Role of Thyroxin in Mammalian Brain Development

Mathematical Modeling of the Role of Thyroxin in Mammalian Brain Development

Afzal Sara, Ahmad Muhammad, Shahid Faryal

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Abstract: it has been known that tri-iodothronine (T3) plays a vital role in fetal brain development however despite its importance no mathematical model has been made for it thus far. Here we present a mathematical model for metabolic pathway of T4 to T3 conversion in brain development and gene regulatory pathway of recently characterized thyroid hormone receptor response genes. Our results show that low transcription of these thyroid hormone receptor response genes is sufficient for fetal brain development. On the other hand, high concentrations of metabolites are required in fetal brain development.

It has been well-established that thyroid hormone plays an essential role in the mammalian brain development. More specifically, Thyroxin (T4) is partially de-iodinized to tri-iodothronine (T3) in brain cells. The T3 acting through the nuclear receptors, Thyroid Receptor Hormone, serve as activating factors for certain downstream genes like Ser3 and hairless that play vital roles in the proper brain development in the mammalian fetus including having involvement in cell differentiation and migration, myelination and signalling.

Synaptotagmin-related gene 1 (Srg1) is one such novel thyroid hormone-responsive gene, which might possibly have a role in synaptic structure maintenance and/or activity^[2].

Hairless, on the other hand is a transcriptional cofactor that may possibly influence the expression of other genes that respond to thyroid hormone^[1].

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Ser3 catalyzes the first step in serine biosynthesis, amongst having other roles^[3]. This particular gene has been chosen for inclusion in our system because of the recently identified role of L-serine in proper functioning of central nervous system (CNS)^[4].

Thyrotropin releasing hormone (TRH) is a functional neurotramsmitter in the CNS of mammals and can also modify the tasks that the CNS performs.

The entire pathway we are observing can be broken down into the following two parts:

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1. Metabolic pathway: which involves deiodinized Thyroxin (T4) and its subsequent conversion to tri-iodothronine (T3) catalysed by enzyme iodothreoninedeiodinase2 (D₂)
2. Gene regulatory pathway: which involves the activation of Thyroid hormone receptor and three of its downstream targets; hairless, synaptotagmin-related gene 1(srg1), 3-phosphoglycerate dehydrogenase (ser3) and thyrotropin releasing hormone (TRH).

This has been clearly depicted in the wiring diagram in Figure 1.

Figure 1: Wiring diagram of Thyroxin activation and interaction with downstream targets

Despite the simplicity of the pathway it has never been modeled in its entirety before, which was partly the inspiration behind choosing to model using a Systems’ approach. Moreover, the steps of this thyroid hormone pathway can be dissected and analyzed using a broad range of topics studied in the Systems’ Biology course, including Michaelis Menton Kinetics and Boolean Networks. Hence, the project will serve as a platform to demonstrate our all-inclusive understanding of the concepts learnt in the course by practically applying combinations of these techniques to the system chosen.

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At a more macroscopic level, modeling of this system can unveil possibilities of developing a deeper understanding of the system dynamics of this thyroid hormone pathway, and an insight into its various potential targets. Analyzing these targets can thus help uncover potential drugs or treatments for various diseases or disorders resulting from improper brain development such as the Pendred’s Syndrome, the Grave’s disease and other psychiatric disorders like Cerebral Palsy, related to hyperthyroidism and hypothyroidism.

Methods and results:

Michaelis Menton Kinetics:

Figure 2: Wiring diagram of conversion of T₄ to T₃ under competitive inhibition

Having found parameters (k_i’s and k_m’s) from literature using experimentally derived values^(8-11), we plotted the trajectories of substrate (T4), enzyme (D₂), ES complex, inhibitor (rT₃D₂) and the IE complex using ODE’s. The ODE system for the dynamics of this reaction reads:

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= k_-1[T₄D₂]+ k_-i[rT₃D₂]- k₁[T₄ ][D₂]

= k_-1[T₄D₂]+ k_-i[rT₃D₂]- k_i[rT₃][D₂]- k₁[T₄][D₂]+ k₂[T₄D₂]

= k₁[T₄ ][D₂]- k_-1[T₄D₂]- k₂[T₄D₂]

-k_-i[rT₃D₂]+ k_i[rT₃][D₂]

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= k₂[T₄D₂]- k_d[T₃]

Following are the trajectories obtained:

Figure 3: Michaelis Menten Kinetics

T₃ interaction with Thyroid Hormone Receptor (THR)

We took a closer look at the T₃’s interaction with its receptor. The mechanism is shown below in figure 4:

Figure 4: T₃ interaction with THR

Using the following ODE’s in ODE45 we modeled this interaction and obtained figure 5 shown below:

V_pi=k_pi

V_di=k_di* R_i

V_ia=k_ia*T₃*R_i

V_ai=k_ai*R_a

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V_da=k_da*R_a

Figure 5: Receptor ligand interaction of T₃

In this graph, the blue line indicates the inactivated receptor while the green line indicates the activated receptor.

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Boolean Network:

A Boolean network of gene regulatory network was constructed and analyzed. The genes in the gene regulatory pathway were assigned symbols for simplicity as follows:

Symbols	Genes How Fast for Rush Jobs? We can crank out a killer paper in 24 hours—quality locked in, no shortcuts. Just set your deadline when you order, and our pros will hustle to deliver, even if you’re racing the clock. Perfect for those last-minute crunches without compromising on the good stuff.
A	THR-T3
B	Hairless
C Can You Handle Complex Subjects? For sure! Our writers with advanced degrees dive into any topic—think quantum physics or medieval lit—with deep research and clear, sharp writing. They’ll tailor it to your academic level, ensuring it’s thorough yet easy to follow, no matter how tricky the subject gets.	SRG1
D	TRH
E	Ser3 How Do You Meet Prof Standards? We stick to your rubric like glue—nailing the structure, depth, and tone your professor wants—then polish it with edits for that extra shine. Our writers know what profs look for, and we double-check every detail to make sure it’s submission-ready and grade-worthy.
	Figure 6: Boolean Graph

Using the formula 2^N, with N indicating the number of involved genes/nodes, we generated 2⁵=32 possible states at time ‘t’ using a method of recursion in MatLAB.

The rules used for generating successive states were devised as follows:

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A(t+1)= not B(t)

B(t+1)= A(t)

C(t+1)= A(t)

D(t+1)= A(t)

E(t+1)= not C(t)

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The rules were applied 20 times to observe patterns in successive states and analyze the states where the system might have converged. However instead of forming a few stable states, our system in fact formed a basin of absorption including the following states:

[1, 1, 1, 1, 1][0, 1, 1, 1, 0][0, 0, 0, 0, 0]

(For detailed results refer to supplementary material 1).

Directed Graph:

A directed graph for the gene regulation pathway was constructed as follows:

Figure 7: Directed Graph

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Nodes/genes=(A,B,C,D,E)

Edges/interactions= {(A, B, +ve), (A, C, +ve), (A, D, +ve), (B, A, -ve), (C, E, -ve)}

Baysian Network:

Next, a Baysian network was constructed for the gene regulatory pathway.

Figure 8: Baysian Network

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The dependencies of genes are indicated for by the following conditional probabilities:

P(X_A|X_B)

P(X_B|X_A)

P(X_C|X_A)

P(X_D|X_A)

P(X_E|X_C)

Transcription of thyroid hormone response genes:

We proceeded to study the transcription of the thyroid hormone response genes, the activator and the repressor.

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Figure 9: Microscopic binding and conformations of promoter^ref.

From figure blah we inferred the basal level mRNA synthesis of T3 responsive genes is approximately 25au, the level of repressor expression is around 2-3a.u. and level of activator expression is 35a.u.

Figure 10: mRNA synthesis of T₃ response genes in frog^[7]

Since we free could not find free energy values for thyroid hormone receptor we instead used the free energy value for the oestrogen hormone receptor whose dynamics resembles those of other members of the nuclear receptor family including the thyroid hormone receptor.

Next, in order to calculate the rate of transcription we used the following equation:

Stochastic modeling:

T₄ + D₂ T₄D₂ T₃ + D₂

D₂rT₃

Individual Reaction: T4 + D2 T4D2

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Rate constant = k1

Transitions

T4 T4 -1

D2 D2 -1

T4D2 T4D2 + 1

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Individual Reaction: T4 D2 T4 + D2

Rate constant = k-1

Transitions

T4 T4 +1

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D2 D2 +1

T4D2 T4D2 – 1

Individual Reaction: T4 D2 T3 + D2

Rate constant = k2

Transitions

D2 D2 +1

T4D2 T4D2 – 1

T3 T3 + 1

Individual Reaction: D2 + rT3 D2rT3

Rate constant = k3

Transitions

D2 D2 -1

Rt3 rT3 – 1

Rt3D2 rT3D2 + 1

Individual Reaction: D2 rT3 D2 + rT3

Rate constant = k-3

Transitions

D2 D2 + 1

Rt3 rT3 + 1

Rt3D2 rT3D2 – 1

We could not, however calculate the propensities due to time constraints on this project, and also because having research on neural cells, we discovered that the volume of neurons has been expriementally measured to be 5964µm^{3 [6]} , and therefore we cannot approximate deterministic rate constants to stochastic propensities because volume is not equal to 1.

Figure 11: Stochastic modeling under Gillespie algorithm

Analysis:

Applying michaelis menton to our metabolic pathway indicated that our system was consistent with an ideal Michaelis menton model.

Due to lack of parameters for the production and degradation of thyroid hormone receptor in active and inactive states, our model did not obey conservation principle of number of receptor molecules as indicated in figure 5.

Our Boolean network results indicate that our gene regulatory pathway forms a basin of absorption which includes the following states:

[1, 1, 1, 1, 1][0, 1, 1, 1, 0][0, 0, 0, 0, 0]

These states can be potential targets of further studies in this field.

Our calculation of gene regulation function depicts that the level of transcription of thyroid hormone receptor response genes is which indicates the brain development occurs at low transcript levels of thyroid hormone receptor response genes. However, it can be interesting to experiment various expression levels of these genes and compare them to the effect on fetal brain development.

Implementation of stochastic modeling using Gillespie revealed that the trend was similar to that of an ideal stochastic model. We also observed that the graph obtained for stochastic simulation was similar to that obtained for deterministic model. This shows that our metabolic pathway involves high number of molecules.

Sources Used

^[1]http://cercor.oxfordjournals.org/content/10/10/939.long

^[2] Thompson CC (1996) Thyroid hormone-responsive genes in developing cerebellum include a novel synaptotagmin and a hairless homolog. J Neurosci 16:7832–7840.

^[3]http://www.yeastgenome.org/cgi-bin/locus.fpl?dbid=S000000883

^[4]http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1223326/

^[6]http://ac.els-cdn.com/S1058674183710165/1-s2.0-S1058674183710165-main.pdf?_tid=f09dd6b0-cef4-11e3-a410-00000aacb35f&acdnat=1398703895_93d6f53032bc3c44d5fe595f81ac0e9b

^[7]http://www.ideal-ageing.eu/uploads/publicaties/2013/2013_grimaldi_biochimbiophysacta.pdf

^[8]http://www.ncbi.nlm.nih.gov/pubmed/7408767

^[9]http://joe.endocrinology-journals.org/content/210/1/125.full.pdf

^[10]http://www.ncbi.nlm.nih.gov/pubmed/3760742

^[11]Ajit Sadana Biosensors Kinetics of Binding and Dissociation Using Fractals 2003.pdf

^[12]http://toxsci.oxfordjournals.org/content/48/1/38.full.pdf

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