Linear Regression model reveals SCFA's relationship with serotonin levels

In our mathematical modeling section, we mainly tried to simulate the in vivo production of butyrate, an important SCFA. Our complex vector system is modeled in a simple way and the expected production of serotonin is based on a Machine Learning model. Data from both our lab experiments revealed relationships between SCFA's with serotonin production and also our part's activity with butyrate production. Our ultimate goal is to find the conditions that our part's final enzyme 'Butyryl-CoA:acetate CoA-transferase' (ButCoAT) produces butyrate, in a way that it can induce serotonin production.

First, the objective of our analysis was to establish a mathematical relationship between the concentration of short-chain fatty acids (SCFAs) and the concentration of serotonin. To achieve this goal, we employed machine learning techniques on the data obtained from our ELISA wet lab experiment. This dataset consisted of pairs of serotonin concentrations and varying SCFA concentrations.

We utilized non-linear regression methods to develop an interpretable function that could predict serotonin concentration based on SCFA concentration. The formula we derived for this purpose is as follows:

$$f(acetate, butyrate) = a \cdot acetate + b \cdot butyrate + \frac{c \cdot acetate}{butyrate^{0.1}} +d$$

Where: \(a = 1.06479392 \cdot 10^{-7}, b = 1.25943986 \cdot 10^{-8}, c = -1.73230132 \cdot 10^{-8}, d = 1.880177124361716\).

The above concentrations are measured in \(nmol/L\).

Our analysis indicates that among the two primary SCFAs we examined, acetate appears to play a slightly more significant role in influencing final serotonin levels. This conclusion is further supported by independent statistical analyses. Notably, the maximum serotonin concentration is achieved when both acetate and butyrate are present at their highest concentrations.

We then used the equation derived from linear-regression to see how much serotonin is produced in different acetate concentrations for a certain butyrate concentration. Results are presented below.

After simulating SCFA's - serotonin relationship, we proceeded to Tukey's post-hoc test on the anova test results (see wet lab experiment section) to support the findings of the machine learning model (that Acetate and Butyrate in high concentrations induce serotonin production). With this test we saw the actual differences in serotonin production between the SCFA combinations. Our reference group was set as: Butyrate = 0 nmol/L and Acetate = 0 nmol/L, which is our control. Tukey's test results for strong p-values are the following:

Table 1: Indications : B00: butyrate = 0 nmol/L, B05: butyrate = 5*10^5 nmol/L, B10: butyrate = 10^6 nmol/L, B25: butyrate = 25*10^5 nmol/L, A00: acetate = 0 nmol/L, A10: acetate = 10^7 nmol/L, A20: acetate = 2*10^7 nmol/L, A30: acetate = 3*10^7 nmol/L . Indications of combination example: B10_A20: butyrate = 10^6 nmol/L (B10) & acetate = 2*10^7 (A20)

Combination	Diff	Lwr	Upr	p adj
B00_A20-B00_A00	2.20884986	1.22238354	3.19531619	0.0000000
B05_A20-B00_A00	1.64867147	0.66220514	2.63513779	0.0000001
B05_A30-B00_A00	2.27137802	1.28491169	3.25784435	0.0000000
B10_A20-B00_A00	3.09467448	2.10820815	4.08114081	0.0000000
B10_A30-B00_A00	1.60324156	0.61677523	2.58970788	0.0000002
B15_A20-B00_A00	1.79875794	0.81229161	2.78522426	0.0000000
B15_A30-B00_A00	2.30538182	1.31891549	3.29184814	0.0000000

By comparing the difference in serotonin mean values we observe that, for a standard butyrate concentration, increasing acetate tends to increase serotonin production, which comes in agreement with the information that our Machine Learning Model provides.

Modeling of part activity and function

After getting information on SCFA's relationship with serotonin we moved on to constructing a basic model of enzyme kinetics with the goal to find how much butyrate is produced with a given amount of acetyl-coa and the 5 enzymes that are produced from our construct. In more detail the model consists of Ordinary Differential Equations (ODEs) that calculate each metabolite's concentration values through different time stamps. It also reveals the activity of Butyryl-CoA:acetate CoA-transferase, the final enzyme of our system. This enzyme produces butyrate and we observe different butyrate values dependent on different substrate concentrations.

The kinetics of our system are described by the Michaelis-Menten enzyme model. The rapid equilibrium assumption and the steady state approximation are some of the models assumptions.

The general reaction scheme of an enzyme-catalyzed reaction is as follows:

$$E + S \rightarrow k_{1}[ES] \rightarrow k_{2}E + P$$

After applying all the assumptions, the following equations, known as Michaelis-Menten equation occurs:

$$V_o = \frac{V_{max}[S]}{K_M + [S]}$$

We acquired the above constants ( Vmax, Km ) through bibliography. If we solve for Substrate the following equation occurs:

$$ \frac{d[S]}{dt} = \frac{-V_0}{V_{max}} \cdot [S]$$

Due to law of mass action, the equation for product concentration is:

$$ \frac{d[P]}{dt} = \frac{V_0}{V_{max}} \cdot [S]$$

We also added a physical constant to every metabolite to decrease its concentration after getting its maximum Steady-State concentration. The decrease constant is referring to factors like metabolite degradation and metabolite removal from our system. The decrease was interpreted as : \( -k_{decrease} \cdot [metabolite] \)

Each enzyme produced by our system has the same ribosome binding sites and is constitutively expressed by a unique promoter too. Thus, transcription and translation constants ( k_transcript , k_translation ) are the same for each enzyme. So, transcription rates \( (k_{transcript} \cdot [vector]) \)and translation rates \( (k_{translation} \cdot [mrna]) \)are added to each part's ODEs. The ordinary differential equations that occur and describe the change (for instance) in butyryl-coa, butyrate and their corresponding enzymes concentrations over time, are in form:

$$\frac{d[mrna_{BHD}]}{dt} = k_{transcript} \cdot [vector] - mrna_{degradation} \cdot [mrna_{BDH}]$$

$$ \frac{d[BHD]}{dt} = k_{translation} \cdot [mrna_{BHD}] - protein_{degradation} \cdot [BHD]$$

$$\frac{d[butyryl_{coa}]}{dt} = \frac{V_0}{V_{max}} \cdot [crotonyl_{coa}] - k_{decrease} \cdot [butyryl_{coa}]$$

$$ \frac{d[mrna_{ButCoAT}]}{dt} = k_{transcript} \cdot [vector] - mrna_{degradation} \cdot [mrna_{ButCoAT}]$$

$$\frac{d[protein_{ButCoAT}]}{dt} = k_{translation} \cdot [mrna_{ButCoAT}] - prot_{degr} \cdot [protein_{ButCoAT}] $$

$$\frac{d[butyrate]}{dt} = \frac{V_0}{V_{max}} \cdot [butyryl_{CoA}] - k_{decrease} \cdot [butyrate]$$

The same set of equations is used for every other metabolite in our system and their corresponding ODEs occur. Thus, we describe each metabolite conversion from one molecule to another through enzymatic catalysis. We were able to observe changes in metabolites concentration over time. By focusing on the most important enzyme of our system, the one that catalyzes the final step, we saw how different butyryl-coa concentrations produce different butyrate concentrations with a non-linear relationship. An optimization of the genetic circuit we want to simulate is shown below.

Below we present the simulation of our system for a certain acetyl-coa concentration.

Focusing on the concentrations of the most important molecules we observe a rapid consumption of Acetyl CoA that results in an early spike of butyrate concentration which is decreased in a span of 25 mins. Also a small amount of Acetoacetyl-CoA is needed in the pathway.

Moreover, we examine the dependance of butyrate on butyryl-coa concentrations. We observe that the production of each of the aforementioned molecules is of course dependent on the concentration of the previous one in the order of the metabolic pathway. For example Butyrate is dependent on the concentration of Butyryl-CoA. A non-linear relationship between substrate (butyryl-CoA) and product (butyrate) is observed. We see early product formation, before steady-state maximum concentration of butyrate, to better examine the fast changes in butyrate concentration.

To conclude, we simulated our system's activity from part's expression to serotonin production. We simulated butyrate production by the Butyryl-CoA:acetate CoA-transferase. Linear Regression formula with butyrate and certain acetate concentrations as input, returned the expected serotonin production through time.