes in basal and peak levels of the ERK phosphorylation [37], the cellular density could affect the efficiency of the pulsatile phosphorylation and the thermodynamic cost.

We confirm that these information-geometric quantities show the thermodynamic properties of the ERK phosphorylation by comparing the Raf inhibitor addition situation. The pulsatile dynamics of the ERK activation is generated by stochastic noise from the Raf and feedback loops [37, 39], and the Gibbs free energy difference of the ERK phosphorylation is induced by stimulus *s*′ from the upstream pathway, including the Raf [39, 40]. Thus, the Raf inhibitor addition reduces the Gibbs free energy change of the ERK phosphorylation and these information-geometric quantities. To ensure these relationships, we measured the ERK phosphorylation dynamics under the Raf inhibitor (SB590885) addition and compared its dynamics with the original dynamics before adding the Raf inhibitor (Fig. 4). Of note, it is well-known that a low dose of the Raf inhibitor could paradoxically activate the ERK signaling through the Raf dimerization [42–44]. The condition of the cell density is medium in this experiment. The application of a low dose (100 nM) of the Raf inhibitor immediately activated the ERK, and the ERK activity demonstrated slower dynamics than that before the Raf inhibitor treatment and after the activation [37] (Fig. 4**a** and Supplementary Video 2). This result implies that the thermodynamic cost of the ERK phosphorylation is immediately increased when adding the Raf inhibitor. After the dynamics of the relaxation on the upstream pathways, the Raf inhibitor addition generally decreases the thermodynamic cost of the ERK phosphorylation.

In Fig. 4**b**, we show the histograms of these informationgeometric quantities and the efficiency , before (*d*′ = pre) and after (*d*′ = post) the Raf inhibitor addition, where the subscript *d*′ ∈ {pre, post} regards the Raf inhibitor addition. The mean value of the action , decreases when the Raf is inhibited , and the mean values differ by one order of magnitude (see also Extended Data Table 3). This result shows the Raf inhibitor addition reduces the thermodynamic cost of the ERK phosphorylation. The mean velocity and length also decrease when the Raf inhibitor is added, , and , while the transition time *t*_{1} – *t*_{0} becomes longer by the Raf inhibitor addition (Extended Data Fig. 3 and see also Extended Data Table 3). Surprisingly, the mean value of the efficiency is robust to the Raf inhibitor addition , while the peak of the histogram can be decreased after adding the Raf inhibitor. Moreover, the efficiency , seems to have no correlation with , nor , (Extended Data Fig. 4 and see also Extended Data Table 4), and this robustness would not come from an artifact correlation between information-geometric quantities. The efficiency compensation of the ERK phosphorylation might exist when the upstream pathways are perturbed by the inhibitor.

In summary, we introduced information-geometric quantities as thermodynamic measures, and evaluated the speed limit’s efficiency for the ERK phosphorylation dynamics in living cells at the single-cell level. Our method quantitatively clarifies the amount of information transferred in living systems based on information geometry and the conversion efficiency from the thermodynamic cost to the intrinsic speed, which complements other studies about measurements of informational quantities in biological systems such as the mutual information in signal transduction [46–48] and the Fisher information matrix on molecular networks [49, 50]. Our method is generally applicable to the activation process, and it widely exists in signaling pathways such as the G-protein coupled receptor pathways for sensation [14] and the receptor tyrosine kinase signaling pathways for cell proliferation [45]. For example, the state change from the active state *A*_{on} to the inactive state *A*_{off} in the G-protein coupled receptor corresponds to its conformational change by small rearrangements accompanying the ligand-binding. It is interesting to evaluate the speed limit’s efficiency for other activation processes on an equal footing with this ERK phosphorylation in living NRK-52E cells. Our approach has great potential for other biological applications, which might clarify an efficient signal transduction mechanism and lead to novel insight into living systems as an information processing unit driven by the thermodynamic cost.

## METHODS

### Measurement of the phosphorylated ERK fraction

We measured the ERK phosphorylation, as previously described [51]. The EKAREV-NLS stable-expression NRK-52E cell lines (NRK-52E/ERKAREV-NLS cells) [37], were used, and the EKAREV-NLS is the genetically encoded ERK sensor used in Komatsu *et al*., 2011 [41].

The NRK-52E/EKAREV-NLS cells were maintained with the Dulbecco’s Modified Eagle Medium (DMEM; ThermoFisher), 10 % Fetal bovine serum (FBS; Sigma), and 10 μg/mL Blasticidin S (Invitrogen). The cells were seeded at a specific concentration (Low: 2.0 × 10^{3} cells/cm^{2}, Medium: 2.0 × 10^{4} cells/cm^{2}, High: 2.0 × 10^{5} cells/cm^{2}) on glassbottom dishes (IWAKI). One day after the seeding, the time-lapse imaging was performed. The culture media was replaced with the FluoroBrite (ThermoFisher), 5 % FBS (Sigma), and 1 × Glutamax (ThermoFisher) 3–6 hours before starting the time-lapse imaging. We used an inverted microscope (IX81; Olympus) equipped with a CCD camera (CoolSNAP K4; Roper Scientific) and an excitation light source (Spectra-X light engine; Lumenncor). Optical filters were as follows: an FF01-438/24 excitation filter (Semrock), an XF2034 (455DRLP) dichroic mirror (Omega Optical), and two emission filters (FF01-483/32 for CFP and FF01-542/27 for YFP (Semrock)). Images were acquired every 20 sec (the exposure time was 100 ms) with binning 8 × 8 on MetaMorph software (Universal Imaging) with an IX2-ZDC laser-based autofocusing system (Olympus). A ×20 lens (UPLSAPO 20X; Olympus, numerical aperture: 0.75) was used. The temperature and CO_{2} concentration were maintained at 37° C and 5 % during the imaging with a stage top incubator (Tokai hit). For the Raf inhibitor experiment, the experiment was performed under the same condition as the medium cell density condition. We applied SB590885 (Selleck Chemicals) (a final concentration is 100 nM) 2 hours after the imaging initiation. The numbers of trials for each experimental condition are two.

We used the same relation between the FRET/CFP ratios of the EKAREV-NLS and the phosphorylated ERK fraction (pTpY-ERK2) from the western blotting in Fig. 2**b** as described previously [37]. In brief, the phosphorylated ERK fraction was quantified by the Phos-tag western blotting [52] in HeLa cells stimulated with different concentrations of 12-O-Tetradecanoylphorbol 13-acetate (TPA; Sigma) for 30 min to induce the ERK phosphorylation. Under the same condition, HeLa cells stably expressing EKAREV-NLS were imaged, followed by the quantification of the average FRET/CFP ratios. Finally, the FRET/CFP ratios were plotted as a function of the phosphorylated ERK fraction with the fitted curve of the Hill equation shown in Fig. 2**b** by the Solver Add-in in Excel (Microsoft).

### Data analysis

For the imaging analysis, Fiji was used. The background was subtracted by the Subtract Background Tool, and after that, the nuclei were tracked with a custom-made tracking program. Only the cells which were tracked over the entire images were used for calculation of information-geometric quantities. The phosphorylated ERK fraction was calculated based on the FRET/CFP ratios with the Hill equation shown in Fig. 2**b**.

For analysis of the time-series phosphorylated ERK fractions, MATLAB 2019b (MathWorks) was used. The low-pass filter, whose cutoff frequency is 0.005 Hz, was used for the cell density experiment data with the designfilt function in the Signal Processing Toolbox to reduce the noise. The low-pass filter, whose cutoff frequency is 1/1200 Hz, was used for the Raf inhibitor addition experiment data to detect the slower dynamics after the Raf inhibitor addition. We identified the activation and inactivation processes from the signs of the first and second derivatives. The first derivatives were numerically calculated using two data points of the phosphorylated fraction, and the second derivatives were numerically calculated using two data points of the first derivatives. We only calculated information-geometric quantities when the phosphorylated ERK fractions between maximum and minimum of the process over 0.01 for the density-experiment data and 0.001 for the Raf inhibitor addition experiment data. The square of an intrinsic speed *ds*^{2}/*dt*^{2} was numerically calculated using two-points of the phosphorylated ERK fraction time series. The action , length , mean velocity , and efficiency *η* were calculated using *ds*^{2}/*dt*^{2} according to the definitions. We used the set of time series for each cell in two trials for each experimental condition to make histograms.

For the statistical analysis, R (version 3.6.3; R project) was used. The Brunner-Munzel test was performed with the brun-nermunzel.test function in the brunnermunzel library (version 1.4.1). The two-sample Kolmogorov–Smirnov test was performed with the ks.test function in the stats library (version 3.6.3). The Pearson correlation coefficients were calculated and tested by the cor.test function in the stats library (version 3.6.3). The Holm method was used to control the family-wise error rates with the p.adjust function in the stats library (version 3.6.3). The sample numbers are shown in Extended Data Table 1.

## DATA AVAILABILITY

The data sets generated and analyzed during the current study are available from the corresponding author on reasonable request.

## CODE AVAILABILITY

The source codes used in the current study are available from the corresponding author on reasonable request.

## AUTHOR CONTRIBUTIONS

S. I. proposed the main method based on information geometry and designed the research. Kazuhiro Aoki performed experiments. Keita Ashida analyzed the data. All authors discussed the analytical results of the experiment and wrote the paper.

## COMPETING INTERESTS STATEMENT

The authors declare no competing interests.

## SUPPLEMENTARY INFORMATION

### Supplementary Note: Chemical thermodynamics and information geometry for switching dynamics between the active and inactive states

This supplementary note briefly discusses a relationship between chemical thermodynamics and information geometry for switching dynamics between the active state *A*_{on} and the inactive state *A*_{off}. Let us consider the chemical reaction for the active and inactive states with stimulus *s*′,
where *k*_{on}(*s*′) and *k*_{off} are rate constants. Let [*A*_{off}] and [*A*_{on}] be concentrations of chemical states *A*_{off} and *A*_{on}, respectively. These concentrations are nonnegative, i.e., [*A*_{off}] ≥ 0 and [*A*_{on}] ≥ 0. In our study, *A*_{on} corresponds to the phosphorylated ERK state, *A*_{off} corresponds to the nonphosphorylated ERK state, and *s*′ corresponds to upstream proteins’ stimulus in the Ras-Raf-MEK-ERK pathway.

At first, we consider switching dynamics between the active and inactive states. The following rate equation describes the time evolution of concentrations [*A*_{off}] and [*A*_{on}],
where *t* is time, the vector ** J** implies the reaction rate, and S is called the stoichiometric matrix. Because the vector
is in the left null space of S such that
the sum of concentrations [

*A*

_{tot}] = [

*A*

_{on}] + [

*A*

_{off}] is conserved where 0 be the zero vector in two dimensions, and the symbol

^{T}implies the transpose of a vector. This conservation law

*d*[

*A*

_{tot}]/

*dt*= 0 corresponds to the conservation of the total mass. If we introduce the concentration fractions we have

*P*

_{0}≥ 0,

*P*

_{1}≥ 0 and

*P*

_{0}+

*P*

_{1}= 1. Then, (

*P*

_{0},

*P*

_{1}) can be regarded as the probability distribution, and its dynamics are described by the master equation, which is equivalent to the rate equation (10), where W is called the rate matrix. The equilibrium distribution is given by which satisfies the detailed balance condition

This equilibrium distribution *P*^{eq} generally depends on stimulus *s*′. Concentrations at equilibrium corresponding to *P*^{eq} are given by

The ratio of two rate constants *k*_{on}(*s*′)/*k*_{off} implies a thermodynamic property of the system. This fact is known as the local detailed balance condition such that
where *R* is the gas constant, *T* is the temperature of the solvent, and and are standard chemical potentials corresponding to the active and inactive states. We now define the chemical potentials *μ*_{on}([*A*_{on}]) and *μ*_{off} ([*A*_{off}]) as

The local detailed balance condition (23) can be regarded as the equivalence of the chemical potential at equilibrium

If we define the vector of the chemical potential at equilibrium as
this vector is in the left null space of S, i.e., *μ*^{eqT}S = 0^{T}. Then, the quantity *μ*^{eqT} [** A**] is conserved,

Next, we discuss the chemical thermodynamics of a dilute solution with the temperature *T* kept constant. The Gibbs free energy per unit volume *G*([** A**]) is defined as
where

*G*

_{0}is a constant. This Gibbs free energy

*G*([

**]) satisfies the relations in chemical thermodynamics,**

*A*From Eq. (28), we obtain
and therefore the Gibbs free energy at equilibrium *G*([** A**]

^{eq}) is expressed as

The Gibbs free energy difference *G*([** A**]) –

*G*([

**]**

*A*^{eq}) is given by

If we use the definition of the *f*-divergence
which is the generalization of the Kullback-Leibler divergence for the positive measure space, the Gibbs free energy difference is given by

From the nonnegativity of the *f*-divergence *D _{f}*([

*A*]||[

*A*]

^{eq}) ≥ 0, we obtain which implies that the Gibbs free energy at equilibrium takes the smallest value. If we consider the conservation of the total mass [

*A*

_{on}]

^{eq}+ [

*A*

_{on}]

^{eq}= [

*A*

_{on}] + [

*A*

_{on}] and introduce the probability distributions

**and**

*P*

*P*^{eq}, the

*f*-divergence is rewritten as the Kullback-Leibler divergence,

The time derivative of the Gibbs free energy is also given by the *f*-divergences
where we used the local detailed balance condition (23), which is equivalent to *μ*_{on}([*A*_{on}]) – *μ*_{off}([*A*_{off}]) = *RT* ln(*J*_{−}/*J*_{+}). From the nonnegativity of the *f*-divergence, we obtain

The entropy production *σ*([** A**]) is defined as

Therefore, *dG*([** A**])/

*dt*≤ 0 can be regarded as the second law of thermodynamics

We here introduce information geometry. In information geometry, the square of line element *ds*^{2} is given by the 2nd order Taylor expansion of the Kullback-Leibler divergence
where *d P* is an infinitesimal difference of two probabilities, which satisfies . The Fisher information of time is also defined as
and its square root implies the intrinsic speed in information geometry. We can also introduce information geometry for chemical reaction networks. The square of line element for chemical reaction networks is given by the 2nd order Taylor expansion of the

*f*-divergence

We obtain the relation between two geometries *ds*^{2} and as

Because the *f*-divergence is related to the Gibbs free energy, we can consider a thermodynamic interpretation of information geometry. Under near-equilibrium condition [** A**] = [

**]**

*A*^{eq}+

*d*[

**], we have**

*A*Therefore, *ds*^{2} can be interpreted as the Gibbs free energy difference under near-equilibrium condition. We also show that the Fisher information of time *ds*^{2}/*dt*^{2} is given by
where the flow is defined as and the force is defined as . In this notation, the entropy production rate is given by . In terms of the entropy production rate, the Fisher information of time is given by

Under near-equilibrium condition *J*_{+} ≃ *J*_{−}, we have with the Onsager’s coefficient *L* = *RT*/(*k*_{off} [*A*_{on}]^{eq}) and

Thus, the Fisher information of time gives the time derivative of the entropy production rate under near-equilibrium condition

We also discuss a thermodynamic meaning of the action for a transition from time *t* = *t*_{0} to *t* = *t*_{1}, defined as

Under near-equilibrium condition, we obtain

In a relaxation process, the system can be in equilibrium at the final time *t* = *t*_{1}, and the entropy production rate vanishes at the final time *σ*|_{t=t1} ≃ 0. Therefore, the action can be proportional to the entropy production rate at the initial time *t* = *t*_{0},
if the system is under near-equilibrium condition, and reach equilibrium at the final time *t* = *t*_{1}. That is why we regard the action as a thermodynamic cost. In an experiment of the FRET measurement, we can easily estimate the Fisher information of time *ds*^{2}/*dt*^{2} because it only consists of measurable quantities, i.e., the nonphosphorylated and phosphorylated ERK fractions *P*_{0} and *P*_{1} and its change speed *dP*_{0}/*dt* and *dP*_{1}/*dt*. On the other hand, it is hard to measure the Gibbs free energy *G* itself because we need prior knowledge about concentrations at equilibrium [*A*_{on}]^{eq} and [*A*_{off}]^{eq} to estimate *G*. It is also hard to measure the entropy production rate *σ* because we need prior knowledge about the rate constants *k*_{on}(*s*′) and *k*_{off} to estimate *σ* via *J*_{+} and *J*_{−}.

Information theoretically, we can discuss the meaning of *ds*^{2}/*dt*^{2} even for a system far from equilibrium. We here introduce the observable (*f*_{0}, *f*_{1}) corresponding to (*P*_{0}, *P*_{1}), and its ensemble average is defined as 〈*f*〉_{t} = *P*_{0}(*t*)*f*_{0} + *P*_{1}(*t*)*f*_{1}. The square of the time derivative of 〈*f*〉_{t} is given by
which implies the time-response of the observable. The variance of the observable is given by
which is the fluctuation of the observable. Thus, the fluctuation-response ratio is calculated as

Therefore, the Fisher information of time implies the fluctuation-response ratio even for a system far from equilibrium.

We finally discuss the length defined as

We here introduce the change of variables and which satisfies the normalization of the probability *P*_{0}(*t*) + *P*_{1}(*t*) = 1. Then, the Fisher information of time is given by
which implies that *ds*^{2} gives a differential geometry of the circle of radius 2. Thus, the length is calculated as

The lower bound is the geodesic length on the circle of radius 2. If we assume that the time evolution of *P*_{1} from time *t*_{0} to *t*_{1} is monotonic, the equality holds . In the main text, we consider for the activation and inactivation processes because *P*_{1} in the activation (inactivation) process is monotonically increasing (decreasing) in time. From the identity
the length can be calculated as
where the angle is known as the Bhattacharyya angle.

Supplementary Video 1: The fluorescence imaging of the phosphorylated ERK in NRK-52E cells at different cell densities. The movies of the low (left), medium (middle), and high (right) densities are arranged in order from the left. The pseudo-color indicates the FRET/CFP ratio, where the green (red) color indicates that the FRET/CFP ratio is 1.0 (2.0) (see also Fig. 2**c**). The timestamps indicate the time from imaging initiation in minutes.

Supplementary Video 2: The fluorescence imaging of the phosphorylated ERK in NRK-52E cells with the Raf inhibitor addition. The Raf inhibitor was added at time 120 min. The pseudo-color indicates the FRET/CFP ratio, where the green (red) color indicates that the FRET/CFP ratio is 1.0 (2.0) (see also Fig. 2**c**). The timestamp indicates the time from imaging initiation in minutes.

## ACKNOWLEDGEMENTS

S. I. is supported by JSPS KAKENHI Grants No. JP19H05796, and JST Presto Grant No. JP18070368, Japan. Kazuhiro Aoki is supported by JSPS KAKENHI Grants No. JP19H05798. S. I. and Keita Ashida thank Kohei Yoshimura for discussions of chemical thermodynamics, S. I thanks Keita Kamino for the discussion of experimental difficulties in measurement of information quantity of living cells, and Kazuhiro Aoki thanks Yohei Kondo for helpful discussions of thermodynamics. S. I. also thanks Kiyoshi Kanazawa for careful reading of this manuscript.