96-Well Quantitative Plate Instructions for Use
1. Purpose
The 96 quantitative plate, used in conjunction with the SeaPlus programmable quantitative sealer, enables simple, rapid, and accurate quantitative detection of total coliforms, Escherichia coli, fecal (thermotolerant) coliforms, enterococci, and Pseudomonas aeruginosa.
2. Detection Principle
Based on the traditional Most Probable Number (MPN) statistical model, the system uses a programmable quantitative sealer to automatically distribute the sample-reagent mixture into 96 independent wells. After incubation, the number of positive wells is converted to an MPN value to determine the bacterial count in the original sample.
3. MPN Formula Explanation
In microbiological enumeration, the Most Probable Number (MPN) is an indirect counting method based on dilution culture and probability of positive reactions. It estimates the most likely concentration of microorganisms using statistical modeling involving the Poisson distribution and likelihood function.
4. Core Assumptions and Mathematical Foundation of MPN
4.1 Application Conditions
Microorganisms are uniformly distributed in the sample
Microbial presence in each dilution tube is an independent event
Each microorganism can grow and produce a positive reaction under the culture conditions
Dilution process is precise with a fixed dilution factor (usually 10-fold)
4.2 Poisson Distribution
The Poisson distribution is used to describe the probability distribution of the number of microorganisms per unit volume. For a culture tube at a specific dilution with a theoretical microbial concentration λ (units: organisms per tube), the probability of finding exactly k microorganisms in the tube is given by:
P(k;λ)=k!e−λ⋅λk Where:
e is the base of the natural logarithm (≈2.71828)
k is the actual number of microorganisms in the tube (0, 1, 2, ...)
λ is the theoretical mean number of microorganisms per tube at that dilution (directly related to the original sample concentration and the dilution factor)5. Core Mathematical Formulation of MPN
5. Core Mathematical Formulation of MPN: Likelihood Function and Optimization
The essence of MPN is to infer the "most probable λ" from the "observed number of positive tubes" and then convert it to the original sample concentration. The core procedure relies on maximizing the likelihood function, with the specific formulas as follows:
5.1 Likelihood Function for a Single Dilution (Fundamentals)
Assume n culture tubes are prepared at a specific dilution (e.g., dilution factor D), with r tubes positive (microbial growth present) and n−r tubes negative (no microbial growth).
Probability of a negative tube: When the number of microorganisms k = 0, substituting into the Poisson distribution formula gives P(negative) = e^(-λ) (since λ^0 = 1 and 0! = 1).
Probability of a positive tube: When the number of microorganisms k ≥ 1, P(positive) = 1 − P(negative) = 1 − e^(-λ).
Therefore, the likelihood function (probability of observing r positive and n−r negative tubes) at this dilution is:
L(λ)=(1−e−λ)r⋅(e−λ)n−r
5.2 Likelihood Function for Multiple Dilutions (Practical Application)
In actual MPN experiments, typically three or more dilutions are set (e.g., D₁ < D₂ < D₃, corresponding to decreasing concentrations), with nᵢ tubes at each dilution and rᵢ positive tubes observed.
Since microbial distribution at each dilution is independent, the total likelihood function is the product of the likelihood functions for each dilution:
L(λ1,λ2,...,λm)=∏i=1m[(1−e−λi)ri⋅(e−λi)ni−ri]
Where:
m is the number of dilutions (e.g., 3)
λᵢ is the theoretical mean number of microorganisms per tube at the i-th dilution
Key relationship: λᵢ = C · V · Dᵢ (C = original concentration, V = inoculation volume, Dᵢ = dilution factor)
5.3 Core MPN Formula: Maximizing Likelihood to Solve for C
The objective is to find the value of C that maximizes the total likelihood function L. Since the likelihood function is in product form, direct differentiation is complex, so we take the natural logarithm to convert it to an additive form:
lnL=∑i=1m[ri⋅ln(1−e−λi)+(ni−ri)⋅(−λi)]
Substituting λᵢ = C · V · Dᵢ and taking the partial derivative with respect to C, setting it to zero gives the likelihood equation:
∑i=1m[1−e−λiri⋅V⋅Di⋅e−λi−(ni−ri)⋅V⋅Di]=0
This transcendental equation cannot be solved directly by algebraic methods and requires numerical solution using iterative methods (e.g., Newton-Raphson method). The resulting C is the MPN value, typically expressed as "organisms per 100 mL" or "organisms per gram" depending on the sample type.
Detection range: 1-2419 MPN per 100 mL water sample.
6. Operating Procedures
Transfer 100 mL of the sample or diluted sample into a sterilized Erlenmeyer flask, then add 2.7 g ± 0.5 g of medium powder and stir thoroughly until completely dissolved.
Hold the quantitative plate vertically with one hand, positioning it so that the wells are oriented upward. Press the upper part of the plate to bend it, pulling the protrusions on the back of the plate toward the fixed side of your palm. If it cannot be bent, use the other hand to open the plate, taking care not to touch the wells, and push it into the palm to avoid secondary contamination.
Pour the mixture of samples and reagent into the quantitative plate.
Place the quantitative plate on the rubber mat so that the wells are centered and the liquid does not contact the protrusions on the back. The holes on the rubber mat should align with those on the quantitative plate, with the openings facing upward. The presence of foam in the liquid is acceptable.
Seal the plate using a programmable quantitative sealer.
Incubate according to the specific time and temperature requirements for the test. For example, when detecting fecal coliforms, place the sealed 96-well quantitative plate in a constant-temperature incubator and incubate at 44.5℃ ± 0.5℃ for 24 hours.
After incubation, count the number of positive wells and compare the results with the MPN table to determine the MPN value of the target bacteria per 100 mL of water sample.
7. Precautions
Maintain sterile technique throughout the procedure.
Strictly control incubation temperature and time to ensure accuracy.
Dispose of positive samples according to laboratory protocols (e.g., place in heat-resistant bags, do not seal tightly, sterilize by autoclaving before disposal).