MentDB WeakMQL Recipes
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BM

Biologist & microbiologist

Growth curves, enzyme kinetics and dilution series.

10 recipes on this page · open in the interactive view

Recipes
  1. Bacterial growth curve: rate, doubling time, carrying capacity
  2. Enzyme kinetics: Km, kcat and catalytic efficiency
  3. Serial dilution plate counts and log reduction
  4. Predator-prey cycles of Lotka-Volterra
  5. qPCR: efficiency from the standard curve and relative expression
  6. Hemocytometer count: Poisson check and concentration
  7. Kleiber's law: allometry of the metabolic rate
  8. Flow cytometry: automatic gating of leukocytes
  9. Photosynthesis-irradiance curve (Jassby-Platt)
  10. Resting membrane potential: Nernst and Goldman
executed on MentDB

Bacterial growth curve: rate, doubling time, carrying capacity

The OD600 of an E. coli culture was read every hour. The script fits the logistic model OD = K/(1 + a.exp(-r.t)) and gives the growth rate, the doubling time in the exponential phase, the carrying capacity, the inflection time and the maximum growth rate r.K/4.

Pipeline
01 · fitLogistic fitstat fit→02 · extractr, a and Kjson select→03 · deriveDoubling time and inflectionmath calc
Script
admin
#1. OD600 of an E. coli culture, time in h;
-> "[t]" "[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12]";
-> "[od]" "[0.03, 0.056, 0.101, 0.182, 0.302, 0.496, 0.723, 1.028, 1.245, 1.466, 1.686, 1.708]";
#2. Logistic fit OD = K / 1 + a.exp of -r.t;
json load "rx_ch_lg" (stat fit [t] [od] "logistic");
-> "[r]" (json select "rx_ch_lg" /b); -> "[a]" (json select "rx_ch_lg" /a); -> "[kk]" (json select "rx_ch_lg" /c);
#3. Doubling time in exponential phase, inflection time, max growth rate;
concat "r = " (math round [r] 3) " 1/h, doubling time = " (math round (math calc "ln(2)/[r]*60") 0) " min, K = " (math round [kk] 2) " OD, inflection at " (math round (math calc "ln([a])/[r]") 2) " h, max rate = " (math round (math calc "[r]*[kk]/4") 3) " OD/h"
Result
mentdb
r = 0.622 1/h, doubling time = 67 min, K = 1.81 OD, inflection at 6.6 h, max rate = 0.281 OD/h
Go further
  • Fit the Gompertz model with stat fit_model to get the lag time.
  • Compare the growth rates of two strains with stat ttest_2 on replicates.
executed on MentDB

Enzyme kinetics: Km, kcat and catalytic efficiency

Initial rates were measured at eight substrate concentrations with 0.050 uM enzyme. The script fits the Michaelis-Menten equation by non-linear least squares (no Lineweaver-Burk bias), gives Km with its error, the turnover number kcat = Vmax/[E] and the catalytic efficiency kcat/Km in 1/M.s.

Pipeline
01 · fitMichaelis-Menten fitstat fit_model→02 · extractVmax, Km and its errorjson select→03 · kcatkcat and kcat/Km with unitsunit calc
Script
admin
#1. Initial rates v in uM/s versus substrate S in uM, enzyme 0.050 uM;
-> "[s]" "[1, 2.5, 5, 10, 20, 40, 80, 160]";
-> "[v]" "[0.3, 0.676, 1.224, 1.738, 2.573, 2.983, 3.752, 3.919]";
#2. Non-linear Michaelis-Menten fit, no Lineweaver-Burk bias;
json load "rx_ch_mm" (stat fit_model "Vmax*S/(Km+S)" "S" [s] [v] "{\"Vmax\": 4, \"Km\": 10}");
-> "[vm]" (json select "rx_ch_mm" /parameters/Vmax); -> "[km]" (json select "rx_ch_mm" /parameters/Km);
#3. Turnover number kcat and catalytic efficiency kcat/Km in 1/M.s;
-> "[kcat]" (math calc "[vm]/0.050");
-> "[eff]" (unit calc (concat [kcat] " / (1 s) / (" [km] " umol/L)") "L/(mol*s)");
-> "[_format]" "sci 3";
concat "Vmax = " (math round [vm] 2) " uM/s, Km = " (math round_uncertainty [km] (json select "rx_ch_mm" /errors/Km)) " uM, kcat = " (math round [kcat] 1) " 1/s, kcat/Km = " (math calc "[eff]") " 1/M.s"
Result
mentdb
Vmax = 4.26 uM/s, Km = 13.8 ± 1.1 uM, kcat = 85.1 1/s, kcat/Km = 6.19E6 1/M.s
Go further
  • Fit a competitive inhibition model on a second series with stat fit_model.
  • Use the [_format] session variable to choose the display (sci, eng, fix).
executed on MentDB

Serial dilution plate counts and log reduction

Triplicate plates (0.1 mL spread) were counted at 10^-6 for the culture and at 10^-2 after a disinfectant. The script computes the CFU/mL with the relative 95 % margin of the triplicate and the log reduction of the treatment.

Pipeline
01 · ciMean count and 95 % intervalstat conf_interval_mean→02 · cfuCFU/mL = count / (dilution.volume)math calc→03 · logLog reductionmath calc
Script
admin
#1. Plate counts, 0.1 mL spread: culture at 10^-6, after disinfection at 10^-2;
-> "[n0]" "[187, 203, 176]"; -> "[n1]" "[42, 51, 38]";
json load "rx_ch_cfu" (stat conf_interval_mean [n0] 0.95);
#2. CFU per mL = mean count / dilution x plated volume;
-> "[c0]" (math calc (concat (json select "rx_ch_cfu" /mean) "/(1e-6*0.1)"));
-> "[c1]" (math calc (concat (stat mean [n1]) "/(1e-2*0.1)"));
#3. Log reduction of the disinfectant, scientific display;
-> "[_format]" "sci 3";
concat "N0 = " (math calc "[c0]") " CFU/mL +- " (math round (math calc (concat "100*" (json select "rx_ch_cfu" /margin) "/" (json select "rx_ch_cfu" /mean))) 1) " %, N = " (math round [c1] 0) " CFU/mL, log reduction = " (math round (math calc "log10([c0]/[c1])") 2)
Result
mentdb
N0 = 1.89E9 CFU/mL +- 17.9 %, N = 43700 CFU/mL, log reduction = 4.64
Go further
  • Use stat dist_inv "poisson" for an exact interval on a single plate.
  • Check the 30-300 countable range before averaging.
executed on MentDB

Predator-prey cycles of Lotka-Volterra

Hares and lynx follow the Lotka-Volterra equations. The script integrates the system, measures the period of the cycle with the autocorrelation of the prey population, and draws the phase portrait (lynx against hares), a closed orbit.

Pipeline
01 · odeIntegrate the systemnum ode→02 · acfAutocorrelation of the preydsp autocorr→03 · periodFirst peak = perioddsp peaks→04 · drawPhase portraitplot ode
Script
admin
#1. Lotka-Volterra predator-prey: hares x, lynx y;
json load "rx_lv" (num ode "{\"x\": \"1.0*x-0.1*x*y\", \"y\": \"0.075*x*y-1.5*y\"}" "{\"x\": 10, \"y\": 5}" 0 60 "{\"points\": 601}");
#2. Period of the cycle from the autocorrelation of the prey, time step 0.1;
-> "[r]" (dsp autocorr (json select "rx_lv" /x) 300);
json load "rx_ac" (dsp peaks [r] "{\"height\": 0.3}");
-> "[T]" (math calc (concat "0.1*" (json select "rx_ac" /indices[0])));
#3. Phase portrait: lynx against hares, a closed orbit;
plot ode (json select "rx_lv" /) "x" "y"
Result
mentdb
The chart is shown in the Plot tab of the editor.
Go further
  • Return [T] instead of the plot to read the period (5.5 time units).
  • Draw both populations against time with plot ode and only the result of num ode.
executed on MentDB

qPCR: efficiency from the standard curve and relative expression

From a dilution series, get the slope of Ct against log10 copies (covariance over variance) and the amplification efficiency; then compare a target gene in treated and control cells, normalized by GAPDH, with the Livak 2^-ddCt and the Pfaffl methods.

Pipeline
01 · slopecov(log Q, Ct) / var(log Q)stat covariance→02 · efficiencyE = 10^(-1/slope)math calc→03 · dCtMeans of the triplicatesstat mean→04 · ratioLivak and Pfafflmath calc
Script
admin
#1. qPCR standard curve: log10 of the copies per reaction and mean Ct;
-> "[lq]" "[7, 6, 5, 4, 3]";
-> "[ct]" "[15.12, 18.49, 21.83, 25.21, 28.60]";
#2. Slope = covariance over variance of log10 Q, amplification efficiency E = 10^-1/slope;
-> "[slope]" (/ (stat covariance [lq] [ct]) (stat variance [lq]));
-> "[E]" (math calc "10^(-1/[slope])");
#3. Target gene and reference gene GAPDH, triplicate Ct, control versus treated cells;
-> "[dT]" (- (stat mean "[24.12, 24.31, 24.05]") (stat mean "[21.93, 22.18, 22.02]"));
-> "[dR]" (- (stat mean "[18.21, 18.10, 18.33]") (stat mean "[18.42, 18.30, 18.51]"));
#4. Livak 2^-ddCt with E = 2, and Pfaffl ratio with the measured efficiency, reference gene E = 1.98;
-> "[livak]" (math calc "2^([dT]-[dR])");
-> "[pfaffl]" (math calc "[E]^[dT]/1.98^[dR]");
concat "slope = " (math round [slope] 3) ", efficiency = " (math round (math calc "100*([E]-1)") 1) " %, R2 = " (math round (math pow (stat correlation [lq] [ct]) 2) 6) "; dCt target = " (math round [dT] 3) ", dCt GAPDH = " (math round [dR] 3) "; fold change Livak = " (math round [livak] 2) ", Pfaffl = " (math round [pfaffl] 2)
Result
mentdb
slope = -3.368, efficiency = 98.1 %, R2 = 0.999993; dCt target = 2.117, dCt GAPDH = -0.197; fold change Livak = 4.97, Pfaffl = 4.86
Go further
  • Test the dCt of the biological replicates with stat ttest_2 before reporting the fold change
  • Get the limit of detection from the Ct of the lowest standards with stat conf_interval_mean
executed on MentDB

Hemocytometer count: Poisson check and concentration

Yeast cells are counted in the 16 small squares of a Neubauer chamber. The script builds the frequency table, checks the Poisson distribution of the counts with the index of dispersion, and gives the cell concentration of the culture.

Pipeline
01 · tableCounts per squarestat frequency→02 · modeMost frequent countsstat mode→03 · dispersionVariance / mean, chi2 teststat pvariance→04 · p-valueUpper tail of chi2(15)stat dist_cdf→05 · concentrationCells per mLmath round_sig
Script
admin
#1. Yeast cells counted in the 16 small squares of a Neubauer chamber, sample diluted 1:10;
-> "[c]" "[7, 5, 9, 6, 8, 4, 7, 6, 10, 5, 7, 8, 6, 9, 7, 6]";
-> "[n]" 16; -> "[m]" (stat mean [c]);
#2. Count distribution and Poisson check: variance over mean near 1, chi-square dispersion test;
-> "[freq]" (stat frequency [c]);
-> "[vm]" (/ (stat variance [c]) [m]);
-> "[chi2]" (* (- [n] 1) [vm]);
-> "[p]" (- 1 (stat dist_cdf "chi2" "[15]" [chi2]));
#3. Concentration: a small square is 1/250 mm3, i.e. 4e-6 mL, times the dilution;
-> "[conc]" (/ (* [m] 10) 0.000004);
concat "mode = " (stat mode [c]) ", mean = " [m] ", population variance = " (stat pvariance [c]) ", variance/mean = " (math round [vm] 3) ", dispersion chi2 = " (math round [chi2] 2) " p = " (math round [p] 3) ", concentration = " (math round_sig [conc] 3) " cells/mL, squares with 7 cells: " (json load select "rx_p4d_freq" [freq] "/[3]/count") ", cumulative frequency up to 7: " (json select "rx_p4d_freq" "/[3]/cumulative_frequency")
Result
mentdb
mode = [6,7], mean = 6.875, population variance = 2.484375, variance/mean = 0.385, dispersion chi2 = 5.78 p = 0.983, concentration = 17200000 cells/mL, squares with 7 cells: 4, cumulative frequency up to 7: 0.6875
Go further
  • A variance/mean far above 1 means clumping: re-vortex the sample and count again.
  • Compute the 95 % Poisson confidence interval of the total count with stat dist_inv chi2.
executed on MentDB

Kleiber's law: allometry of the metabolic rate

Basal metabolic rates of seven mammals from the mouse to the cow are fitted with the allometric law B = a.M^b, then used for a 25 kg dog with its metabolic body mass M^3/4.

Pipeline
01 · fitLog-log power lawstat fit→02 · reada, b and r2json load select→03 · massM^3/4 as a 4th rootmath nroot→04 · predictBMR of the dogmath power
Script
admin
#1. Basal metabolic rate in W versus body mass in kg of 7 mammals, mouse to cow;
-> "[m]" "[0.021, 0.29, 2.5, 11, 70, 160, 600]";
-> "[b]" "[0.17, 1.45, 7.6, 22.5, 82, 150, 410]";
#2. Allometric law B = a.M^b fitted by log-log regression;
-> "[fit]" (stat fit [m] [b] "power");
-> "[a]" (json load select "rx_p4d_klb" [fit] "/a");
-> "[e]" (json select "rx_p4d_klb" "/b");
#3. Kleiber metabolic body mass M^3/4 as the fourth root of M^3, for a 25 kg dog;
-> "[mb]" (math nroot (math power 25 3) 4);
-> "[pred]" (* [a] (math power 25 [e]));
#4. Mass-specific rate of the mouse against the cow;
-> "[ratio]" (/ (/ 0.17 0.021) (/ 410 600));
concat "B = " (math round [a] 3) " M^" (math round [e] 3) " (Kleiber 3/4), r2 = " (math round (json select "rx_p4d_klb" "/r2") 4) ", dog 25 kg: M^0.75 = " (math round [mb] 2) " kg^0.75, BMR = " (math round [pred] 1) " W, mouse burns " (math round [ratio] 1) " times more per kg than the cow"
Result
mentdb
B = 3.465 M^0.752 (Kleiber 3/4), r2 = 0.9978, dog 25 kg: M^0.75 = 11.18 kg^0.75, BMR = 39 W, mouse burns 11.8 times more per kg than the cow
Go further
  • Plot the log-log data with pa xy_scatter to show the 3/4 slope.
  • Compare with the 2/3 surface law with stat fit_model and a fixed exponent.
executed on MentDB

Flow cytometry: automatic gating of leukocytes

Forward and side scatter events are grouped with a density-based clustering, which finds the lymphocyte, monocyte and granulocyte populations, leaves the debris out and gives the leukocyte differential.

Pipeline
01 · loadDensity clustering, radius 0.45ml cluster load_from_json→02 · countNumber of populationsml cluster nb→03 · sizesEvents per populationml cluster nb_point→04 · inspectOne eventml cluster point_get→05 · closeFree the memoryml cluster close
Script
admin
#1. Flow cytometry: forward and side scatter of 20 leukocytes in channel units / 100, plus 2 debris events;
-> "[ev]" "[[2.1, 1.0], [2.3, 1.2], [2.0, 0.9], [2.2, 1.1], [2.4, 1.0], [2.1, 1.3], [2.2, 0.8], [2.3, 0.9], [3.6, 2.2], [3.8, 2.4], [3.5, 2.1], [3.7, 2.5], [4.4, 5.8], [4.6, 6.1], [4.3, 5.9], [4.5, 6.3], [4.7, 6.0], [4.4, 6.2], [4.6, 5.7], [4.5, 6.0], [0.4, 3.9], [6.8, 0.3]]";
#2. Density-based gating: neighbours within 0.45, at least 3 events per population;
ml cluster load_from_json "rx_p4d_fcm" 0.45 3 [ev];
-> "[k]" (ml cluster nb "rx_p4d_fcm");
#3. Size of each population and leukocyte differential;
-> "[n0]" (ml cluster nb_point "rx_p4d_fcm" 0);
-> "[n1]" (ml cluster nb_point "rx_p4d_fcm" 1);
-> "[n2]" (ml cluster nb_point "rx_p4d_fcm" 2);
-> "[tot]" (+ [n0] (+ [n1] [n2]));
-> "[first]" (ml cluster point_get "rx_p4d_fcm" 2 0);
ml cluster close "rx_p4d_fcm";
concat [k] " populations, sizes " [n0] ", " [n1] ", " [n2] " = " (math round (* 100 (/ [n0] [tot])) 1) " %, " (math round (* 100 (/ [n1] [tot])) 1) " %, " (math round (* 100 (/ [n2] [tot])) 1) " %, " (- 22 [tot]) " events left as debris, first event of population 3: " (string replace (string replace (string replace [first] [_n_] "") " " "") "," ", ")
Result
mentdb
3 populations, sizes 8, 4, 8 = 40 %, 20 %, 40 %, 2 events left as debris, first event of population 3: [4.4, 5.8]
Go further
  • Show the gates with ml cluster xy_scatter before closing the cluster.
  • Change the radius and the minimum points to see when monocytes merge with lymphocytes.
executed on MentDB

Photosynthesis-irradiance curve (Jassby-Platt)

Gross oxygen production of a diatom culture is measured at seven irradiances. The script fits the Jassby-Platt tanh model and derives the initial slope alpha, Pmax, the saturation onset Ik and the irradiance for 90 % of Pmax.

Pipeline
01 · fitP = Pmax.tanh(alpha I / Pmax)stat fit_model→02 · parametersPmax and alphajson load select→03 · IkPmax / alphamath calc→04 · I90Ik.atanh(0.9)math atanh→05 · checktanh(1)math tanh
Script
admin
#1. Photosynthesis-irradiance curve of a diatom culture, gross O2 production in umol/mg Chl/h;
-> "[i]" "[0, 25, 50, 100, 200, 400, 800]";
-> "[p]" "[0, 21.5, 40.2, 66.1, 88.9, 97.8, 99.6]";
#2. Jassby-Platt model P = Pmax.tanh of alpha.I/Pmax;
-> "[fit]" (stat fit_model "pm*tanh(al*x/pm)" "x" [i] [p] "{\"pm\": 100, \"al\": 1}");
-> "[pm]" (json load select "rx_p4d_pi" [fit] "/parameters/pm");
-> "[al]" (json select "rx_p4d_pi" "/parameters/al");
#3. Saturation onset Ik = Pmax/alpha and irradiance for 90 % of Pmax: Ik.atanh 0.9;
-> "[ik]" (/ [pm] [al]);
-> "[i90]" (* [ik] (math atanh 0.9));
#4. Check the model at Ik: tanh 1 = 76 %;
concat "Pmax = " (math round [pm] 1) ", alpha = " (math round [al] 3) ", Ik = " (math round [ik] 1) " umol photons/m2/s, I90 = " (math round [i90] 1) ", P at Ik = " (math round (* 100 (math tanh 1)) 1) " % of Pmax, r2 = " (math round (json select "rx_p4d_pi" "/r2") 4)
Result
mentdb
Pmax = 98, alpha = 0.816, Ik = 120 umol photons/m2/s, I90 = 176.7, P at Ik = 76.2 % of Pmax, r2 = 0.9986
Go further
  • Add a photoinhibition term (Platt 1980) with stat fit_model and a third parameter beta.
  • Integrate the daily production over a sine light curve with num integrate.
executed on MentDB

Resting membrane potential: Nernst and Goldman

For a mammalian neuron at 37 C, the script computes RT/F from the CODATA constants, the Nernst potential of K+ and the Goldman-Hodgkin-Katz resting potential, then the driving force and the direction of the K+ current.

Pipeline
01 · temperature37 C in kelvinunit convert→02 · RT/FCODATA _R and _Fmath calc→03 · potentialsNernst and GHKmath log→04 · directionSign of the driving forcemath signum→05 · referenceFaraday constantconstant physics_f
Script
admin
#1. Mammalian neuron at 37 C: concentrations in mM, inside and outside, relative permeabilities K:Na:Cl = 1:0.04:0.45;
-> "[ki]" 140; -> "[ko]" 5; -> "[nai]" 15; -> "[nao]" 145; -> "[cli]" 10; -> "[clo]" 110;
-> "[t]" (unit convert 37 "degC" "K");
#2. RT/F in mV with the CODATA constants of math calc;
-> "[rtf]" (math calc "1000*_R*[t]/_F");
#3. Nernst potential of K+ and Goldman-Hodgkin-Katz resting potential;
-> "[ek]" (* [rtf] (math log (/ [ko] [ki])));
-> "[vm]" (* [rtf] (math log (/ (+ [ko] (+ (* 0.04 [nao]) (* 0.45 [cli]))) (+ [ki] (+ (* 0.04 [nai]) (* 0.45 [clo]))))));
#4. Driving force on K+;
concat "RT/F = " (math round [rtf] 2) " mV (F = " (constant physics_f) "), E_K = " (math round [ek] 1) " mV, GHK Vm = " (math round [vm] 1) " mV, K+ driving force = " (math round (- [vm] [ek]) 1) " mV, sign " (math signum (- [vm] [ek])) " = outward K+ current"
Result
mentdb
RT/F = 26.73 mV (F = 96485.33212331001 C/mol), E_K = -89.1 mV, GHK Vm = -67.3 mV, K+ driving force = 21.7 mV, sign 1 = outward K+ current
Go further
  • Change the external K+ (hyperkalemia, 8 mM) and see the depolarisation.
  • Solve the K+ permeability ratio that gives -70 mV with num solve.

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