Global quantitative understanding of yeast’s fate decision making in response to sexual deception

Cell cycle arrest and polarized cell growth are commonly used to qualitatively characterize the yeast’s fate in response to sexual deception. Biologically, the cellular decisions governing these fates can be viewed as cascading signals along response pathways. However, it is unknown how quantitatively the cell fate decision-making evolves over time during the macroscopic and microscopic overall response. Here, by observing multi-dimensional responses at the single-cell level, we find that yeast cells can take on a variety of destinies. Multiple states are revealed, along with kinetic switching rates and pathways among them, giving rise to a quantitative landscape of mating response. The applications of landscape and flux theory to such biological systems enable us to quantify the non-equilibrium dynamics of the yeast cell’s mating responses. Furthermore, our experimental results establish the first global quantitative demonstration and understanding of the molecular mechanisms underlying the formation of these states, supporting the real-time synchronization of intracellular signaling with their physiological growth and morphological functions. To further elucidate these microscopic mechanisms, we conduct biochemical reaction simulations to demonstrate the emergence of these states. These findings provide new insights into the global signaling mechanisms underlying how the yeast “think” and do in response to sexual deception.

" respectively represent different negative feedback adjustment pathways; " " " " 140 represent the two signal pathways of the polar growth. 141

142
Quantifying the cellular mating decision 143 Studies have shown that the yeast, whose mating decision is an all-or-none  150 Observing the single cell of yeast with a total internal reflection fluorescence 151 microscope, the abundant expression of Fus3p showed that it can gradually reach an 152 "adaptation state" ("a-b" stage) after being stimulated by the pheromone for a certain 153 period of times ( Figure 2B). This trend of the rising first and then stabilizing can be 154 understood as the "psychological state" of the yeast cell from being "excited" to 155 restoring the "calm". But if the cells resumed budding ("b-c" stage), the expression of 156 Fus3p was greatly reduced until it returns to near the "initial state" ("0-a" stage).
with the pheromone stimulation time at 0.7μM. The x-axis represents the time of incubation with 191 the pheromone-containing medium; the y-axis represents the fluorescence intensity of 192 the three times "a", "b" and "c" in the figure show the fluorescence intensity values of pheromone 193 stimulation for 160 minutes, 1520 minutes and 1820 minutes respectively, and the pictures next to 194 them are the single-cell photos of yeast taken through the microscope at that point. (C) Statistical 195 graphs of Fus3-GFP fluorescence intensity at different pheromone concentrations. The yeast cells 196 were cultured in YPD medium containing different concentrations of the pheromone for 24h; the 197 black curve represents the overall fluorescence intensity statistics; the green curve represents the 198 "initial steady state" fluorescence intensity; the red curve represents the "adapted steady state" The online version of this article includes the following source data and video(s) for figure 2: 206 Source data 1. Data from flow cytometry assays associated with Figure 2C. 207 Two cell fate decision states reflected on gene expressions 210 To explore the cell fate decision-making of the yeast during the "calming period" 211 (non-equilibrium steady states), we plotted the fluorescence intensity trajectories of 212 . CC-BY 4.0 International license available under a was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprint (which this version posted July 4, 2022. ;https://doi.org/10.1101https://doi.org/10. /2022 If our dynamic molecular mechanism is correct, functional consumption and  Biologically, this switching of the two fates shows the yeast's two decision-254 makings, looking for a mate or waiting for a mate. Immersed in the low state ( , ) 255 does not directly restore budding but chooses to continue intermittent growth, which 256 . CC-BY 4.0 International license available under a was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprint (which this version posted July 4, 2022. ;https://doi.org/10.1101https://doi.org/10. /2022 means that the yeast cell doesn't give up mating even if its spouse missed their 257 appointment. On the other hand, the yeast is also actively saving itself, accumulating 258 the counter-attack power through many negative feedback regulators, to escape the 259 deception of lies. This cellular behavior that not only waits hopefully, but also actively 260 strengthens its resistance to prevent disappointment, which seems to be paradoxical. 261 In fact, it shows that the yeast is not "losing its mind" as a result of being deceived, 262 but rather is making the "sane" decisions. was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprint (which this version posted July 4, 2022. ;https://doi.org/10.1101https://doi.org/10. /2022 dividing line of time, and the 600 minutes is also the starting point of steady-state data; (B) Three-267 dimensional distribution graph of Fus3p fluorescence intensity inside and outside the nucleus of 268 yeast cells in the stationary phase under the different pheromone concentrations. On the left is the 269 3D distribution of fluorescence or the 3D population landscape, on the middle is the 2D 270 histograms or the 2D underlying potential landscapes U in exponential scale (defined as p ~ e -U ), 271 which is also the population landscape; on the right is the 2D underlying potential landscapes U 272 ( ln ). (C) Schematic diagram of the response of yeast cells to pheromone stimulation. The 273 solid line is the path that plays a major role in the polar growth, and the dashed line has a smaller 274 weight on the polar growth than the solid line; the "Outer_P " represents the indirect pathway of 275 the Fus3p in the cytoplasm to the nucleus for inhibiting the cell cycle; the "Inner_P " represents 276 the indirect pathway for the Fus3p in the nucleus for the cell polar growth; the "Outer_P " 277 represents the direct pathway of the Fus3p in the cytoplasm for the cell polar growth; the 278 "Inner_P " represents the direct pathway for the Fus3p in the nucleus to transfer to the cytoplasm 279 for the cell polar growth; the " " represent the inhibitory effects of the negative feedback 280 regulation; the two gray dashed lines represent the cell membrane and the nuclear membrane, 281 respectively. (D) A simplified schematic diagram of the molecular mechanism of Fus3p's self-282 activation and self-inhibition. A and B respectively represent two genes that interact with Fus3p in 283 the signaling pathway; the solid line represents self-activation; the dashed line represents self-284 inhibition. (E) The fluorescence intensity trajectories of Fus3p inside and outside the nucleus over 285 time. The red line represents the fitting line of the high state and the low state. 286 The online version of this article includes the following source data and figure supplement(s) for 287 figure 3: 288 . CC-BY 4.0 International license available under a was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprint (which this version posted July 4, 2022. ;https://doi.org/10.1101https://doi.org/10. /2022 Source data 1. Data required to create the landscape for Figure 3B. Uncovering the decision landscapes in the pheromone dimension 296 To further understand the nature of the bistable state, we counted the apparent 297 characteristics of the double peaks, such as the area overlap ratio ( ) and the spacing 298 between the double peaks ( Figure 4A). From the calculation results, we can see that 299 as the pheromone concentration increases, the area overlap ratio ( ) between the two 300 states first increases (0.7μM-1.0μM) and then decreases (2.0μM-3.0μM), while the 301 bimodal distance first decreases and then increases. As the peak distance decreases, 302 the area overlap ratio of the double peaks increases, reflecting the gradual closeness of 303 the two gene expression states.

304
To uncover the capability of the switching between these two decisions, the 305 hidden Markov chain model (HMM) is used to fit the experimental data perform the 306 bi-stable analysis on the decision-making landscape. The steady state probability can 307 be used to quantify the population landscape or the potential landscape (Fang et CC-BY 4.0 International license available under a was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprint (which this version posted July 4, 2022. ;https://doi.org/10.1101https://doi.org/10. /2022 . As seen clearly, the population and potential landscapes ( and ) at

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According to the results, the trend of the barrier height versus pheromone 317 concentration is essentially consistent with that of its residence time ( Figure 4B). The 318 greater the barrier height in the decision-making is, the greater the resistance to 319 switching is. A slight increase in barrier height significantly increases the residence 320 time of the "high-state" for polar growth at low doses (0.7-0.8μM). At the high doses

321
(1.0-3.0μM), the residence time also decreases gradually as the barrier height 322 decreases. From a biological perspective, this increase at low doses is due to the was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprint (which this version posted July 4, 2022. ; https://doi.org/10.1101/2022.07.04.498707 doi: bioRxiv preprint 4D). Due to the relatively greater resistance to the switching from the high-state to the 333 low-state at the low doses, the yeast is more inclined to stay in the high state to look 334 for mating actively, while the opposite is true at the high doses. was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprint (which this version posted July 4, 2022. ; https://doi.org/10.1101/2022.07.04.498707 doi: bioRxiv preprint Source data 1. Data tables related to quantifications in Figure 4A. unequivocally that when the size ratio is 0.27, cellular growth is slower, allowing for 370 more opportunities to observe this ratio distribution. That is, the rate of growth or the 371 . CC-BY 4.0 International license available under a was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprint (which this version posted July 4, 2022. ; https://doi.org/10. 1101/2022 capacity for growth at various locations within the cell are not exactly identical.

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Therefore, rather than using the cell's area, the total length, or the arithmetic mean of 373 the length, we chose to describe the cell's morphology using the reciprocal of the 374 radius in the cell-filled circle multiplied by the number of filled circles, i.e. four growth rates into two categories: the fast growth rates and the slow growth rates.

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The fast growth rates included the length-first rate one (representing the longitudinal 390 growth rate) and the width-first rate one (representing the lateral growth rate); the 391 slow growth rates included two non-obvious rates (representing the slightly length-

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. CC-BY 4.0 International license available under a was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprint (which this version posted July 4, 2022. ; https://doi.org/10. 1101/2022 To explore the cellular decisions behind these four growth rates, the response ). Therefore, we suggest that " 1 " is more directional than the " 2 " 401 (longitudinal growth). In summary, we propose that the molecular mechanism of the 402 length-first rate is mainly described by the graph " 2 ", the width-first rate is mainly 403 illustrated by the graph " 1 2 ", and the non-obvious rates are mainly shown by the 404 graphs " 0 " & " 1 " ( Figure 5F). Meanwhile, the experimental evidences of these 405 four molecular mechanisms have been indirectly confirmed in the following self-406 consistent analysis.

407
. CC-BY 4.0 International license available under a was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprint (which this version posted July 4, 2022. ; https://doi.org/10.1101/2022.07.04.498707 doi: bioRxiv preprint was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprint (which this version posted July 4, 2022. ; https://doi.org/10.1101/2022.07.04.498707 doi: bioRxiv preprint growth of yeast cells. Figure 2C includes the same additional annotations. 419 The online version of this article includes the following source data and figure supplement(s) for 420 Source data 1. The ratio data of the smallest filled circle to the largest filled circle in Figure 5C. 422 Source data 2. Cell growth rate statistics associated with Figure 5E. Experimental evidence for the molecular mechanisms 428 To validate the above-mentioned molecular mechanism, we investigated the link 429 between the growth rates and dual fluorescence gene expression states. By taking the 430 absolute value of the growth rate, we directly divided into the "Fast growth rate" and "High-state" fluorescence data should be completely included in the "Fast-state" 436 growth rate data, and the "Low-state" fluorescence data should also be completely 437 included in the "Slow-state" growth rate data. However, the behavior we have 438 observed is obviously not the case. When using the different colors to represent the 439 fluorescence states, we can clearly see that the low state (red dots), high state (blue 440 . CC-BY 4.0 International license available under a was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprint (which this version posted July 4, 2022. ; https://doi.org/10.1101/2022.07.04.498707 doi: bioRxiv preprint dots), and the overlapping data (green dots) are all included in the two growth rates.

441
The biological meaning of these overlapping data is the fast growth of the yeast in a 442 low-fluorescence state ("FL") and the slow growth of the yeast in a high-fluorescence 443 state ("SH").

444
For the dynamical understanding of the cellular decision-making, the mate- intensity had certain probability of belonging to the "Low-state", on the contrary, the 458 small fluorescence intensity also had certain probability of belonging to the "High-459 state" (Figure 6C). Therefore, the "FL" represents the state of "Fast" growth rate and 460 "Large-intensity" in the "Low-state" fluorescence; the "SH" represents the state of 461 "Slow" growth rate and "Small-intensity" in the "High-state" fluorescence.

462
. CC-BY 4.0 International license available under a was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprint (which this version posted July 4, 2022. ; https://doi.org/10. 1101/2022 To better describe the overlapping effect of the fluorescence data, we create a 463 schematic diagram that illustrates the mixed area of fluorescence intensity ( Figure   464 6D). Among them, the " ", " ", " " and " " respectively represent the 465 proportion of the state of "FL", "SH", "FH" and "SL" in the growth rate data; the 466 " " and " " is the number of the "Fast-state" growth rate data and the "Slow-state" 467 growth rate data respectively. Subsequently, we calculate the overlap ratio distribution was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprint (which this version posted July 4, 2022. ; https://doi.org/10.1101/2022.07.04.498707 doi: bioRxiv preprint intensity is fitted, the upper one is the "High-state", the lower one is the "Low-state"; "a" and "b" 485 are the two points in the "High-state" and the "Low-state" fluorescence intensity, respectively. (D) 486 Diagram depicting the relationship between data of various dimensions. The " : " represents 487 the ratio of the "Low-state" to the "High-state" of the fluorescence intensity in the "Fast-state" 488 growth rate data; the " : " represents the ratio of the "Low-state" to the "High-state" of the 489 fluorescence intensity in the "Slow-state" growth rate data; " " is the number of the "Fast-state" 490 growth rate data; the " " is the number of the "Slow-state" growth rate data. 491 The online version of this article includes the following source data and figure supplement(s) for 492 figure 6: 493 Source data 1. Statistics of the absolute growth rates of cells associated with Figure 6A. 494 Source data 2. The cell growth rate and fluorescence intensity data presented in Figure 6B. directions. Therefore, we propose that the cell morphology cannot be determined 513 solely by the action of a single growth pathway, such as " 1 2 ", " 2 ", " 1 ", " 0 " 514 ( Figure 5F), but should be the result of their combined action.

515
To understand the molecular mechanisms of the four cell morphological fates 516 more intuitively, we used the growth ability represented by the growth rate of the cells 517 in different directions to count the synergistic effect of the minor axis and major axis 518 of the cell ( Figure 7C). From the Figure, we can see that there are four distinct states 519 in the cooperative distribution of the two data, which strongly suggests that the ability 520 to grow from different directions is a key determinant forming the cell morphology.

521
. CC-BY 4.0 International license available under a was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprint (which this version posted July 4, 2022. ; https://doi.org/10.1101/2022.07.04.498707 doi: bioRxiv preprint Importantly, it is entirely due to the four combinations of these two growth abilities 522 that make the cells to form four stable morphological fates. These two growth abilities 523 show the comprehensive considerations of the yeast in the decision-making process.

592
Meanwhile, the dynamical process of the cell growth is simulated using the 593 Bni1p generated in the reactions. Although both the growth pathways (" 1 ", " 2 ") are 594 involved in the growth process of the cellular length and width, the weights of the 595 pathways that grow in the two directions are significantly different. Therefore, we 596 directly define the "Bni1_in" produced by the " 1 " that plays a major role as the 597 growth of the major axis, and the "Bni1_out" produced by the " 2 " as the growth of 598 . CC-BY 4.0 International license available under a was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprint (which this version posted July 4, 2022. ; https://doi.org/10.1101/2022.07.04.498707 doi: bioRxiv preprint the minor axis.

599
In this simulation, the denotes the radial length of the yeast cell, while the " " 600 denotes the transverse length of the yeast cell ( Figure 8A). Each iteration increases 601 the radial and transverse lengths of the yeast cell by " " and " ", respectively.

602
" " is proportional to " Bni1p_in" and inversely proportional to " ", meanwhile, 603 " " is proportional to " Bni1p_out" and inversely proportional to " ". That is The distribution of the cell morphology ( ) using simulation. 616 The online version of this article includes the following source data for figure 8: 617 . CC-BY 4.0 International license available under a was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprint (which this version posted July 4, 2022. ; https://doi.org/10.1101/2022.07.04.498707 doi: bioRxiv preprint

620
In this study, we used the biological and physical tools to quantitatively uncover 621 and understand the cell fate decision-making of the yeast after encountering sexual 622 deception. To understand the "psychological process" of its internal activities, a total 623 internal reflection microscope and a microfluidic system were used to observe the  The ascending phase of the Fus3p's curve is the "excitement period" to adapt to the 628 external stimuli, while the steady phase of the curve shows the "calming period" of 629 the yeast cell. Due to the "calming period" (adaptation steady period) of the yeast 630 during the "adaptation period" under the 0.6μM pheromone being relatively short, we 631 selected five concentrations that can capture "enough attentions" by the yeast as the 632 external stimulus conditions.

633
If the yeast, like other higher organisms, becomes "angry" after encountering 634 sexual deception, is its subsequent polar growth an irrational behavior? Some studies 635 believe that the establishment of polarity helps organisms to survive better in nature the "excitement period", it seems that the description of the polar growth as searching 642 for a mate is a rational behavior at this time. However, how to understand the 643 "psychological state" of the polar growth after the yeast has restored to the "calm"? 644 We found that the cellular decision-making landscape was presented during the 645 "calming time". When the Fus3p is in the low state (L 1 , L 2 ), the yeast cell will stay in 646 place to wait for mating. In the high state (H 1 , H 2 ), the yeast cell will actively grow in 647 polarity to look for its mate. This switching between the waiting for mating and 648 looking for mating indicates that the decision-making of the yeast cell is "sane" rather 649 than impulsive behavior after being "angry".

650
When the yeast is determined to switch its decision from one to another, how 651 strong is the resistance to switch, how fast is the conversion and how long is the 652 waiting time for each behavior? To understand these problems that the yeast needs to 653 face, we further quantitatively explored the barrier height, switching rate and the 654 residence time in the decision-making landscape of the yeast at different doses 655 (Figures 4B and C). Comparing the various shapes of the landscapes, the intuition 656 that the greater the barrier height characterizing the potential landscape topography is, 657 the longer the residence time of the state is (Fang et al., 2019; Wang et al., 2011), can 658 be fully confirmed by our experimental data.

659
In the process of the polar growth, which direction should be grown and how fast 660 it should be growing are questions that the cell has to think about. To explore this 661 . CC-BY 4.0 International license available under a was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprint (which this version posted July 4, 2022. ; https://doi.org/10.1101/2022.07.04.498707 doi: bioRxiv preprint cellular decision-making of the yeast when looking for its mate, the growth direction 662 is divided into the longitudinal growth and the lateral growth, and the growth rate is 663 divided into the fast-speed growth and the low-speed growth. We suggest that the 664 underlying molecular mechanism of these decisions is due to the different response was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprint (which this version posted July 4, 2022. ;https://doi.org/10.1101https://doi.org/10. /2022 high doses (3μM). The non-monotonic trend of the characteristics such as the average 684 cell length and the overlap rate infers that the "psychological activity" of the yeast is 685 awakened gradually at the high doses, which is quantitatively confirmed by this net 686 flux study. Considering that the higher the dose, the longer the "calming time", it 687 seems that even if the polar growth was purposely reduced, the yeast still did not give 688 up waiting for mating. was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprint (which this version posted July 4, 2022. ; https://doi.org/10.1101/2022.07.04.498707 doi: bioRxiv preprint  The stimulus source we chose is pheromone, an alpha factor peptide hormone, 713 whose molecular weight is 1683.98 (GenScript). We weigh a certain amount of 714 powdered pheromone and dissolve them in YPD and YNB liquid medium 715 respectively to prepare 1000μM pheromone medium, and then gradually dilute it to was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprint (which this version posted July 4, 2022. ; https://doi.org/10.1101/2022.07.04.498707 doi: bioRxiv preprint time is the quotient of the total residence time and the number of state changes. was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprint (which this version posted July 4, 2022. ; https://doi.org/10.1101/2022.07.04.498707 doi: bioRxiv preprint are two types of cells. One is the original undeformed cells (the initial mother cells at 818 the stage of "0-a" & the new daughter cells at the stage of "b-c" in the Figure 2B). 819 The other is the deformed cells (the deformed mother cells at the stage of "a-c" in the 820 Figure 2B). 821 Through the statistics of the data stimulated by different pheromone was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprint (which this version posted July 4, 2022. ;https://doi.org/10.1101https://doi.org/10. /2022 coefficient" of the intranuclear and extranuclear fluorescence intensity will be 0 (Lee 846 Rodgers and Nicewander, 1988), which provides the experimental evidence that the 847 ( , ) and ( , ) states do not appear.

848
Derivation of the transition rates 849 The master equation can be written as 850 .

851
Here P 1 and P 2 are the probabilities of the low expression state and high expression 852 state, respectively, while (i, j=1, 2) is the transition rate from to . We can

868
So we can get the transition rate as:

869
. CC-BY 4.0 International license available under a was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprint (which this version posted July 4, 2022. ;https://doi.org/10.1101https://doi.org/10. /2022  . This is because bears a resemblance to the harmonic mean.  the "P 1 " is more directional than the " " (longitudinal growth) ( Figure 3C).

895
. CC-BY 4.0 International license available under a was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprint (which this version posted July 4, 2022. ; https://doi.org/10.1101/2022.07.04.498707 doi: bioRxiv preprint which explains why the two are neither independent nor exactly correlated. The other most important reason for above is the overlapping effect of the dual fluorescence 925 states. From the fitting process of the fluorescence two-state, we know that there is 926 certain amount of mixed data in the main part of the "two hills" (Figure 3B). In the 927 overlapping area of the "two hills", the fluorescence intensity of the "High-state" is 928 not necessarily higher than that of the "Low-state", for example, the fluorescence 929 intensity of the "b-point" in the "Low-state" is greater than the "a-point" in the "High-930 state" (Figure 6C).

931
In order to better describe the overlapping effect of the fluorescence data, we 932 draw a schematic diagram to explain the mixed area of the fluorescence intensity 933 ( Figure 6-figure supplement 4). Meanwhile, if we want to distinguish the state of 934 the fluorescence intensity in the overlapping area (such as the "a" intensity in the 935 Figure 6C), we need to compare the probability that the intensity belongs to the high 936 state or the low state, that is, or multiplied by their respective transition 937 probabilities. As a result, the large fluorescence intensity has certain probability of 938 belonging to the "Low-state", on the contrary, the small fluorescence intensity also 939 has certain probability of belonging to the "High-state". Therefore, the "FL" 940 represents the state of "Fast" growth rate and "Large-intensity" in the "Low-state" 941 fluorescence; the "SH" represents the state of "Slow" growth rate and "Small-942 intensity" in the "High-state" fluorescence. Where represents the probability of state , and the transition probability 950 represents the transition probability from state to state . For steady state, we set the 951 . CC-BY 4.0 International license available under a was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made , which is the long time limit. The steady state flux between state and can be 953 defined as: . If the steady state of the system is in equilibrium 954 state, the flux between any two nodes in the system is zero, that is the detailed balance 955 condition. For the general biological system, it does not necessarily satisfy the 956 detailed balance condition ( 0), the system is in non-equilibrium steady state, 957 there will be at least one net flux among states.

958
In order to study the non-equilibrium steady states, we can separate the 959 dynamical process into two parts, one is the detailed balance part and the other is the 960 detailed balance-breaking kinetic process.
Then, solving these linear equations one can get the net flux values of the non- was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprint (which this version posted July 4, 2022. was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprint (which this version posted July 4, 2022.