Transition Yet Energy Consuming for Tumor Growth Regulated by the Colored Microenvironment

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Introduction
Transition or phenotypic switching is a remarkable characteristic of tumors.After undergoing multistep processes so-called invasion-metastasis cascade, the tumor achieves metastasis and propagates from primary tumor cells to distant organs [1][2][3][4].Primary tumor cells frst locally invade normal tissues surrounding them and then will proliferate and colonize at a distant site by intravasation into and extravasation from the systemic circulation; here, the metastatic cells also depend on an often-foreign cell microenvironment [5][6][7], in which environmental factors often play critical roles in the sense of stochastic bifurcation by regulating each biochemical step of the tumor cell development, including mediating local invasion, cooperating some competing endogenous, and activation of the signal pathway.Much progress has been made in understanding the relationship between tumor transition and environmental white fuctuations (a classical framework of thermodynamic fuctuations or Brownian motions [8][9][10][11]).However, the correlation between internal and external stochastic fuctuations has a substantial time comparable to the cell cycle (these fuctuations are always called "colored") [12][13][14][15], and their interplay interferes with classical thermodynamic equilibrium.Despite this general description, how the process of tumor transition and metastasis is achieved and how much energy it costs are all elusive, moreover, the roles of regulation from colored noise sources in tumor development, including the interactions of AT/CT (auto/cross-correlation time) of colored noises, remain not fully understood.
In general, cell fate and decision are determined by cellular phenotype defned as the number of peaks in the steady distribution of attractors in a phase plane, and tumor cell development will switch between these attractors in line with stochastic bifurcation regulated by the fuctuation microenvironment [16,17].In the landscape of tumor development, the attractor may exist in a discrete form in space, and the tumor cell fate depends on the dwell time of each attractor, which is a main source of stochasticity or noise [18][19][20].Te classical thermodynamic balance follows the classical Markov assumption and ubiquitous principle; that is, the dwell time of the system in every attractor follows exponential distribution and has no memory [21][22][23][24].However, the results of recent biological experiments on cancer metastasis have indicated that the long noncoding RNA (lncRNA) MALAT1 regulates tumor diferentiation (as evidenced by cystic and encapsulated tumor appearance) by localizing to nuclear speckles and altering transcription based on initial descriptions [25][26][27].Te fuctuations in the abundance of MALAT1 form an external signal to regulate the system dwell time into a nonexponential distribution and then alter the promoter activity, resulting in the memory (self-autocorrelation).Tis implies that the gene expression noise is in nature colored (i.e., the noise has nonzero correlation time) [26,28].From the perspective of biological function, Shahrezaei and Swain verifed that the memory from colored noise can improve response sensitivity, by amplifying stochasticity in coherent feedforward loops while attenuating noise in incoherent feedforward loops [22].Tis means that the AT/CT may act as a "fne-tuner" of complex cellular systems to regulate cell phenotype and transition.Essentially, the colored fuctuation environment breaks the Markov assumption, leading to a non-Markovian jump process.Deciphering efectively how the AT/CT regulates cell phenotype switching is an important and challenging task [29][30][31].Fortunately, Novikov's theorem provides an excellent method to transform a complex system embedding in a colored microscopic environment into the Markov jump process to address the efect of the AT/CT on cell phenotypic diversity, which is more directly efective than the stationary generalized chemical-master equation (sgCME) [30,[32][33][34].Hence, it is signifcant to study the efect of colored noise (focusing on AT/CT regulation from the time viewpoint) on tumor cell transition.
Essentially, the existence of AT/CT may induce the change of dwell time of every attractor and render the jump process between distinct attractors to produce the molecular memory, breaking the detailed balance in tumor development.Tese transitions between metastable states need to dissipate the free energy [35][36][37][38][39]. Here, free energy is a physical concept described as the work done by the transport of a bead from one well to the other and back, measured by entropy production that illustrates the irreversibility of the jump stochastic process according to Landauer's principle [40][41][42][43].Te decomposition of circumfuence is a key step for calculating entropy production.However, in the tumor development process, AT/ CT of fuctuations originating from the abundance of MALAT1 may have a nonzero time lag in every attractor, rendering cycle fow decomposition difcult.Here, investigating the equivalence from the mesoscopic scale, we propose the approximation algorithm to estimate the equivalent switching rate among the distinct attractors by employing the technology of a linear mapping approximation [40,41,44,45].Te advantage of this algorithm is that we can yield directly the probability net fux of every attractor but also obtain the entropy production rate for a given complex multistable system [44,45].Terefore, clarifying how much free energy is consumed from the viewpoint of nonequilibrium and emphasizing the regulated function of AT/CT for achieving tumor cell transition and metastasis are important for dynamic intervention in tumor development.
Inspired by the signifcant experiments on the lncRNA MALAT1in lung cancer metastasis [26,46], we introduce a mechanistic model with a colored microenvironment due to fuctuations in the abundance of MALAT1 to emphasize the regulation of AT/CT on tumor transition and metastasis.Moreover, we mainly focus on the mechanism of how to achieve the biological function (phase switching) and what cost is involved, beyond the previous reports only on the stochastic resonance [47,48].According to the dynamic analysis in multitime scales, we could uncover that the AT/CT and NS/CS can regulate tumor cell dynamic evolution process determined largely by the types of noise.For the landscape of tumor development, amplifying the NS of additive noise can suppress tumor cell metastasis, while enhancing the AT of multiplicative noise may have a dual function, that is, it can not only induce the tumor cell switching but also promote the killing of cancer cells; Te biological function is achieved by mediating the stochastic bistability regime triggered by the NS and AT.Te CS of the noises can also have a double function, depending on positive or negative values, i.e., enhancing the CS (from a negative correlation to a positive correlation) can induce the MFPT from the primary tumor to the malignant tumor to have a minimum value and a maximum value, but the CT always has a minimum value.More importantly, the development state of tumor cells jumping between distinct attractors needs to dissipate more energy with increasing the CT in a multiplicative environment, and there exists a trade-of by regulating NS and AT in the additive noisy environment.In addition, the energy consumption is monotonically decreasing with increasing CT, and the CS can promote this dependence relationship.
Tis, then, signal sensing is the most ubiquitous process in tumor cell development, the question of what mechanism is behind the complex regulation, including how to induce the phenotypic switching in a noisy environment and what cost to consume, is all not previously known.Focusing on these hotspots is important for elucidating the mechanism of tumor cell metastasis and dynamic evolution.

Model Description.
Here, considering the attack of immune cytotoxic cells, we model the growth process of cancerous tissue, to focus on the NS and AT/CT of the colored stochastic fuctuations, on tumor cell development.It is known that, due to the existence of the immune T lymphocytes (efector cells), the development of reducing tumor cells (target cells) may be simulated as a biochemical process including the following reactions [49,50]: where X denotes the tumor cell that proliferates spontaneously at a rate λ, and their local interactions with cytotoxic cells E 0 (efector cells) are illustrated by a kinetics parameter k 1 , implying the rate of binding of immune cells to the complex E that subsequently dissociates at a rate k 2 , refer to Figure 1.Also, this dissociation results in a product P that represents dead or nonreplicating tumor cells.k 3 is the degradation rate of P and A is the normal cell carnifcation coefcient.Tis simple motif, which can be used to model many motifs in response to various possible cell microenvironments [51][52][53], has been extensively studied.
To introduce noisy sources in the above tumorgrowth model, we simply introduce the background of the small molecule MALAT1 of the lncRNAs, the frst one found in cancer metastasis.Tis small molecule is the main biomarker cell in lung cancer metastases [2,25] and can regulate tumor diferentiation (e.g., the development of cystic and encapsulated tumor appearance) by localizing to nuclear speckles.Recent experimental results have indicated that in the mouse mammary tumor virus-(MMTV-) polyomavirus middle T antigen (PyMT) model of human luminal B breast cancer, promoter deletion or ASO-mediated knock-down of MALAT1 can result in the decrease of lung metastases and the elevation of E-cadherin (supporting an epithelial phenotype).Most cancer cells are likely kept in blood vessels compared with control cells, implying that the cancer cells do not invade distal lung sites form micrometastases, i.e., inefective colonization following MALAT1 knockout [7,46], and also implying that the tumor cell transition is indeed due to the colored noise from fuctuations in the MALAT1 abundance (specifcally, these fuctuations are due to lncRNA MALAT1 localizing to nuclear speckles and may be modeled as dichotomous noise that has been proved to have some memory [54,55]).Here, we try to answer the traditional issues of whether noise is harmful or benefcial for cell development, emphasizing how AT/ CT and NS afect the phenotype shift and metastasis of tumor cells.

Mathematical Models.
Without loss of generality, the number of immune cells satisfes the conservative condition: Ten, the tumor development described by equation ( 1) is essential in a series of ordinary diferential equations (ODEs) that demonstrates the transient evolution process of each species (for more details, refer to Appendix A).
where N stands for the maximum number of normal cells, the resulting kinetics can be dimensionless directly by setting and equation ( 2) has the following equivalent forms: In general, the parameter β > 1 holds, meaning that the product of the rate of efector cell binding to cancer cells, and gives a general assumption that the total number of conjugate cells per unit volume is greater than the rate of cancer cell proliferation.Terefore, we can divide the whole tumor cell development network into three types of motif: recognition, apoptosis procedure, and apoptosis [1,3,4,56], meaning there may be a huge diference in time scale between the three processes.We also can treat the system as the coupled of fast and slow subsystems [42,[57][58][59][60][61].According to the method of separating time scales and quasi-steadystate approximation [31], we can yield a steady probability distribution for biomarker protein (Figure 1(a)).Here, we emphasize mainly the concentration of the tumor cells.Let the second and third of (3) be equal to zero, we can easily get the expressions of steady state y, z in terms of x.If these expressions are substituted into the slow equation, then we yield the following reduced system (for more details, refer to Appendix A): where x represents the actual concentration of the tumor cell, β is the immune rate, and θ is the constant parameter.Obviously, this motif, coupling a positive and negative feedback loop, would have two steady states depending on diferent parameter values.In addition, the deterministic potential function defned by the deterministic force f(x) in ( 4) is given by Complexity 3 from which we obtain two steady attractors x 1 � 0 (the extinction state of the tumor) and

􏽱
)/2θ > 0 (the stable state of the tumor), and one unstable steady state

􏽱
)/2θ.Of note, the system parameters are set according to the recent experiment on lung cancer evolution [26], and the values are listed in the note of numerical results.Using these data, we can show that there are two stable attractors in the given system and its potential function has double wells (Figure 2).Tese two stable steady states correspond, respectively, to the tumor cell extinction state (x 1 � 0) and the stable tumor state (x 2 � 7.2659), and the unstable state 1.7341.Moreover, the system has a bistable region in the space of parameters.
Next, we introduce a stochastic model.Of note, the stochastic environment originating from the fuctuations in the MALAT1 will directly infuence the tumor number and alter the tumor's immune rate, being the main source of extrinsic noise.Also, this extrinsic noise may be colored noise for the reasons stated above.Te fuctuations afecting the immune rate β result in multidimensional noises, that is, additive noise η(t) and multiplicative noise dx/dt � x(1 − θx) − βx/x + 1 − x/x + 1ξ(t) + η(t), which can be viewed as the capability for expansionary transfer of the tumor cells.Te consideration of these factors combined with deterministic equation (4) leads to the more complex forms as follows: In equation (6), two noises are all assumed to be Gausscolored noises and nonzero autocorrelation, that is,  4 Complexity Here, symbol 〈•〉 represents the mean or expectation, ∆t is the time window, τ 1 , τ 2 are AT, D, α are the NS, and λ, τ 3 are CS and CT, of the corresponding noise process.
Note that equation ( 6) is used to describe the tumor development as shown in Figure 1.Te existence of colored noise embedded in tumor development impels the dwell time in each attractor to drift from the classical Markov assumption; that is, it is impossible to obtain directly the stationary probability distribution by the two types of the classical stochastic integral.Obviously, the dwell time in each attractor of nonexponential distribution means that the gene expression must have some so-called molecular memory [26,28,[30][31][32].In particular, if τ i ⟶ 0, the colored noise degenerates directly into the Gaussian white noise, and we can solve the above equation by the Ito stochastic integral.In fact, analytical results were obtained for similar cases [31,[62][63][64].Here, we investigate mainly the biological function of colored noise, i.e., the efects of this noise on phenotypic transition and stability of tumor cells.

Analytical Results: Te Steady Distribution and Its Peaks.
Generally, a cell's phenotype may be partly refected by the number of peaks and their steep degrees in a stationary probability distribution.Considering the infuence of the noisy environment on tumor cell development, we employ Novikov's theorem [65,66] to transform the non-Markov process into the approximation Fokker-Planck equation (aFPE) to emphasize the regulation of AT on tumor phenotype [65,66].According to equations ( 6) and ( 7), we can derive an aFPE (refer to Appendix B for derivation).Te parameters are from the experiment on lung cancer [26], and θ � 0.1 s-1copy-1 and β � 2.26 s-1copy-1.Here, x 1 � 0 nm is the extinction state of the tumor cell, x 2 � 7.2659 nm represents the tumor state, and x u � 1.7341 nm is the transient state. where | x�x s denotes the derivation of force at the attractor x s , and it must satisfy the inequality 1 ) that limits the ranges of the correlation times, and Equation ( 8) can be rewritten as where with According to equations ( 10)-( 13), the steady distribution where N is a normalization constant and the stochastic potential function x u is given by From this equation, we can obtain the following explicit expression of dP st (x)/dx � 0: where Te stochastic potential given by equation ( 16) can provide the information of stability of the corresponding probability distribution, similar to the case of the potential of the deterministic system described by equation (5).Here, we are interested in understanding how colored noise can induce the transition between stationary states.For the deterministic system described by equation ( 4), this induction can be achieved by analyzing the existence of a critical point or bifurcation point (x u ) and its stability in a given parameter interval.In the stochastic case, however, the stability of the steady distribution can be illustrated by the number of peaks of the corresponding stochastic potential function, obtained from the equation dP st (x)/dx � 0 [9].Moreover, we fnd that the bifurcation condition can also be written as the following algebraic equation:

Tumor Phenotypic Diversity from the Interplay of AT/CT and NS.
Te tumor development process is essentially noisy and exhibits distinct phenotypes.Recently, the results on lncRNA MALAT1 have indicated that correlation function AT/CT is changeable, implying that we can use this changeable AT/CT in tumor development to achieve phenotypic diversity [67,68].In order to investigate the regulation efect of colored noise (i.e., D, α, λ, τ 1 , τ 2 , and τ 3 ) on state transition, we analyze the equations of probability distributions (i.e., equations ( 10)-( 15)).For clarity, we distinguish three types of cases: (i) the noise is present only in the immune rate, i.e., only multiplicative noise; (ii) only additive noise; (iii) two kinds of noisy sources are simultaneously present.In Table 1, we present these three cases of the parameter values.

Efect of Multiplicative Noise on State Transition.
Here, we consider that the correlated Gaussian noise ξ(t) appears in the immune rate in equation ( 6), i.e., the corresponding stochastic equation takes the form From this equation, we can show that a small fuctuation in the immune rate may result in a sudden transition because of the appearance of multiplicative noise.Fixed control parameters θ and β, changing the multiplicative noise ξ(t), may reshape the phenotype of stationary probability distribution (equation ( 14)), referring to Figure 3. Tis fgure demonstrates that the steady distribution P st (x) that is in the form of a function of the density of tumor cell x may change its shape with changing the colored noise environment.Specifcally, the probability of a stable tumor state (high state) reduces with increasing AT (the blue to green line), implying that increasing the AT may help to kill the cancer cell.Meanwhile, the probability at extinction state (low state, i.e., x � 0) falls rapidly.In Figure 3(b), the AT has a similar but more signifcant efect on the stationary probability distribution when the NS increases from 0.2 to 0.7.Tis means that a larger NS can amplify the efect of AT on cell phenotype.In fact, comparing Figures 3(a) with 3(b), it is obvious that the probability of the unstable attractor increases gradually with increasing the AT, and the NS can make this diference more apparent, implying that the AT is a main factor promoting the tumor cells to difuse and the NS accelerates the difusion.In a word, the multiplicative noise may have a dual function in tumor cell development when the AT and NS change; that is, it can promote directly tumor cell difusion and also contribute to killing the tumor cells at the same time.
From equations ( 18) and ( 19), we can show that the extremum values of the steady distribution satisfy the following condition: According to this equation, we plot the extremum of the stationary probability distribution function as a function of the immune rate β as shown in Figure 4.
After determining the extrema of the stationary distribution function, we next investigate critical transition (i.e., fnd the tipping point at which abrupt state changes in the tumor cell development), which can be achieved by changing AT and NS. Figure 4 indicates that the efect of the NS and AT on the stationary distribution is nearly opposite [70].Specifcally, when the AT is fxed at 0.01, increasing the    [69].Te other parameter values are θ � 0.1 s − 1 copy − 1 and β � 2.26 s − 1 copy − 1 , which satisfy the bistable condition in Figure 2.
Table 1: Parameter values of colored noise correspond to three types of cases: case (i) only multiplicative noise, case (ii) only additive noise, and case (iii) the mixture of multiplicative noise and additive noise.(19) subjected to the noise only in the immune rate β, i.e., only multiplicative noise is considered.(a) For a fxed AT τ 1 � 0.01, increasing the NS from 0.2 to 2 to 4 (from red line to green line to blue line); (b) for a fxed NS D � 2, increasing the AT from 0.01 to 1 to 10 (from green line to blue line to red line).Te other parameter value is θ � 0.1 s − 1 copy − 1 , and the dashed line denotes an unstable attractor.8 Complexity NS from 0.2 to 2 to 4 (referring to Figure 4(a), from red line to green line to blue line) may broaden the bistable regime due to changes in the position of the unstable attractor (refer to the dashed line).Tis would hint that the fact that the regulation of the NS may be equivalent to that of positive feedback in the case of gene expression since positive feedback may induce bistability [30].However, the way of phenotype diversity by regulating AT would be diferent.Figure 4(b) shows that when the NS is fxed at 2, the bistable regime of the underlying system reduces with increasing the AT from 0.01 to 1 to 10 (referring to Figure 4

Efect of Additive Noise on State Transition.
In contrast to the case of multiplicative noise, we can more directly show the biological function of additive noise.Keeping the NS fxed and increasing the AT reduce substantially the probability of tumor state but increase the probability of extinction state (Figure 5), implying that the regulation of AT in an additive noisy environment may inhibit the development of tumor cells.However, upon combining Figures 5(a) with 5(b), it can be observed that the changing trend of the stationary probability distribution is nearly the same even if the NS changes from 0.2 (Figure 5(a)) to 0.7 (Figure 5(b)), and the probability of extinction state shows a larger increase in Figure 5(a) than that is in Figure 5(b), indicating that when the NS is small, the phenotype of the system is more sensitive to the AT, and the NS may attenuate the efect of the AT.

Joint Efect of Multiplicative and Additive Noise on State
Transition.Here, we analyze the Langevin equation (6) where multiplicative noise and additive noise are simultaneously present, i.e., Case (iii).In this case, we need to consider the CT λ between multiplicative noise and additive noise.Tis consideration involves feedback regulation and nonlinear factor (see equation ( 6)), i.e., the tumor cell concentration is chemically coupled to the immune rate β [7,46].Moreover, in lncRNA MALAT1 regulation, the noise of these two kinds would be not independent, so there would exist some correlation between them [26].In fact, the analytic solution for aFPE (equations ( 10)-( 17)) is the function of CT (τ 3 ) and CS (λ) between the two correlation noises ξ(t) and η(t), as well as AT (τ 1 τ 2 ) and NS (D, α) (referring to the efective drift and difusion terms in aFPE (equations ( 10)-( 13)).Focusing on the regulation of CT and CS on the stationary distribution, we plot Figure 6 to decipher how the phenotypic switching occurs by varying CT and CS.We observe from Figure 6 that by increasing the CS from − 0.5 to 0.5, the probability of tumor state increases while the probability of extinction state reduces to a smaller value if other parameter values are all fxed, indicating that the CS may promote the tumor cell expression (tumor state).Hence, the positive CS means that it positively regulates the tumor cell expression, but the negative CS means that it promotes the tumor cell extinction (referring to Figure 6(a)); that is, the CS may be taken as an efective factor controlling tumor development.
Te efect of the CS on cell phenotype is twofold; that is, when the multiplicative (ξ(t)) and additive (η(t)) noise is positively correlated (e.g., λ � 0.2, referring to Figure 6(b)) and the CT increases from 0.01 to 2 (blue to red and then to green line), the probability of extinction state decreases to near zero whereas the probability of high expression state increases.Tis implies that CT is a positive factor in tumor development since it promotes tumor transition and metastasis.In contrast, the negative correlation of external noise (i.e., CS < 0) has the opposite efect on tumor cell development.Figure 6(c) demonstrates that with increasing the CT, the probability of the tumor state decreases (from the blue line to the red to the green), but the probability of the extinction state has a signifcant increase.Combining Figures 6(b) and 6(c) together indicates that the development of tumor cells depends largely on the correlation of diferent external noise sources.Tis may explain why diferent therapeutic efects appear in the same immunotherapy.In addition, there may exist a trade-of between regulating CT and CS in tumor development, and it provides a new way to achieve cell phenotypic diversity in that CS dynamically regulates the tumor cell phenotype.

Te Controllability of Mean First Passage Time (MFPT).
Decoding the controllability of MFPT could quantify the efects of noises on tumor cell metastasis and transition between alternative attractors in landscape space.Here, we focus mainly on the regulation of AT/CT on tumor development to decipher the biological function of noises.According to the Krame formula, we can obtain the analytic expression of MFPT T 1,2 � T x 1 ⟶x 2 of the system jumping from one attractor x 1,2 to another attractor x 2,1 by using the steepest descent method [71,72], that is, and similarly, Complexity where V denotes the potential function defned by equation ( 5) U denotes the stochastic potential function (equation ( 16)), M is NS of the corresponding noise, and x 1 denotes the extinction state of tumor cell, while x 2 denotes the tumor state and x u denotes critical point that is also an equilibrium state for tumor cell development (Figure 2).Tere exist natural restrictions on the energy barrier height for changeable parameters (i.e., D, α, λ, τ 1 , τ 2 , τ 3 ), that is, max(D, α) Te numeric results of MFPT between the distinct attractors are given to exhibit the regulation of the interplay of two noises (see the above three cases).First, we investigate the efect of a single noise source, i.e., the system is in Case (i) or in Case (ii), referring to Figure 7, where Figures 7(a) and 7(b) illustrate how increasing AT (τ 1 ) and NS (D) of multiplicative noise regulates MFPT.Specifcally, Figure 7(a) shows that the MFPT is monotonically decreasing with increasing the AT and NS with the tumor cell transition to the tumor state (x 2 ) from the extinction state (x 1 ).Also, Figure 7(b) demonstrates the inverse change tendency, and from this fgure, we clearly see that the MFPT is monotonically increasing with extending AT and NS (referring to Figure S1).Meanwhile, the change of the MFPT in an additive noisy environment is nearly the same as the Terefore, the MFPT can be viewed as a monotonic function of the AT and NS, implying that a noisy environment may promote the state switching from the extinction state to the tumor state but suppress the inverse process.In other words, cell canceration is easier than its treatment and would need enough cost (for example, consumption of energy) to achieve the treatment.Furthermore, by comparing the subplots of Figure 7, we can see that the system has a faster response rate in the multiplicative noise case as shown in Figures 7(a) and 7(c) than in the additive noise case as shown in Figures 7(c) and 7(d) due to the diference in steepness of curves, indicating that the tumor is more sensitive to multiplicative noise than additive noise or that the multiplicative noise environment would more easily induce tumor cells.
Next, we investigate the efect of two interplaying noisy sources on the MFPT.For this, consider the system of Case (iii).It shows that there is a signifcant diference in the MFPTs between the single noise and Case (iii) (Figure 8).Specifcally, MFPT illustrating the extinction state to the tumor state is no longer a monotonic function but exhibits diferent behaviors, refer to Figures 8(a)-8(d).Figure 8(a) indicates that the MFPT has a peak and decreases with decreasing the CS.In fact, if the CS (λ) is fxed, the MFPT is frst an increasing and then a decreasing function of the NS (D) (referring to Figures S2(a)-S2(d) for more details), i.e., the MFPT increases when the NS (D) is small but MFPT decreases when the NS (D) is greater than about 0.05.Tis maybe explains why the CS has double functions on cell phenotype (referring to Figure 6).However, MFPT is a monotonic increasing function of the CS (λ) with fxing the NS (D), meaning that the negative correlation between two noise sources can be viewed as an efective enhancer for achieving cell tumorigenesis due to the small MFPT.In contrast, the positive correlation between the two noise sources can be viewed as an efective repressor for inhibiting cell tumor generation due to the increasing MFPT.
Moreover, we consider the interplay between the CS (λ) and the NS (α). Figure 8 shows that the MFPT has minimum and maximum values, respectively, when the two noises are positively correlated (e.g., when λ is approximately equal to 0.5).Furthermore, the minimum value may vanish with decreasing the CS, referring to Figure 8(c).However, the CT (τ 3 ) always keeps the minimum value (referring to Figures 8(b) and 8(d) and Figure S2 wherein more details are shown), meaning that the time infuencing the duration of the CT (τ 3 ) is longer than that of infuencing the duration of the CS (λ).As is well known, the CT is essentially a time lag between two noisy sources (implying that the process has memory).Terefore, the above result implies that the memory efect may persist longer than limited stimuli.Te combination of Figures 8(a)-8(d) indicates that the MFPT decreases when the NS (D, α) is large enough, meaning that the cells are easier to switch to the tumor in a large noisy environment.
For tumor treatment, we consider the reverse switching process of the tumor development in the complex noisy environment, referring to Figures 8(e)-8(h).Note that because of the interplay of the CS (λ) and the NS (D, α), the dynamic behavior of the MFPT all depends on the CS (λ); that is, when the correlation of both noises is positive, the MFPT is frst an increasing and then a decreasing function of the NS (D, α), but when the correlation is negative, the MFPT is always a decreasing function of the NS (D, α) (referring to Figures 8(e) and 8(g) and Figures S2(e) and S2(g) wherein more details are provided), meaning that the tumor development can exhibit diferent dynamic characteristics, which would explain why the stationary probability distribution has quite diferent change trends regulated by the CS (Figures 6(b) and 6(c)).Although the absolute value of the MFPT is less regulated by the NS (α) than by the NS (D) (referring to Figures 8(e) and 8(g), the duration time of the peak value is longer and the response is slower than in the case of additive noise.Te cost of time lag for the underlying system would be fatal to tumor treatment [3,4].Moreover, the regulation by the CT may amplify the time diference, referring to Figures 8(f ) and 8(h), which shows that the curve of the MFPT has a maximum value but the surface has a smaller curvature and a larger steepness than the case of additive noise.Terefore, the tumor state switching from a high state to a low state is more sensitive to multiplicative noise, but whether this is a reason for treatment needs to be confrmed by clinical tests [3,53,73].

Energy Consumption of Phase Transition Regulated by AT/CT.
To dissect the internal driving mechanism of tumor transition or metastasis, we further investigate what is the cost of keeping the tumor stable and whether there is a costbeneft relationship for tumor development.Generally, energy consumption, measured by entropy production rate (nonnegative value), can directly measure the degree of far from equilibrium [40,43].Te nonequilibrium or irreversible character of the stochastic process is in two ways, that is, by contact with heat baths at distinct temperatures other than one heat bath or by nonconservative forces [43,74].Here, the aFPE equation of the tumor system is complex (equation ( 10)), and the difusive coefcient B(x) is the function of the position of the particle, also regulated by NS/CS and AT/CT from the fuctuations in MATLAT1 abundance.Moreover, the fuctuation of colored noise or the existence of memory induces the system into a non-Markovian.Te key step of decomposing the fux fow in distinct attractors for measuring energy consumption is to become difcult [74,75].Terefore, it is necessary to propose a method of an efective topology network to calculate the fux fow in each attractor.
Te particle jumping in the phase space is essentially mesoscopic stochastic if the interval time is enough small.It is not suitable to use the framework of the fuctuation-dissipation theorem; here, we directly model this process in the form of a chemical-master equation (CME) to calculate the fux cycle for every steady state [76,77].Simply, we can compare the steady distribution of the aFPE system (equation ( 14)) with the stationary solution of the classic two states CME to reconstruct the biochemical reaction network in mesoscopic time scale.12 Complexity In this way, we can obtain an efective topology network for the tumor development system, and there are at least three advantages; that is, (i) it grasps the main character of our system, including switching between two stable attractors, stochastic gene expression, and some regulators from the environment, to obtain a very simpler calculating form for measuring energy cost than other results of entropy production for nonlinear FP equations, referring to Figure 9; (ii) we can unify four diferent types of time scales, including the force system, two types of noises, and their interaction; (iii) we transform the non-Markovian process into the framework of Markov  Complexity process, and it is possible to quantify the regulation of the noisy environment.
Without loss of generality, we defne the transition rate between two stable equilibrium states to be equal to the reciprocal of MFPT and propose a stochastic system with the feedback loop, referring to Figure 9(a).Te topological equivalent structure is illustrated by the following CME, which is renewed of the system (3) on a mesoscopic scale.
where P off (m, t) and P on (m, t) denote the two factorial probabilities that the gene is in the corresponding states.
According to the conservative principle of probability, we have P � P 0 + P 1 , also given the statistical distribution of distinct attractors.It is worth mentioning that model ( 23) is indeed a Markovian process [22,78], illustrating the particle jumps and resides in the neighbor hall of stable attractors.Te analytical solution of CME ( 23) is difcult because of the existence of nonlinear feedback, even if some simplifed results have been reported [22,78].Four sets of parameters, transition rate, birth rate, degradation rate, and feedback strength, are included in the mesoscopic system.Here, applying the Magnus expansion constraint to the conditional mean-feld approximation [44], we employ the linear mapping approximation (LMA) to simplify the CME equations (the main goal is to eliminate the efect of the nonlinearity due to the existence of feedback term) (referring to Appendix C) and to estimate efectively the parameter values.To do so, we can obtain the equivalent transition rate between two attractors σ b � σ b 〈n p n g 〉/〈n g 〉 by setting the steady moment equation to zero and have Tus, equation ( 19) can be reduced to yielding a steady distribution where and is a generating function corresponding to the probability distribution.
Te frst principle tells us that the equivalent network (equation ( 23)) must have the same analytical solution as that of equation (14).Equation ( 26) provides an efective theoretical framework to obtain the estimated value for parameters in the prespecifed error bound by the nonlinear ftting [64,65].For the related results, one may refer to Figure S3 in Appendix E. Numeric solutions indicate that the ftting with the theoretical solution of equation ( 14) is well and the error is controlled.Terefore, we also calculate the factorial probabilities P off (m, t) and P on (m, t), illustrating the probability the particle dwells in each attractor.In this way, we can yield an efective Markovian jump process with discrete phase space, labeled by i { }.Te transition probability of the state jumping from i to j is denoted by k(i, j), and the stationary distribution of state i is denoted by P(i), and we can yield directly the energy cost EP (also the entropy production rate) as follows [40,43]: Substitution into the equation ( 26), we have where P off (m) and P on (m) represent the stationary distribution in two attractors (of-state and on-state), respectively.Equation (29) gives the measure of the energy cost of a particle jumping between two attractors in the nonequilibrium state.Terefore, we investigate the energetic cost of tumor phase switching in a colored microenvironment to elucidate how energy consumption is reshaped by noise in three diferent cases as shown in Figure 10.
In Figure 10, we observe that by increasing the AT (τ 1 ) of multiplicative noise, the energy consumption increases overall, referring to Figure 10

Complexity
(α) is smaller than that at the critical point, the energy consumption becomes larger and larger with increasing the AT (τ 2 ). Figure 5 shows that the bimodality becomes more and more apparent with increasing the AT (τ 2 ) when the NS (α) is fxed at a given level, meaning that the system assumes more energy to achieve the phenotypic transition of the tumor system.When the NS (α) is larger than that at the critical point, the change tendency of energy consumption is opposite to that in the case of small noise.Tat is, by increasing the AT (τ 2 ), energy consumption becomes smaller and smaller and fnally tends to a stable value, referring to Figure 10(b).Since the AT (τ 2 ) increase may induce bimodality (refer to Figure 5(b)), a larger AT (τ 2 ) may reduce energy consumption even if the system phenotypes switch in a larger noise environment.Tis is counterintuitive because the existing results indicated that a system would consume more energy in a larger noise environment [38].Te reason why there is such a diferent result is that tumor development is essential to a non-Markovian process due to the AT (τ 2 ), that is, the AT or memory may resist extracellular fuctuations to save energy.Also, there is indeed a trade-of between energy consumption and a noisy environment mediated by the AT.Te CT (τ 3 ) and CS (λ) between the two noises may change the energy consumption of the tumor development system.In fact, we have known that the positive and negative CS (λ) can induce phenotypic switching and lead to presenting a diferent change pattern, referring to Figures 6(b) and 6(c), but the energy consumption is a decreasing function of the CT (τ 3 ) and has an intersection point in which the CS (λ) is equal to zero, i.e., the noisy processes are uncorrelated, referring to Figure 10(c).Tis means that the CS (λ) would be an efective mechanism for regulating energy consumption.Comparing the positive value with the negative value of the CS (λ) shows that the energy consumption has a little increase for a negative value of the CS (λ) but has an apparent reduction for a positive value of the CS (λ), indicating the positive CS (λ) of the two noises may induce the system to save energy consumption.Te combination of Figures 6(b) and 10(c) shows that the cell phenotype may switch to the unimodal state and the probability of tumor state dominates the stationary steady distribution with increasing CT (τ 3 ); that is, the positive CS (λ) of noises promotes the tumor neoplasia at a lower energy dissipation.However, if the CS (λ) is negative, the energy consumption will remain at a high level because the system recovers its bimodality, and the probability of the extinction state will become larger and larger with increasing the CT (τ 3 ).Tis means that achieving the high state switching to the extinction state is at the cost of energy consumption, which may explain why the tumor is difcult to treat.

Conclusions and Discussions
Cancer transition and metastasis are essentially stochastic and regulated by its microenvironment characterized by extracellular noises from diferent sources.Te fuctuations in lncRNAs would be a main noisy source of cancer development under immune surveillance [7,64].Recently, by ASO-mediated knock-down of MALAT1 in mouse mammary tumor virus PyMT of human luminal B breast cancer, Li et al. [7] confrmed that the fuctuation of lncRNA MALAT1 is correlated with lung cancer metastases through localizing to nuclear speckles and decreasing the elevated Ecadherin, afecting the cells invade distal lung sites.Terefore, the stochasticity of expression in MALAT1 abundance forms a colored microenvironment.Here, we propose an efective stochastic model of a cancer development system attacked by immune cytotoxic cells to investigate the mechanism of MATLAT1 marker fuctuation that regulates tumor development in two distinct ways (from time and space, achieving by regulating AT/CT and NS/CS), emphasizing cell phenotypic diversity and energy consumption.It declares that increasing AT of multiplicative noise has a dual biological function; that is, it can not only promote the tumor cells to difuse but also make contributions to killing the cancer cells, and NS can enhance this transition.Te reason why the transition can be achieved is that the stochastic bistability regime is mediated by changing the unstable attractor and NS may broaden the regime, meaning that this regulation is equivalent to the positive feedback in gene expression.While increasing AT of additive noise may reduce the steady probability of tumor state, i.e., AT may inhibit the tumor transition but NS of additive noise may attenuate this transition.However, the phenotype depends on the correlation between noises.Specifcally, when CS is larger than zero (positive correlation), CT is a stimulus efective for tumor development, but it may decrease the probability of tumor metastasis if CS is negative, and there is a hedging efect between CT and CS on cell phenotype switching.Also, the positive CS may induce MFPT to have a minimum value and a maximum value, and CT always keeps a minimum value in the corresponding plane.Furthermore, tumor development regulated by noise-induced factors may consume energy to achieve cell transition and metastasis.From the viewpoint of the mesoscopic scale, we reconstruct the efective topology network in the form of CME to integrate the diferent time scales in tumor  16 Complexity development and calculate the energy cost.We show that energy dissipation depends on the type of noise.Tere exists a trade-of between phase transition and energy cost for additive noise by regulating NS and AT in an additive noisy environment, but extending AT of multiplicative noise may increase efectively energy dissipation; the energy cost is a decreasing function of CT, and the CS may amplify this diference, depending on its sign.Tese results indicate that the biological function of colored noises can be achieved in time and space scenarios by changing AT/CT and NS/CS.Equations ( 10)-( 14) demonstrate that the molecular memory induced by AT/CT may violate the detailed balance [53,78,79].For the reduced system of tumor development, there are multiple heat baths here (referring to equation (10)), meaning the non-Markovian tumor system is drifting by driving forces at diferent levels and its biological function is achieved at the expense of energy [80][81][82][83][84].In our model, we can meditate AT/CT and NS/CS to regulate the whole expression process, i.e., there is the third method to regulate gene expressions.For more complex expression networks, one would expect that our results here, qualitative and model-free, such as AT, can be a "fne-tuner" for phase switching at the cost of energy, still hold.
Obviously, the system parameters D, α, λ, τ 1 , τ 2 , τ 3 can determine directly the value sensitivity index by the steady-state probability (Equation ( 14)) when the other conditions are the same.Of note, the time scale of two weighted indicators MFPTand EP has wide variations, and we must rescale them by dividing the minimal value of every trajectory and obtain Figure 11 (refer to Appendix D).It indicates that the sensitivity index of every regulation parameter on the phenotypic diversity is signifcantly diferent.Te relative fold of regulation additive noise is up to about 5.814-fold, implying the tumor system is more efective for NS of additive noise than other noise resources (CS 1.733-fold).Also, the efect of CTcan reach about 1.325-fold relative to AT of additive noise (calibration as reference 1), demonstrating that it may be an efective therapeutic schedule for tumor treatment called coherent difraction therapy [88,89].
Generally, gene expression may follow certain design principles for optimal evolutionary ftness [38,90].One of these constraints may be energy efciency.From the perspective of evolution theory, there is a trend of evolution for genes towards resource conservation to maximize the energy available to cells for biosyntheses, growth, and division, due to the limited system of energy obtained [91].Consistently, lots of researchers have reported that the actual evolutionary path of genes is much lower than others' path in energy landscapes to make more efective use of the energy obtained [37][38][39].Owing to the trade-of between dynamic regulation and energy efciency, this may be an optimal design, that is, the so-called minimum energy consumption principle.Of note, here, the minimum consumption is also a relative concept.Actually, the energy consumption is zero when the system is detailed balance, while the energy consumption will be positive when the system is in a nonequilibrium state.Terefore, it is interesting to decipher whether there exists an optimal path consuming the least energy to achieve the same biological function in the next research.
Moreover, the relationship between the phase transition and energy cost (as shown in Figures 4 and 10) shows that energy consumption is mainly used for broadening the stochastic bistability regime but for not maintaining relative stability, and the memory may accelerate energy dissipation, but CS may reduce energy consumption.Tese results imply the cell always consumes energy to achieve its phenotype adaptability, and it is also intelligent to take advantage of the correlation of the extracellular noisy environment to save energy dissipation.Tere may be a ftness potential landscape for tumor development in the above sense of the principle of "minimization," which is worth further investigation.

Figure 1 :
Figure 1: Te schematic diagram of tumor development in a colored noise environment.(a) Tumor cells' interaction with cytotoxic cells and the formation of the proliferation complex are regulated by the lncRNAs throughout the expression process; (b) separating the tumor cell progress into the fast and slow processes (reaction rates not shown).

Figure 2 :
Figure 2: Te steady-state solution (x i ) of tumor cell development (a) and the potential function (b).Te parameters are from the experiment on lung cancer[26], and θ � 0.1 s-1copy-1 and β � 2.26 s-1copy-1.Here, x 1 � 0 nm is the extinction state of the tumor cell, x 2 � 7.2659 nm represents the tumor state, and x u � 1.7341 nm is the transient state. Complexity

Figure 3 :
Figure 3: Te regulation of the AT and NS as functions of the density of tumor cells on the stationary probability distribution in a multiplicative noise environment.(a) If the NS is fxed at 0.2, the AT increases from 0.01 to 2; (b) if the NS is fxed at 0.7, the AT increases from 0.01 to 2. Te circle line represents the stochastic simulation [69].Te other parameter values are θ � 0.1 s − 1 copy − 1 and β � 2.26 s − 1 copy − 1 , which satisfy the bistable condition in Figure 2.

Figure 4 :
Figure4: Extrema of the steady distribution of the model(19) subjected to the noise only in the immune rate β, i.e., only multiplicative noise is considered.(a) For a fxed AT τ 1 � 0.01, increasing the NS from 0.2 to 2 to 4 (from red line to green line to blue line); (b) for a fxed NS D � 2, increasing the AT from 0.01 to 1 to 10 (from green line to blue line to red line).Te other parameter value is θ � 0.1 s − 1 copy − 1 , and the dashed line denotes an unstable attractor.

Figure 5 :Figure 6 :
Figure 5: Te regulation of AT and NS as a function of the density of tumor cells on the stationary probability distribution in an additive noise environment.(a) Te NS is fxed at α � 0.2, the AT increases from 0.01 to 2; (b) the NS is fxed at α � 0.7, the AT increases from 0.01 to 2. Te circle line represents the stochastic simulation.Te other parameters are the same as in Figure 3.

Figure 7 :Figure 8 :
Figure 7: Te mean frst passage times (MFPTs) are afected by a single noisy source.(a, b) Te multiplicative noise and (c, d) the additive noise.(a, c) Te transition from extinction state (x 1 ) to tumor state (x 2 ) and (b, d) an inverse process.Te other parameter values are the same as in Figure 3.

Figure 9 :Figure 10 :
Figure 9: States switching and equivalent networks.(a) Te tumor development switches between two stable equilibrium states, extinction state (of-state) and tumor state (on-state); (b) the corresponding CME; (c) the jumping process among the distinct states, forming a cycle fow.

Figure 11 :
Figure 11: Te sensitivity index defned the weighted sum of MFPT and EP, illustrating the efective regulation of every parameter.
(19) from green line to blue line to red line), implying that the AT may attenuate the regulation of NS.Terefore, the AT and NS of colored noise may have a hedge in regulating the bistable regime of the tumor development system.Comparing Figures4(a) with 4(b), it is clear that the high state (tumor state) and the low state (extinction state) are nearly unchanged and the zero is always a stable attractor of the system due to the structure of equation(19).Te unstable state in tumor development is transient, implying that this transient state may switch not only to extinction state (low state) but also to tumor state (high state) in a given level of probability, depending on the type of noise, and the AT and NS may mediate the probability of achieving the tumor cell difusion or vanishing, thus explaining why the multiplicative noise has a dual function in tumor cell development.