Quantifying nonequilibrium dynamics and thermodynamics of cell fate decision making in yeast under pheromone induction

Inducing haploid Saccharomyces cerevisiae with pheromone, a sex hormone that promotes mating causes yeast cells to cease asexual reproduction in favor of heterosexual cell fusion, resulting in diploids.^1–3 Due to the inability of yeast cells to move like flagellated bacteria, they can only grow in the direction of a high pheromone concentration after inhibiting their own cell cycle when stimulated by pheromone continuously.^4–6 Fus3p, an essential MAP kinase that shuttled back and forth to the nuclear membrane, was responsible for all signal transduction in the cellular response.^7–15 It mediates polar growth through two distinct molecular pathways: P₁ pathway (P₁: Fus3 → Far1 → Cdc24 → Cdc42 → Bni1 → polar growth)^7,16–18 and P₂ pathway (P₂: Fus3 → Bni1 → polar growth).^8,19 Different levels of negative feedback regulation were induced by high- and low-dose pheromone in the biological processes underlying cell response (Fig. 1). Once pheromone concentrations surpass a certain threshold, the yeast cells activate stronger signaling inhibition, including the indirect inhibition of Fus3 by Ste12.^20,21 While the signaling pathways by which cells respond to pheromone have received significant attention in biology,^1,22–24 little is known about the cell fate decision-making that occurs during cellular polar growth.²⁵ Quantifying the dynamics and thermodynamics of the cell fate decision-making reflected in such biological processes as cell growth and morphology remains a significant challenge.

Although the interacting potential energy landscapes can quantify the global behavior and the local dynamics in equilibrium, these dynamical rules are not directly applicable to the living biological systems.^26,27 A living biological system itself is not in equilibrium due to its continuous exchange of information, matter or energy with the external environment.^26,28,29 Indeed, in non-equilibrium systems, the driving forces for the dynamics can be quantified in terms of the underlying landscape's gradient and the curl probability fluxes.^31–34 Such a dynamical complex system is intrinsically open, breaking the thermodynamic equilibrium in which detailed balance can be maintained.^33,34 This underlying landscape, defined by the non-equilibrium potential U, is inextricably linked to the steady-state probability distribution, which quantifies the probability of hills and valleys forming in specific states. Meanwhile, the probabilistic steady-state fluxes are forces with curl or rotating properties that can be used to precisely measure the degree of the detailed balance breaking.^26,27 Such non-zero curl fluxes, by breaking the detailed balance, can have a significant effect on both the energy dissipation in biological systems and the irreversibility of dynamical processes.^35,36 This is because the curl flux is the dynamical origin of entropy production or dissipation, and the entropy production rate quantifies the thermodynamic cost of maintaining a non-equilibrium system's steady state.^34,37 As a result, the system's detailed balance is disrupted, resulting in time reversal asymmetry and determining the direction of time.

Along with describing dynamics, the mathematical theory of Markov processes has increasingly been used to develop a thermodynamic theory describing the transformation of energy between all forms in recent years.^38–41 Markov systems with discrete state variables and continuous time enable thermodynamics to be related to dynamics' theory of potential energy landscapes.^39,41 In general, the cell morphology characterized by area or perimeter is the result of cumulative and irreversible cell growth. If the Markov dynamics can capture these processes, then one can link the nonequilibrium dynamics and nonequilibrium thermodynamics to reveal the underlying biological process.^{26,34,42–44}

Here, microscopy was used in this study to monitor the living state of single-celled yeast cultured in an isothermal microfluidic device in real time. We observed four distinct cell morphological fates using a specific form (H_n) to characterize cell morphology.²⁵ Markov processes are used to describe changes in a cellular morphological system. By quantifying the landscape of cell morphological fates at various concentrations, we found that between 2.0 and 3.0 μM, there were distinct tendencies for the phase transformations in cell morphology toward a dominant single fate. To elucidate the non-equilibrium dynamics of cellular fate decision-making, the probabilistic fluxes of the cell morphological landscape were quantified. Moreover, the quantification of non-equilibrium thermodynamics confirmed that the induction of high-dose pheromones costs yeast cells more energy, supporting the molecular mechanism of greater negative feedback at high doses. The mutual validation of the biological experiments and physical theories, as well as the dynamics and thermodynamics, confirmed the viability and precision of using a non-equilibrium statistical physical theory to quantify living biological systems. This attempt to establish the physical link between molecular events and cellular characteristics across scales has significant implications for future research into the non-equilibrium nature of cell fate decision-making in living biological systems.

A molecular network containing both the positive feedback to activate Fus3 and the negative feedback (I₁–I₃) to directly or indirectly inhibit Fus3 can, in principle, cause yeast cells to reach a bistable state over time (Fig. 1). To determine whether the yeast cells can achieve a non-equilibrium steady state in response to the external stimuli, the expressions of FUS3-GFP in single yeast cells were monitored over time [Fig. 2(a) and Figs. S1 and S2].²⁵ After approximately 600 min of cultivation, the gene expression levels of Fus3 outside the nucleus indicate that the yeast cells have entered a steady state, in which the fluorescence intensity of Fus3 fluctuates around a certain value. Meanwhile, the population landscape of the Fus3 (cytoplasmic and nuclear fluorescent signals) reveals the emergence of the bistable state in the non-equilibrium steady-state phase [Figs. 2(b) and S3].²⁵ 

Given that Fus3-activated Far1 can indicate the polarity direction of the cell growth, P₁ pathway makes cell growth more directional than P₂ pathway does.^7,16,19,45 As Fus3 switches between its two stable states, the relative weight of the two pathways of polar growth (P₁ and P₂) alters, thereby influencing the process of cell polar growth. When observing yeast cell growth under a microscope in real time, we noticed two alternating patterns in the polar growth of yeast cells: first, more lateral than longitudinal growth, followed by more longitudinal than lateral growth. Therefore, the quantification of cell growth should take into account of the growth in length and width of the cell's various parts, rather than relying solely on area or perimeter. We segmented yeast cells using a circular fill pattern [Fig. 2(c)] and employed a value similar to the harmonic mean (H_n) to characterize cell morphology.²⁵ H_n equals the sum of the reciprocal radii of the filled circles multiplied by the number of filled circles, i.e., $H n=n( 1 R 1+ 1 R 2+⋯+ 1 R n)$⁠. The cell morphological trajectories reveal a refined cell growth process in which H_n increases as the cell grows longer along the long axis and decreases as the cell grows wider along the short axis [Fig. 2(d) and Fig. S4].²⁵ Around 600 min, the cell morphology achieves a specific steady state that corresponds to a growth mode in which different morphologies can alternate.

By calculating the distribution of H_n at various concentrations, four distinct cell morphological fates in yeast cells at steady state (data after 600 min) were identified [Fig. 2(e)].²⁵ These four morphological fates can, of course, be observed at any time during the non-equilibrium steady-state period due to the abundance of statistical data volumes. To facilitate the comparison of cell morphology at different pheromone concentrations, the morphology distribution was normalized, i.e., $∫ P x d x=1$⁠. Notably, as pheromone concentrations increased, the four morphological fates observed at low pheromone doses (0.7–2.0 μM) tend to merge into a single dominant fate at high pheromone dose (3 μM). The proportion of “State₂” significantly increases, while the proportion of other states decreases correspondingly. That is, at doses near 3 μM, the tendency of the phase transition in cell morphology appears to emerge.

In fact, this tendency of phase transition in cell morphology is observable during microfluidic yeast cell cultivation. The size of cell morphology was significantly greater at concentrations of 0.7–2.0 μM pheromone than that at a concentration of 3.0 μM [Fig. 2(f)].²⁵ From the perspective of the mechanical network wiring, the morphological change at 3.0 μM is caused by the difference in yeast cells' negative feedback regulation. Once the pheromone concentration surpasses a certain threshold, Ste12 will enter the cytoplasm from the nucleus in the yeast cell to activate Msg5, thereby inhibiting the activity and the expression of Fus3.^20,21 Thus, the high-dose pheromone-induced inhibitors (I₁–I₃) inhibit the expression level and the activity of Fus3 more effectively than the low-dose inhibitors (I₁ and I₂), leading to variable degrees of polar growth (Fig. 1). This stronger negative feedback regulation induced by this high-dose pheromone caused a phase transition in the cell's morphological system, resulting in a change in the scales used to measure the four morphological fates. Consequently, the coordinates of the cell morphology statistics chart deviated from the coordinates of other concentrations at 3.0 μM.

The nonequilibrium potential function or landscape (⁠ $U$⁠) can be defined as the logarithm of the probability distribution at steady state [Fig. 2(d)], that is $U= − ln P s s.$^26,30,48 It is analogous to the Boltzmann distribution law $P∼exp − β U$ that links the potential energy to equilibrium probabilities.^41,49 If $U= − ln P s s$⁠, we have $D i j=min T i j , T j i e − Δ U i j$⁠, here $Δ U i j$ stands for the difference or gradient in $U$⁠. Therefore, the time-reversible part (⁠ $D$⁠) of the driving force corresponds to the gradient of the non-equilibrium steady-state potential (⁠ $∇U$⁠). Biologically, such a potential landscape quantifies the weights of the four cell morphological states in the non-equilibrium stable state. Meanwhile, the nonequilibrium weight landscape, which is determined by the steady-state probabilities in stochastic dynamics, attracts the cell morphological states into stable landscape basins with lower potential or greater probability.

As a result, the curl flux decreased gradually at 0.7–2.0 μM, but significantly increased at 3.0 μM [Fig. 3(a) and Tables S1–S5]. It appears that the sudden increase in the cycle flux or the degree of nonequilibrium is consistent with the distinct morphological changes or trends of phase transition near the 3.0 μM pheromone concentration. This can be understood as follows: while the potential landscape tends to attract the system to the attractor state, the cycle flux being rotational tends to destabilize the current attractor states and this gives the opportunities of repopulating the new attractor states. In other words, the nonequilibrium characterized by the cycle flux can provide the dynamical origin of nonequilibrium phase transition.

To better explain the thermodynamic origin of the cell morphological system, the chemical flux (⁠ $J i j ′$⁠) can be viewed as the circuit's current (⁠ $I$⁠) and the chemical potential (⁠ $A i j$⁠) as the circuit's voltage (⁠ $V$⁠). The results demonstrate that both the chemical currents and chemical voltages against pheromone concentrations have similar behaviors as the curl steady state probability flux against the pheromone concentrations corresponding to the states of various cell morphologies, i.e., the significant increase in the concentration at 3.0 μM [Figs. 3(b) and 3(c)]. Meanwhile, we determined the Pearson correlation coefficients of net flux and chemical current, net flux and chemical voltage to be 0.9936 and 0.9351, respectively [Figs. 4(a) and 4(b)]. Overall, the chemical voltage and chemical current form a dual for determining the underlying biochemical kinetics and the nonequilibrium thermodynamics.

As a result, the value of EPR decreased gradually between 0.7 and 2.0 μM and increased significantly between 2.0 and 3.0 μM [Fig. 3(d)]. At 2.0–3.0 μM, it appears that the significant change in the entropy production rate can provide the necessary thermodynamic cost for the phase transitions of the cell morphology. By calculating the correlation between probabilistic net flux and EPR at various concentrations, we determined their Pearson correlation coefficient to be 0.9954 [Fig. 4(c)]. This shows the strong correlation between the flux and the entropy production rate, which relates the dynamical origin and thermodynamic origin of the nonequilibrium morphological phase transitions. In addition, this confirms the prediction that cells with shorter polar growth consume more energy than cells with longer polar growth when negative feedback regulation is stronger.

According to the fluctuation theorem, a system becomes irreversible when its entropy production is nonzero.^60,61 As the irreversibility increases, the energy dissipation emerges in the cellular morphological fate decision processes. When the non-conservative forces are present, the net flux breaks the detailed balance, resulting in the emergence of time-reversal asymmetry for this non-equilibrium system.^59,62

As can be seen from the results, the value of the averaged $Δ C 3$ decreased gradually between 0.7 and 2.0 μM and increased significantly between 2.0 and 3.0 μM, which is consistent with the probability net flux change trend with pheromone concentration [Fig. 3(e)]. At 0.7–2.0 μM, the little drop or near-constant averaged $Δ C 3$ suggests that the difference between the temporal forward and backward three-point correlations is small. Meanwhile, the significant increase in the averaged $Δ C 3$ at 2.0–3.0 μM indicates the presence of a significant net flux, which breaks the detailed balance to a greater extent. We determined the Pearson correlation coefficients of the net flux and the averaged $Δ C 3$ to be 0.9700 [Fig. 4(d)]. Thus, the curl flux serves as a quantitative indicator of the degree to which the detailed balances are broken, and the time is irreversible. This shows that the tendency of nonequilibrium phase transition can also be characterized by the significant changes in the time irreversibility. Since the time irreversibility can be quantified by the high order correlations directly from the observational time series, this provides a practical way of predicting the onset of nonequilibrium phase transition even when the underlying mechanism is unknown.

The induction of pheromone caused yeast cells to undergo significant responses ranging from gene expressions to cell morphological and physiological changes. Considering that the molecular mechanism of cell polar growth involves the direction of polarity, we chose a characterization order parameter (H_n) that can account for the growth of different parts in yeast. The significant advantage of this order parameter is that it is particularly sensitive to morphological changes at various locations of the cell, allowing differential non-directional normal growth and directional polar growth (i.e., lateral growth and longitudinal growth) to be characterized in real time. Unlike conventional quantitative methods such as area and perimeter, this method of cellular characterization breaks down the notion that the cellular morphology can only accumulate irreversibly over time.

Four distinct cell morphological fates were distinguished by fitting cell morphological trajectories in non-equilibrium steady state over time using a hidden Markov chain model. The statistics of the observed time series can give rise to the landscape quantifying the stability of the state attractors. To quantify the underlying dynamics of the cell fate decision-making, we decomposed the driving force for cell morphological transition into the potential landscape and the flux landscape. Although the asymmetric flux created by the four cell morphological system states is not unique, it can be represented by the three basic loop or cycle fluxes. To quantify the level of global detailed balance breaking, we used the average of these three loops fluxes to indicate the morphological system's degree of non-equilibrium. The results demonstrated that the average net flux around a pheromone concentration of 3.0 μM also undergoes a significant phase transition [Fig. 3(a)].

The probability flux associated with the steady-state chemical flux in the biochemical kinetic system is frequently caused by the energy input from ATP hydrolysis. To visualize the microscopic chemical fluxes in the cell morphological systems, the four cell morphologies can be viewed as reactants or products in the biochemical reactions catalyzed by ATP and growth-related proteins such as Fus3p, i.e., $State ( i ) ⇔ ATP State ( j )$⁠. Given that the H_n-defined cell morphology is sensitive to both width and length, the ATP-driven reactions can be subdivided into two biochemical reactions characterized by their relative growth capacities, namely, $State ( i ) → k width > length State ( j )$ and $State ( j ) → k width < lengt h State ( i )$⁠, where the $k width > length$ and $k width < length$ represent the capacity of lateral and longitudinal growth, respectively. Physically, the net flux between different morphological states is directly proportional to the difference between their transition probabilities. Thus, the high net flux of 3 μM can be explained, from a biological standpoint, by the large difference in the concentration of the morphological states, which is a result of different growth capacities [Fig. 3(a)].

To determine whether biochemical reactions result in significant thermodynamic changes, we quantified the chemical flux and chemical potential in the cell-morphological system [Figs. 3(b) and 3(c)]. It is known in physics that if a system undergoes a non-equilibrium phase transition, its energy-entropy balance and free energy will be significantly altered.^63–65 With the current biological technology, it is still challenging of accurately measuring the energy required for polar growth in a single cell. To quantify the thermodynamics of cell fate decision-making, we characterized the switching of cell morphology fate as a generalized biochemical reaction. We reveal that higher chemical voltages induced by high dose of pheromone resulted in higher chemical currents, which will trigger a greater net input and, thus, more degrees of the detailed balance breaking, leading to higher probabilistic fluxes and the thermodynamic costs associated with them.

Meanwhile, the thermodynamic cost was quantified by the entropy production rate (EPR) of the cell morphological system. The significant increase in EPR at a pheromone dose of 3.0 μM confirmed the peculiar behavior that the cells with shorter polar growth consume more energy than the cells with longer polar growth. Biologically, the yeast induced by a high dose pheromone simultaneously executes positive feedback regulation to promote growth and negative feedback regulation to inhibit growth. This is equivalent to ride a bicycle with the brakes engaged, so the thermodynamic cost should be higher. Furthermore, the time irreversibility characterized by the difference in forward and backward in time cross correlations directly from the experimentally observed time series provides a practical way of quantifying the nonequilibrium and predicting the nonequilibrium morphological phase transition. In summary, we applied non-equilibrium statistical physics to a living biological system in an attempt to reveal the biological phase transition of polar growth induced by high-low dose pheromone. Quantifying the non-equilibrium dynamics and thermodynamics involved in cell fate decision-making sheds new light on the molecular mechanisms underlying pheromone-induced cellular responses.

In this section, we provide a statement regarding the source of data used in this study. Our study utilized the same original data as our previously published work.²⁵ However, in contrast to the previous article that focused on investigating cellular fate decisions and response to signaling molecule concentrations using kinetics and energy landscape analysis, the current study takes a different perspective with a focus on establishing an understanding of non-equilibrium processes through considerations of dynamics and thermodynamics. We infer system' physical quantities and their correlations through such analysis. Specifically, in this study, we explored the same experimental data from a novel physics-oriented perspective, building upon the previous biological research to provide new insight into the field.

Saccharomyces cerevisiae S288C (ATCC 201388: MATa his3Δ1 leu2Δ0 met15Δ0 ura3Δ0) is the yeast strain used in this experiment.^66, FUS3-GFP, whose C-terminus was fused with GFP as the reporter protein, was chosen as the research object from the YeastGFP database.^67,68 We recovered the strain by incubating the FUS3-GFP frozen strain overnight in YPD liquid medium. Then, we isolated single colonies on YPD agar plates by inoculating them. Before observing the yeast cultured on the microfluidic plate under a microscope in real time, we selected a single colony and grew it overnight (approximately 14 h) in 5 ml YNB. The incubator's temperature was 30 °C, and its rotational speed was 250 rpm. Following is the formula for the culture medium: YPD liquid medium [Yeast extract (10 g/l), Peptone (20 g/l), and Dextrose (20 g/l)], YPD agar plate [Yeast extract (10 g/l), Peptone (20 g/l), Dextrose (20 g/l), and Agar (10 g/l)], and YNB liquid medium [Yeast Nitrogen Base Without Amino Acids (6.8 g/l), Dextrose (5 g/l), and Uracil (76 mg/l), 50 × MEM Amino Acids (20 ml/l)].

We selected pheromone, an alpha factor peptide hormone with a molecular weight of 1683.98, as the stimulus source (GenScript).^69–71 On a clean bench, the weighed pheromone was dissolved in YNB liquid medium to produce a solution containing 1000 μM pheromone. Then, the microfluidic culture media containing different concentrations of pheromones were created by diluting the pheromone solution with YNB liquid medium to the optimal concentration for the experiments.

Using an inverted fluorescence microscope (Ti-E, Nikon) with an automated stage and focus and a high NA oil-immersion objective (1.45 NA, 100×), the fluorescence values of single cells were determined. The fluorescence signals were collected by a cooled EM-CCD camera using a 488 nm laser with an output power of 30 mW (only 10% of the laser beam entering the microscope objective) (897U, Andor). All images were captured utilizing both bright field and fluorescent field imaging. All bright-field and fluorescence-field images were acquired with Nikon software. Combining manual and automated analysis with the custom Matlab codes, we performed the data analysis. Multiple trajectories were acquired using time-lapse microscopy, in which fluorescent images were captured periodically and recorded every 10 min.

To investigate the underlying mechanism of the bimodality, we collected real-time fluorescence intensity trajectories [Fig. 2(a) and Figs. S1 and S2), which demonstrate that the yeast response reaches a steady state after approximately 600 min. This steady state indicates that neither the intensity of fluorescence inside nor outside the yeast cell nucleus increases or decreases significantly. The inner and outer fluorescence intensity histograms were derived from the steady-state fluorescence trajectories [Fig. 2(b) and Fig. S3]. The H_n in the yeast characterizes the cell morphology characteristics of yeast. The hidden Markov chain model (HMM) can be used to provide quantitative analysis of all cell morphology trajectories.

Using a hidden Markov model, it was possible to differentiate the cell state from the trajectory. A global maximum likelihood estimate of HMM parameters was performed on each individual time trace. Multiple initial parameters were chosen at random to initiate the iterative HMM analysis and guarantee the convergence to the global minimum. At the end of each iteration, the Baum–Welch algorithm was employed to recalculate the parameters. In each iteration, a steady-state condition was enforced on the re-estimated parameters. HMM can be applied directly to both high-dimensional and two-dimensional data. Therefore, the inner and outer fluorescence data can be trained by the HMM, and the results indicate that yeast fluorescence has a bimodal distribution. Consequently, the cell morphology trajectory data can be trained using HMM, and it is shown that the yeast cell shape has a distribution of four steady-state peaks.

S.L., Q.L., and E.W. acknowledge the support of the National Natural Science Foundation of China with Grant No. 21721003.

The authors have no conflicts to disclose.

Since this study did not involve humans or animals, ethical approval was not required.

Sheng Li: Conceptualization (equal); Data curation (lead); Formal analysis (equal); Investigation (lead); Methodology (equal); Resources (lead); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (lead); Writing – review & editing (equal). Qiong Liu: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Resources (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (equal). Erkang Wang: Conceptualization (equal); Formal analysis (equal); Funding acquisition (lead); Investigation (equal); Methodology (equal); Project administration (lead); Resources (equal); Supervision (equal); Validation (equal); Visualization (equal). Jin Wang: Conceptualization (lead); Data curation (equal); Formal analysis (lead); Investigation (equal); Methodology (lead); Project administration (equal); Resources (equal); Software (lead); Supervision (lead); Validation (lead); Visualization (lead); Writing – original draft (equal); Writing – review & editing (lead).

The data that support the findings of this study are available from the corresponding author upon reasonable request.