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Identifying Drug Targets with a Reduced Model for Competitive Inhibitor Stimulation

Garrett Young1, Mitchell Riley1,2, Colleen Mitchell1

1Department of Mathematics, University of Iowa; 2Department of Internal Medicine, University of Iowa

Background

Methods

Conclusions

Discussion

  • Terazosin (TZ) originally intended for benign prostatic hyperplasia
  • Demonstrated to be neuroprotective at moderate doses [1].
  • Leads to an increase in adenosine triphosphate (ATP)
  • Crystal structure suggests it acts as a competitive inhibitor when binding to the enzyme phosphoglycerate kinase 1 (PGK1)
  • How can competitive inhibition lead to stimulation?
  • Nondimensionalization
  • Fast slow analysis of multi-time scale system
  • Quasi-steady state approximations
  • Equilibrium approximations
  • A carefully considered grouping
  • Reduced model has an analytic solution
  • Allows further exploration of parameters and their effects on competitive inhibitor stimulation
  • Determine specific regimes that should guarantee stimulation
  • The model can help with further insight into drug development by answering the following questions:
    • When might reaction rate parameters lead to stimulatory behavior?
    • Is the amount of substrate present enough to see product stimulation?
    • Are there other enzymes that drugs could bind to and produce the same result?
    • Will this other drug cause stimulation when binding to this enzyme or to this other enzyme?

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  • The reduction can also be used to further explore larger biochemical reactions by being easier to implement, thus allowing for insight into larger impacts from drugs with this behavior.

Results

  • Nondimensionalized version of the model in [3]
  • Model qualitatively replicates in vivo and in vitro experiment results
  • Model of 15 nonlinear differential equations, no analytical solution
  • Can a reduction aid in determining when any drugs in development will have this stimulatory effect?
  • Peeking into different time frames reveals enzyme complexes, substrates, and products evolve on three distinct time scales.

References

  • Applying the previously mentioned methods to the nondimensionalized system drastically reduces the system to one differential equation with an analytical solution leading to reaction rate.

 

 

  • The reduced model maintains qualitative behavior that the full model expresses

Figure 1: The original model in [3] confirmed experimental studies reporting that moderate doses increased ATP production and that larger doses slowed the production of ATP

Figure 2: The mass action model from [3] has been nondimensionalized. It describes the interaction of PGK1 with substrates, products, and TZ. Each reaction is reversible, the interactions with substrates are in the top left diamond, while the interactions with products (ATP being the one of interest) are in the top right diamond, and TZ is shown in green.

Figure 3: Simulations of the nondimensionalized model show three distinct time scales. The left panel shows that the products and substrates change on the super slow scale (over an hour). The middle panel shows the evolution of enzyme over one minute (slow scale). The right panel shows fastest changes of the enzyme complexes in one second.

 

[1] Rong Cai, Yu Zhang, Jacob E Simmering, Jordan L Schultz, Yuhong Li,Irene Fernandez-Carasa, Antonella Consiglio, Angel Raya, Philip M Polgreen,Nandakumar S Narayanan, Yuan Yanpeng, Zhiguo Chen, Wenting Zu, Yanping Han,Chunyue Zhao, Lafang Gao, Xunming Ji, Michael J Welsh, and Lei Liu. “Enhancingglycolysis attenuates Parkinson’s disease progression in models and clinical databases”. In:The Journal of clinical investigation 129.10 (2019), pp. 4539–4549.

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[2] Xinping Chen, Chunyue Zhao, Xiaolong Li, Tao Wang, Yizhou Li, Cheng Cao, Yuehe Ding,Mengqiu Dong, Lorenzo Finci, Jia-huai Wang, Xiaoyu Li, and Lei Liu. “Terazosin activatesPgk1 and Hsp90 to promote stress resistance”. In: Nature chemical biology 11.1 (2015),pp. 19–25.

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[3] Mitchell J Riley, Colleen C Mitchell, Sarah E Ernst, Eric B Taylor, and Michael J Welsh.“A model for stimulation of enzyme activity by a competitive inhibitor based on theinteraction of terazosin and phosphoglycerate kinase 1”. In: Proceedings of the NationalAcademy of Sciences 121.9 (2024), e2318956121.

Figure 4: The left panel shows the nondimensional model’s response due to changes in drug dose and changes in μ. The right panel is doing that for the reduced model along with plotting the optimal dose (white) and maximal dose for stimulation (black).

Figure 5: The top two heatmaps are when z0 = 50nM, and the bottom two are when z0 = 50μM. The left two panels are heatmaps generated by the nondimensional model when altering values for μ and ω and the color comes from looking at the ratio when comparing the amount of v3 at the provided dose compared to no dose. These plots were generated from values when ξ = 300, the equivalent of one minute in the original model from [3]. The two panels on the right are generated using the reduced model and comparing the reaction rate with the provided drug dose compared to no dose.

Figure 6: The left panel shows the parameter space in red where stimulation can be observed. The blue area represents where stimulation will not occur. The yellow star is the parameter values for TZ, and the remaining dots are selected values within the stimulating region, non-stimulating region, and on the boundary. The right panel shows the dose response experienced at the marked parameter values.

Full Nondimensionalized Model

Reduced Model