Determining Growth Rates of Microorganism Populations
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The document summarizes research on determining the initial population and growth constant of multiple non-interacting microorganism groups from total population time series data. It describes using the logistic growth equation to model microorganism population over time based on parameters like initial population, growth rate and maximum population. Methods are presented to calculate the growth rate and initial population for a single group from population data over time. However, the document notes it is difficult to determine parameters for multiple interacting populations uniquely due to non-unique total population solutions.
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Determining Growth Rates of Microorganism Populations
- 1. 27-28 June 2018 Jakarta, Indonesia Tarumanegara International Conference on the Applications of Social Sciences and Humanitis (TICASH 2019) 1 Determining Initial Population and Growth Constant of Multi-group Non-interacting Microorganism from Total Population Time Series: Inversion of Artificial Data Sparisoma Viridi1 , Souvia Rahimah2 1 FMIPA, Institut Teknologi Bandung, Bandung 40132, Indonesia 2 FTP, Universitas Padjadjaran, Sumedang 45363, Indonesia 1 dudung@fi.itb.ac.id, 2 souvia@unpad.ac.id 20190627_3
- 2. 27-28 June 2018 Jakarta, Indonesia Tarumanegara International Conference on the Applications of Social Sciences and Humanitis (TICASH 2019) 2 Outline • Introduction • (Numerical) solution • System • Results and discussion • Summaries
- 3. 27-28 June 2018 Jakarta, Indonesia Tarumanegara International Conference on the Applications of Social Sciences and Humanitis (TICASH 2019) 3 Introduction
- 4. Logistic equation • General form of equation , where Nmax is maximum population number, k is growth rate, and α, β, γ > 0 27-28 June 2018 Jakarta, Indonesia Tarumanegara International Conference on the Applications of Social Sciences and Humanitis (TICASH 2019) 4 γβ α −= max 1 N N kN dt dN A. Tsoularis, J. Wallace, "Analysis of logistic growth models", Mathematical Biosciences [Math. Biosci.], vol. 179, no. 1, pp. 21-55, Jul–Aug 2002, url https://doi.org/10.1016/S0025-5564(02)00096-2
- 5. Logistic eqn.: Well-known form • With α, β, γ = 1, previous equation becomes , which is more well-known • It is characterized by k, N0, and Nmax 27-28 June 2018 Jakarta, Indonesia Tarumanegara International Conference on the Applications of Social Sciences and Humanitis (TICASH 2019) 5 −= max 1 N N kN dt dN .
- 6. 27-28 June 2018 Jakarta, Indonesia Tarumanegara International Conference on the Applications of Social Sciences and Humanitis (TICASH 2019) 6 (Numerical) solution
- 7. (Numerical) solution • Using finite difference method with forward scheme, the solution is • t = 0 → N(0) = N0 • t = ∞ → N(∞) = Nmax 27-28 June 2018 Jakarta, Indonesia Tarumanegara International Conference on the Applications of Social Sciences and Humanitis (TICASH 2019) 7 ( ) ( ) ( ) ( ) t N tN tkNtNttN ∆ −+=∆+ max 1
- 8. Influence of k • N0 = 20, Nmax = 100, k = 2, 1, 0, -1 27-28 June 2018 Jakarta, Indonesia Tarumanegara International Conference on the Applications of Social Sciences and Humanitis (TICASH 2019) 8 . 0 20 40 60 80 100 120 140 0 1 2 3 4 5 6 7 8 9 10 N t S1 S2 S3 S4
- 9. Influence of Nmax • N0 = 1, k = 1, Nmax = 100, 80, 60, 40 27-28 June 2018 Jakarta, Indonesia Tarumanegara International Conference on the Applications of Social Sciences and Humanitis (TICASH 2019) 9 . 0 20 40 60 80 100 120 140 0 1 2 3 4 5 6 7 8 9 10 N t S1 S2 S3 S4
- 10. Influence of N0 27-28 June 2018 Jakarta, Indonesia Tarumanegara International Conference on the Applications of Social Sciences and Humanitis (TICASH 2019) 10 . • Nmax = 100, k = 1, N0 = 140, 100, 400, 1 0 20 40 60 80 100 120 140 0 1 2 3 4 5 6 7 8 9 10 N t S1 S2 S3 S4
- 11. 27-28 June 2018 Jakarta, Indonesia Tarumanegara International Conference on the Applications of Social Sciences and Humanitis (TICASH 2019) 11 System
- 12. System • Population of microorganism in an isolated environment • Initial population number N0 is known • Growth rate k is not known • Population number Nn is known at certain time t = tn • There are might more than one kind of microorganism 27-28 June 2018 Jakarta, Indonesia Tarumanegara International Conference on the Applications of Social Sciences and Humanitis (TICASH 2019) 12
- 13. Growth range • Ranges: Exponential, linear, saturated 27-28 June 2018 Jakarta, Indonesia Tarumanegara International Conference on the Applications of Social Sciences and Humanitis (TICASH 2019) 13 N = 34.70t- 67.98 R² = 0.999 N = 5.183e0.687t R² = 0.998 N = 180 - 6454.e-0.82t R² = 0.998 0 40 80 120 160 200 0 1 2 3 4 5 6 7 8 9 10 N t
- 14. Relation with logistic equation 27-28 June 2018 Jakarta, Indonesia Tarumanegara International Conference on the Applications of Social Sciences and Humanitis (TICASH 2019) 14 −= max 1 N N kN dt dN tkA eNN 0= tk C c eNNN − −= max tkNN BB += Exponential Linear Saturated
- 15. 27-28 June 2018 Jakarta, Indonesia Tarumanegara International Conference on the Applications of Social Sciences and Humanitis (TICASH 2019) 15 Results and discussion
- 16. Use in getting parameters • Growth range must be determined carefully • -> Range A: at least two observation points • -> Range B: at least two observation points • -> Range C: at least three observation points 27-28 June 2018 Jakarta, Indonesia Tarumanegara International Conference on the Applications of Social Sciences and Humanitis (TICASH 2019) 16
- 17. Range A • At initial observation time t = 0, N = N0 • At a certain time t1, • It can be found that the growth rate is 27-28 June 2018 Jakarta, Indonesia Tarumanegara International Conference on the Applications of Social Sciences and Humanitis (TICASH 2019) 17 1 01 tkA eNN = = 0 1 1 ln 1 N N t kA
- 18. Range B • At initial observation time t = t1, N1= NB + kBt1 • At a certain time t2, • It can be found that the growth rate is 27-28 June 2018 Jakarta, Indonesia Tarumanegara International Conference on the Applications of Social Sciences and Humanitis (TICASH 2019) 18 22 tkNN BB += 12 12 tt NN kB − − =
- 19. Range C • At initial observation time t = t2, • At a certain time t3, • Then 27-28 June 2018 Jakarta, Indonesia Tarumanegara International Conference on the Applications of Social Sciences and Humanitis (TICASH 2019) 19 23 32 tk C tk CC e N NN e −− − − = 2 max2 tk C C eNNN − −= 3 max3 tk C C eNNN − −=
- 20. Range C (cont.) • By setting t2 = 0 and t3 = t3 – t2 (*) * This can used also in previous range 27-28 June 2018 Jakarta, Indonesia Tarumanegara International Conference on the Applications of Social Sciences and Humanitis (TICASH 2019) 20 −− = C C C NNN N t k 323 ln 1
- 21. Initial population • Using calculated growth k in each range, initial population in each range N0 can be calculated using appropriate equation (slide 14) 27-28 June 2018 Jakarta, Indonesia Tarumanegara International Conference on the Applications of Social Sciences and Humanitis (TICASH 2019) 21
- 22. Problem • What if there are more than one type of popu-lation with different initial population number and also growth rate? Can this problem be solved with this method? 27-28 June 2018 Jakarta, Indonesia Tarumanegara International Conference on the Applications of Social Sciences and Humanitis (TICASH 2019) 22
- 23. Not-so-unique 27-28 June 2018 Jakarta, Indonesia Tarumanegara International Conference on the Applications of Social Sciences and Humanitis (TICASH 2019) 23 0 200 400 600 800 1000 1200 0 2 4 6 8 10 12 14 16 18 20 N t S1 S2 S3 S4 SS
- 24. 27-28 June 2018 Jakarta, Indonesia Tarumanegara International Conference on the Applications of Social Sciences and Humanitis (TICASH 2019) 24 Summaries
- 25. Summaries • Growth rate of microorganism population can be calculated using two observations points in exponential, linear, and saturated ranges • Calculation is simpler when at each range the time is translate to 0 • There is still no fine solution to get the para- meters for multi-group microorganism popu- lation, due to not-so-unique total population 27-28 June 2018 Jakarta, Indonesia Tarumanegara International Conference on the Applications of Social Sciences and Humanitis (TICASH 2019) 25
- 26. Acknowledgements • This research is supported by a research grant in 2019 27-28 June 2018 Jakarta, Indonesia Tarumanegara International Conference on the Applications of Social Sciences and Humanitis (TICASH 2019) 26
- 27. 27-28 June 2018 Jakarta, Indonesia Tarumanegara International Conference on the Applications of Social Sciences and Humanitis (TICASH 2019) 27 https://osf.io/nse6q/
- 28. 27-28 June 2018 Jakarta, Indonesia Tarumanegara International Conference on the Applications of Social Sciences and Humanitis (TICASH 2019) 28 Thank you