Mathematical Expression of Microbial growth

1. Mathematical Expression of Growth Jamal Abdulaziz M.Sc. Microbiology

2. Microbial Growth • Growth:an increase in the number of cells, not an increase in size of the cells. • Bacterial species only maintained if population continues to grow • Generation:growth by binary fission • Growth rate: cell number/time • or cell mass/time • Generation time:time it takes for a cell to divide and the population to double; most are 1-3 hours. • Generation times vary markedly with the species of microorganism and environmental conditions; they can range from 10 minutes for a few bacteria to several days with some eukaryotic microorganisms.

3. Growth is an orderly increase in the quantity of cellular constituents. It depends upon the ability of the cell to form new protoplasm from nutrients available in the environment. In most bacteria, growth involves increase in cell mass and number of ribosome, duplication of the bacterial chromosome, synthesis of new cell wall and plasma membrane, partitioning of the two chromosomes, septum formation, and cell division. This asexual process of reproduction is called binary fission

4. Binary fission • The normal reproductive method of bacteria is transverse binary fission in which a single cell divides into two identical cells after developing a cross wall (transverse) septum It is an asexual reproductive process.Thus, bacteria increase their numbers by geometric progression or exponential growth, i.e. doubling of bacterial population every generation as : 1, 2, 4, 8, etc. or 20, 21, 22, 23.......2n (where n = the number of generations).

5. What occurs during binary fission?

6. Exponential Growth by Binary Fission • DNA replication • Cell elongation • Septum formation • Septum completion leads to separation or further division • Process repeats

7. The Growth Cycle • The population growth is studied by analyzing the growth curve of a microbial culture. • The standard bacterial growth curve describes various stages of growth ,a pure culture of bacteria will go through beginning with the addition of cells to sterile media and ending with the death of all of the cells present.

8. The Lag Phase • Immediately after inoculation of the cells into fresh medium, the population remains temporarily unchanged. • There is normally a brief period of adaptation by the cells to the new conditions.. Although there is no apparent cell division occurring, the cells may be growing in volume or mass, synthesizing (enzymes to digest the nutrients), proteins, RNA, etc., and increasing in metabolic activity. The rate of growth begins to increase towards the end of this phase.

9. The Log (Logarithmic / Exponential) phase • There is a rapid period of growth during this phase due to the fact that: • Bacteria have developed the necessary enzymes and there are plenty of nutrients. • There are few waste products being produced. • The rate of cell division is currently at its maximum with the number of bacteria doubling as often as every 20 minutes.

10. Stationary Phase • Exponential growth cannot be continued forever in a batch culture (e.g. a closed system such as a test tube or flask). There is a rapid period of growth during this phase. • Population growth is limited by one of three factors: • exhaustion of available nutrients • 2. accumulation of inhibitory metabolites or end products • 3. exhaustion of space, in this case called a lack of "biological space".

11. The Death (Decline) Phase • During this phase more bacteria are dying than are being produced. This is because: • Very few nutrients are left. • Many bacteria are poisoned by the waste produced by such large numbers • Thus the rate of growth is falling. • the number of viable cells decreases geometrically (exponentially), essentially the reverse of growth during the log phase.

13. Typical growth curve for a bacterial population Or minutes

14. Growth Rate and Generation Time The time required for a bacterial cell to divide, i.e. a population to double, during log-phase, is known as the generation time for a population, it is often called “Mean Generation Time” Under a given set of growthconditions(medium,Temperature,pH, etc. Each bacterial species has a genetically determined generation time, the growth can be quantitatively analyzed by the determination of generation time. Therefore, generation time is a unit to measure bacterial growth.

15. CalculationofGeneration(doubling)Time • The relationship between the number of bacteria in a population at a given time(Nt),the original number of bacterial cells in the population (N0), and the number of divisions those bacteria haveundergone during that time (n) can be expressed by the following equation: • Nt= N0 x 2n • No = # of cells in population initially • Nt = # of cells in population at time t • n = number of generations that have occurred

16. The generation time is given by the formula: • G = t/n where, • G = generation time • t = time interval in hours or minutes • n = number of generations • Solving for (n): • Nt = NOx 2n • log (Nt = NOX 2n) • log Nt-log NO= n log 2 • (log Nt- log NO)/ log2 = n • n=(log Nt- log NO)/log2 , log2=0.301 • n= (log Nt- log NO)\0.301

17. Therefore G=t\n • G = t\log Nt- log NO)\0.301= • G=3.3t\logNt-logN0 • G = t\3.3 log Nt/N0 • This is the Mathematical Expression of Growth

22. Calculation of Generation Time • Example: What is the generation time of a bacterial population that increases from 10,000 cells to 10,000,000 cells in four hours of growth? • G=? • No=10,000=10^4 • Nt=10,000,000=10^7 • t=4Hrs =240 minutes G =        t_____       3.3 log b/B G =    240 minutes       3.3 log 10^7/10^4 G =   240 minutes           3.3 x 3 G = 24 minutes \doubling • B=N0,b=Nt • We can calculate the Number of generations using n=3.3 (log Nt- log NO) • n= 3.3(log10^7-log10^4)=10

24. Generation Times of some microorganisms

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