To engineer our SMART yeast, we developed two distinct yet complementary mathematical models. The Stochastic Model validates the fundamental feasibility of our genetic oscillator, while the Deterministic Model provides a framework to interpret its output, much like a doctor reading a pulse to diagnose an ailment.
Based on Gillespie Algorithm
To simulate the random, noisy nature of biochemical reactions inside a single cell and verify if our designed genetic oscillator could theoretically sustain oscillations over time.
Feasibility Confirmed. The model successfully produced oscillating fluorescence patterns, providing the theoretical confidence needed to proceed with wet lab construction. This significantly accelerated our design-build-test cycle.
Based on Differential Equations
To establish a clear mathematical relationship between system parameters (like stress levels) and the observable output (the "pulse" waveform), allowing us to infer the internal state of the yeast from its fluorescent signal.
Stress Inference Achieved. By fitting experimental data to our equations, we can determine the type and extent of stress the yeast is experiencing. This model forms the core of our "pulse diagnosis" analogy, turning raw data into actionable insights.
Given that our purpose is to predict whether the oscillator could operate normally in a long period of time, which is feasible for dynamic models but extremely challenging for descriptive models, we decided to build a dynamic model based on the mechanism of oscillator. Based on the structure of the gene oscillator, we divide the chemical reactions in the system into six categories: the binding of proteins to DNA, the dissociation of proteins from DNA, transcription, translation, mRNA degradation and protein degradation. The transcription of DNA will be inhibit if it is combined with the protein expressed by the previous gene. And the transcription will recover as long as the combination is dissociate.
By calculating the occurrence trend of each reaction at a certain moment and randomly selecting a reaction at that moment, we successfully simulated the situation in actual cells and verified the feasibility of the gene oscillator. Also, to simulate the circumstances under which stress occurs, based on our gene oscillator model, we choose to increase the transcription rate when a specific gene is suppressed in order to simulate the scenario where the inhibition of a specific gene is alleviated under stress.
By selecting appropriate parameters and attempting to run our stochastic model, we observed that the output fluorescence intensity-time curve(Fig. 1) could approximately produce oscillations, preliminarily demonstrating the feasibility of our designed oscillator and allowing for the construction and experimental validation of the oscillator in wet experiments. With the help of the stochastic model, we could validate the feasibility of a oscillator in a few days rather than conducting an experiment that would take several weeks, which significantly improved our efficiency of our screening oscillators.
To determine the category of pressure that our cells are suffering from, both descriptive models like machine learning models and dynamic models can be raised. However, when taking the fact into consideration that we have difficulty collecting sufficient data for training descriptive models in such a limited time, the dynamic models become the only choice for us. And given that dynamic models are built based on the mechanism of the system, this approach can make our research more interpretable.
Three repressor-protein concentrations,
, and their corresponding mRNA concentrations,
were treated as continuous dynamical variables. Based on the differential equation group used to describe gene expression in gene oscillators, we could draw a protein concentration - time curve(Fig. 2). The differential equations are as follows:
(
= 1,2,3 for a gene oscillator consisting of 3 genes)
where
is the degration rate constant of protein and
is that of corresponding mRNA. 
and
are the translation rate constant and transcription constant of the corresponding gene.
And
is the combined parameter,
is the binding parameter, which represents the binding ability of corresponding protein.
Andis the total protein concentration of the protein corresponding to the ith gene,is the concentration of mRNA corresponding to the ith gene.
And
are constants, where
is the Hill coefficient,
is the total gene concentration,
is the dissociation constant,
is the number of genes contained in the oscillator (for our project,=3).
The leakage
(≪1) is the ratio of the rate of blocked transcription to unsuppressed transcription,
is substituted with
.
Then we tried to adjust those parameters that are usually variable, including , 
and , and observed the changes in the shape of the curve.
represents the ratio of the degration rate constant of protein and corresponding mRNA. We discovered that a larger β will lead to a shorter period of the curve. This could be explained by the view that a larger β means that the corresponding protein will degrade faster in contrast to the circumstance with the same degration rate of corresponding mRNA. This indicates that the inhibition will be lifted more quickly, which will accelerate the cycle.
represents the ratio of the rate of blocked transcription to unsuppressed transcription. And we discovered that the change of δ would have various effects on the shape of the curve. First, for our ternary oscillator, increasing the δ corresponding to a certain gene will lower the average concentration of the protein corresponding to its next gene, while increasing the average concentrations of its own and the protein corresponding to its previous gene. In addition, an excessively large δ may cause the amplitude to gradually decrease until the oscillation stops.
We speculate that the translation of the curves is due to the increase of δ in gene 3, which lowers the degree of suppression of gene 3, leading to an increase in expression levels, which in turn enhances its inhibitory effect on the expression of gene 1. Since the expression of gene 1 is limited, it weakens its inhibition on gene 2. Therefore, we observe an upward shift of the curves corresponding to genes 2 and 3 and a downward shift of the curve corresponding to gene 1.
And if δ continues increasing, the expression of gene 3 under suppression will gradually approach that in normal situations, which will lead to the disruption of the structure of the gene oscillators. The constant expression of gene 3 will result in a constant expression of gene 1, which in turn will lead to an invariant expression of gene 2, thereby causing the oscillator to fail and leading to the stable expression of each gene.
The fluctuations in fluorescence intensity are similar to the rise and fall of a pulse. The changes in amplitude and frequency can be analogized to the distinctions in Traditional Chinese Medicine theory between floating and sinking pulses, as well as the differences between rapid and slow pulses. For example, when inflammation is caused by external factors, leading to symptoms such as increased myocardial contractility due to sympathetic nervous excitement, or increased vascular permeability due to cytokines released during infection, the floating pulse sensation will be strengthened. The doctor of traditional Chinese medicine can gather these information to comprehensively assess the patient's condition.
Based on a similar idea, we believe that the feature of the curve, like amplitude and frequency, could also reflect the condition of the system, just like the pulse patterns in traditional Chinese medicine can reflect a patient's illness.
Oscillator amplitude corresponds to TCM "Shi Mai" (Excess Pulse) and "Xu Mai" (Deficiency Pulse). Shi Mai (Excess Pulse) is defined as "a forceful pulse that remains firm even when pressed firmly": it feels strong and solid, and its firmness is still palpable even with heavy pressure. Xu Mai (Deficiency Pulse) is described as "a weak pulse that feels empty when pressed": it beats faintly, lacking force whether pressed lightly or heavily, as if "hollow inside." In the oscillator system, waveforms with large amplitude are labeled as the yeast’s "Shi Mai," representing strong interconnections between the oscillator’s components (i.e., high oscillator vitality); waveforms with small amplitude are labeled as the yeast’s "Xu Mai," indicating weak interconnections (low vitality). In TCM, Shi Mai and Xu Mai reflect whether a patient’s qi and blood are abundant (roughly equivalent to "physical strength"), and for yeast, their "Shi Mai" and "Xu Mai" similarly indicate the oscillator’s vitality.
Oscillator frequency corresponds to TCM "Shu Mai" (Rapid Pulse) and "Chi Mai" (Slow Pulse). Shu Mai (Rapid Pulse) refers to "a rapid pulse that beats over 6 times per respiration cycle," while Chi Mai (Slow Pulse) means "a slow pulse that beats only about 3 times per respiration cycle." In the oscillator system, the yeast’s "Shu Mai" corresponds to high oscillation frequency, reflecting fast synthesis and degradation rates of stress-resistant proteins and corresponding mRNA; the yeast’s "Chi Mai" corresponds to low oscillation frequency, indicating slow rates of these processes.
The average value of the oscillator’s stress-resistant protein expression corresponds to TCM "Fu Mai" (Floating Pulse) and "Chen Mai" (Sinking Pulse). Fu Mai (Floating Pulse) is characterized by "being felt immediately with light pressure, but fading slightly with heavy pressure," as if the pulse is close to the skin surface. Chen Mai (Sinking Pulse), by contrast, is "felt only with heavy pressure, not detectable with light pressure": opposite to Fu Mai, it feels as if the pulse is close to the bones or deep within the body. In the oscillator system, the yeast’s "Fu Mai" corresponds to a high average value (with the waveform shifting upward overall), indicating high average expression of stress-resistant proteins and strong stress resistance in yeast; the yeast’s "Chen Mai" corresponds to a low average value (with the waveform shifting downward overall), meaning low expression of stress-resistant proteins and weak stress resistance.
When "taking the pulse" of yeast (i.e., analyzing its oscillator waveform), changes in its corresponding "pulse type" can reveal shifts in both the stress environment and the yeast’s stress resistance, enabling targeted "treatment" (adjustment of its living conditions).
Our deterministic model can predict the parameters in the equations and assess they are normal or not by collecting apparent fluorescence intensity oscillation data, thereby inferring deeper internal 'ailments' of the system and subsequently deducing the 'cause of disease,' i.e., the type of stress, and then assists in selecting appropriate 'treatment methods'.
With the patterns we discovered, we further used least squares to fit the parameters in the equation based on the fluorescence intensity data observed at different times during the experiment.
And furthermore, we propose a method to infer the stress situation in the system based on the parameters in the equation that we inferred before, which is going to occur or is hard to realize timely via conventional methods. This idea mainly based on the actual meaning of each parameter in the differential equations. For example, the parameterrepresents the ratio of the rate of blocked transcription to unsuppressed transcription. So if the increase of δ is detected, we can consider that the corresponding gene repression has been lifted, which implies the presence of the corresponding stress in the system. Also, via analyzing the extent of the increase in δ, we could determine the extent of the stress, too.
With the type of stress and its extent revealed, it will become much easier to adjust the fermentation parameters on time so as to ensure that the system keeps operating optimally and the highest production efficiency is reached.
And in future, if sufficient data is produced and collected, data-driven machine learning methods could be introduced to better analyze the type of stress and its extent. For example, time series-based machine learning methods, like Transformer Network, can utilize model parameters from several time points over a period to better predict the type and severity of stress currently experienced by the system and further predict the stress that the system may encounter in the future, thereby helping to achieve better regulation of the system.
Our team has pioneered the application of TCM’s "syndrome differentiation and treatment" concept and pulse diagnosis to microscopic organisms. By using "pulse types" to detect subtle cues of yeast’s stress resistance, we can adjust its environment for "syndrome-specific treatment." Additionally, the yeast "treated" by our young "TCM practitioners" gains the ability to self-regulate and adapt to stressful environments—using the oscillator structure to maintain its own "balance" and resist adverse conditions.