GROWTH EQUATIONS Biot 309 Fall, 2013 N = (log10Nf – log10No ) / .301 N Number of generaDons Nf Final ConcentraDon of Cells No Original concentraDon of cells .301 Conversion Factor to Convert Log2 to Log10 •  Logarithm is a funcDon that gives the exponent in the equaDon bn = x. It is usually wriRen as logb x = n. •  For example: 34 = 81 Therefore log3 81 = 4 Measure Culture at 9:00 a.m. Measure Culture at 3:00 p.m. 10,000 cells / ml 100,000 cells / ml Calculate N (number of genera=ons) N = (log10Nf – log10No ) / .301 N = (5-­‐4)/0.301 N = 1/0.301 N = 3.33 Genera=ons HOW MANY HOURS DID THIS TAKE? HOW MANY MINUTES? CONTINUED … Measure Culture at 9:00 a.m. 10,000 cells / ml Measure Culture at 3:00 p.m. 100,000 cells / ml Calculate N (number of genera=ons) HOW MANY HOURS DID THIS TAKE? HOW MANY MINUTES? N = 3.33 Genera=ons in 6 Hours 6 Hours x 60 min/hr = 360 Minutes WHAT IS THE DOUBLING TIME or the GENERATION TIME? Measure Culture at 9:00 a.m. 10,000 cells / ml Measure Culture at 3:00 p.m. 100,000 cells / ml Calculate N (number of genera=ons) HOW MANY HOURS DID THIS TAKE? HOW MANY MINUTES? N = 3.33 Genera=ons in 6 Hours 6 Hours x 60 min/hr = 360 Minutes WHAT IS THE DOUBLING TIME or the GENERATION TIME? Genera=on Time = 360 Minutes / 3.33 Genera=ons N = 108 Minutes = 1 genera=on =me Measure Culture at 9:00 a.m. 10,000 cells / ml Measure Culture at Noon 1,000,000 cells / ml Calculate N and GeneraDon Time
Measure Culture at 9:00 a.m. Measure Culture at Noon
Calculate N 10,000 cells / ml 1,000,000 cells / ml N = (log10Nf – log10No ) / .301 N = (6-­‐4)/0.301 N = 2/0.301 N = 6.64 Genera=ons in 3 Hours We Know:
3 Hours = 180 Minutes Therefore:
Genera=on Time = 180 Minutes / 6.64 Genera=ons N = 27 Minutes to Generate Measure Culture at 9:00 a.m. Measure Culture at 1:00 p.m. 2000 cells / ml 18,000 cells / ml Calculate N and GeneraDon Time
Measure Culture at 9:00 a.m. Measure Culture at 1:00 p.m. Calculate N 2000 cells / ml 18,000 cells / ml N = (log10Nf – log10No ) / .301 N = (4.25-­‐3.30)/0.301 N = .95/0.301 N = 3.16 Genera=ons in 4 Hours We Know:
4 Hours = 240 Minutes Therefore: Genera=on Time = 240 Minutes / 3.16 Genera=ons N = 75.9 Minutes to Generate N = 1.27 Hours to Generate An inoculum of 107 bacterial cells was inoculated into medium and growth monitored. No change was seen for 18 min; then growth occurred rapidly. A`er a further 76 min., the popula=on had increased to 4.32 x 108. What is the doubling =me of the culture? An inoculum of 107 bacterial cells was inoculated into medium and growth monitored. No change was seen for 18 min; then growth occurred rapidly. A`er a further 76 min., the popula=on had increased to 4.32 x 108. What is the doubling =me of the culture? KNOWNS: N0= 107 t = 76’ Nf = 4.32 x 108 N= # generaDons N=(log Nf -­‐ logN0)/.301 = (log4.32 x 108 – log 107)/0.301 = [(8 + 0.636) -­‐ 7]/ 0.301 = 1.636/0.301 = 5.44 = N # generaDons G=t/N = 76/5.44 = 14’ You perform a serial diluDon and determine that the original number of cells in your sample was 12, 000. How many bacteria will be present in 12 hours if the generaDon Dme is 15 minutes? (assume unlimited food and clean environment)? You perform a serial diluDon and determine that the original number of cells in your sample was 12, 000. How many bacteria will be present in 12 hours if the generaDon Dme is 15 minutes? (assume unlimited food and clean environment)? SETUP:IDENTIFY WHAT YOU KNOW = 12,000 Time = 12 h Genera=on Time = 15’ = =me/genera=ons SOLVE FOR GENERATIONS: 0.25h = 12/genera=ons Genera=ons = 12/0.25 = 48 N=(log Nf -­‐ logN0)/.301 48= (logNf-­‐ log12,000)/.301 48x.301= (logNf-­‐ log12,000) = (logNf -­‐ ( 4x0792)) = logNf-­‐0.317 14.448 + 0.317 = logNf 14.765 = logNf TAKE an=log of both sides number of generaDons in 12 hours = 4 generaDons per hour x 12h = 48 generaDons 12,000 x 248 = 3.4 x 1018 bacterial cells You determine that a coconut cream pie has 3 million (3 x 106) Staph. aureus cells in it. You es=mate that the food preparer did not wash his hands and probably inoculated the cream with 500 Staph. aureus. He also forgot to refrigerate it. If the pie was made 6 hours ago, how many genera=ons have occurred? You determine that a coconut cream pie has 3 million (3 x 106) Staph. aureus cells in it. You es=mate that the food preparer did not wash his hands and probably inoculated the cream with 500 Staph. aureus. He also forgot to refrigerate it. If the pie was made 6 hours ago, how many genera=ons have occurred? Knowns: Nf = 3 x 106 N0 = 500 = 5x102 Time = 6 h N=(log Nf -­‐ logN0)/.301 = (log 3x106 – log 5x102) / 0.301 N= (6.+.4771) – (2 + .6990)/0.301 = (6.4771-­‐2.6990)/.301 = 3.778/0.301 = 12.5 genera=ons How long is each genera=on? You determine that a coconut cream pie has 3 million (3 x 106) Staph. aureus cells in it. You es=mate that the food preparer did not wash his hands and probably inoculated the cream with 500 Staph. aureus. He also forgot to refrigerate it. If the pie was made 6 hours ago, how many genera=ons have occurred? Knowns: Nf = 3 x 106 N0 = 500 = 5x102 Time = 6 h N=(log Nf -­‐ logN0)/.301 = (log 3x106 – log 5x102) / 0.301 N= (6.+.4771) – (2 + .6990)/0.301 = (6.4771-­‐2.6990)/.301 = 3.778/0.301 = 12.5 genera=ons How long is each genera=on? 6x60 = 360’ 360’/12.55g = 28.7 minutes Using the genera=on =me from problem 2, how many bacteria would be present a`er 8 hours at room temperature? Using the genera=on =me from problem 2, how many bacteria would be present a`er 8 hours at room temperature? Calculate new Nf = major answer GeneraDon (doubling) Dme = 28.7’ 8h = 480’ N0 = 5x102 SOLVE FOR Nf Calculate new N (intermediate answer) = 480’/28.7 = 16.7 genera=ons N=(log Nf -­‐ logN0)/.301 16.7x 0.301 = (logNf – log5x102) = logNf – 2.6990 5.027 + 2.699 = 7.726 = logNf TAKE ANTILOG of both sides An=log 7.726 = Nf = 5.33 x 107 You determine that a coconut cream pie has 3 million (3 x 106) Staph. aureus cells in it. You es=mate that the food preparer did not wash his hands and probably inoculated the cream with 500 Staph. aureus. He also forgot to refrigerate it. G = t_____ B= beginning b=end 3.3 log b/B If the pie was made 6 hours ago, how many genera=ons have occurred? KNOWNS: B= 3 x 106 b = 5x102 t= 6 h = 6 x 60 = 360’ G = 360/ 3.3 x log (3x106/5x102 ) =360/ 13.134 = 12.5 genera=ons How long is each genera=on? 6x60 = 360’ 360’/12.552g = 28.7 minutes Let’s say that flesh ea=ng Strep. pyogenes divides every 10 minutes at body temperature. You fall down and scrape your knee and get infected with 5 Strep. pyogenes cells. A`er 4 hours, without medical interven=on, how many bacteria will be ravaging your body? Knowns: G (genera=on or doubling =me = 10” N0 = 5 SOLVE FOR N G= t/N 10’ = (4 x 60”/N N= t/G = (4x 60)/10 = 24 genera=ons Solve for Nf using N= (logNf – log N0)/0.301 SUBSTITUTE 24= (log Nf – log 5)/0.301 (24 x 0.301) +0.7 = log Nf = 7.92 log Nf = 7.92 TAKE ANTILOG OF BOTH SIDES Nf = 8.32 x 107 Strep. Pyrogenes a`er 4 h Let’s say that for every 1 million bacteria, a cen=meter of flesh is consumed. A`er 4 hours, how much =ssue would be lost? Are you s=ll alive and would you want to be? (This problem is not fact based.) •  Batch Culture: inoculate into medium –  AcDve exponenDal growth for limited period of Dme b/c nutrients run out •  ConDnuous Culture: keeps culture in log phase, especially important for producDon –  Remove old medium and add new on a regular basis –  Monitor and adjust pH Total Cell Counts •  microscopically •  volume in grid known •  count cells  cells/
volume •  Living + dead cells = problem MEASURING BACTERIA