Serial Dilution Explained: The Formula, a Worked Series and Where Errors Creep In

A serial dilution is a stepwise series in which each solution is made from the one before it, so the concentration falls by the same factor at every step. Laboratories use it to reach very low concentrations for a standard series, a concentration-response curve or a colony count. The arithmetic is simple, but errors build along the series.

This guide gives the dilution factor formula, a worked microplate table, the arithmetic of pipetting error, peptide loss to plastic, log-spaced series, and when independent dilutions suit better.

The serial dilution formula and dilution factor

The dilution factor is the total volume after the transfer divided by the volume transferred. Moving 40 µL into 160 µL gives a total of 200 µL, so the factor is 200 / 40, or 5.

Factors multiply along a series. Three tenfold steps give a cumulative factor of 10 × 10 × 10, or 1,000, and the final well holds one-thousandth of the starting concentration.

How C1V1 = C2V2 fits in

The relationship C₁V₁ = C₂V₂ says the amount of analyte is unchanged by adding liquid. Moving 100 µL of a 100 µM solution to a final volume of 200 µL gives 100 × 100 = C₂ × 200, so C₂ is 50 µM. For one-off calculations involving molecular weight, the site’s peptide calculator handles the arithmetic.

A worked example: twofold dilution down a plate row

Take a generic standard at 100 µM and a row of eight wells in a microplate. Well 1 holds 200 µL of the standard, and wells 2 to 8 each hold 100 µL of assay buffer. Move 100 µL from well 1 to well 2, mix, take a fresh tip, then move 100 µL from well 2 to well 3, and continue along the row. Remove and discard 100 µL from well 8 so that all eight wells end at the same volume.

The fourth column is the serial dilution formula, C_n = 100 / 2ⁿ. The last two columns are illustrative and are explained in the next section. If an equal volume of assay reagent is later added to each well, every value here halves, so report the final in-well concentration.

The same logic for colony counts

In microbiology, a tenfold series thins a culture until one plate carries a countable number of colonies. An American Society for Microbiology method advises selecting the plate with 30 to 300 colonies [1], though recommended windows vary: a study of plating accuracy lists ranges including 25–250 and 15–300, and reports that ISO 7218 lowered the lower limit to 10 colonies per plate [2].

If a plate from the 10⁻⁵ dilution receives 100 µL and grows 42 colonies, the undiluted sample held about 42 / 10⁻⁵, or 4.2 × 10⁶ colony-forming units (CFU), per 100 µL.

How error builds along a serial dilution

Pipetting error comes in two kinds, which is why the ISO 8655-6 gravimetric procedure evaluates systematic error and random error as separate results [3].

Random error grows slowly. A concentration at step n is the stock multiplied by n transfer ratios. The law of propagation of uncertainty combines independent contributions as a sum of variances [4], and for a product of ratios that means relative variances add. If each step carries an independent relative error u, the relative error after n steps is u × √n. At ±1% per step, seven transfers give ±2.6%.

Systematic error grows quickly. A bias repeats in the same direction at every step, so it multiplies. If every transfer ratio runs 2% low, the concentration after n steps is 0.98ⁿ of nominal. After seven transfers that is 0.868, or 13.2% low. At 5% per step it reaches about 30% low.

What the published modelling shows

Hanson, Ekins and Chodera modelled an eight-point, 1:2 tip-based series, using manufacturer specifications for the liquid handler and a measured effect in which residual liquid in washed fixed tips dilutes the sample. The concentration error grew with each step, and the relative bias reached almost −50% in the final well, with longer series approaching −100%. The bias shifted the modelled potency estimates toward weaker values by about 0.25 log₁₀ units. Without that effect, the model’s pipetting bias averaged out, since it was assumed to be zero on average, so one repeating fault accounted for the accumulated bias [5].

Hedges derived variance estimates from a pipette’s specification alone and advised treating the resulting confidence intervals as minimum values [6].

Why colony counts tolerate this better

Plate counts are read on a logarithmic scale, and their largest error is often not the pipette. Jongenburger and colleagues found that low colony counts, which follow Poisson statistics, dominated uncertainty at low numbers, while pipetting, weighing and counting errors were small under well-controlled practice [2]. By that logic, the 42-colony plate above carries a counting error of about √42 / 42, or 15%. Martini and colleagues likewise expect compounding pipette error to be negligible with calibrated equipment, while noting that one failing O-ring can bias a column of a multichannel transfer [7]. Ben-David and Davidson’s simulations stayed within about ±0.1 log₁₀ of reference values even with dilution errors up to 10% [8].

Assays that report on a continuous scale have less room: a 13% shift can be material for a calibration curve. Our guides to analytical method validation and Michaelis-Menten kinetics show where the concentration axis feeds into reported parameters, and a Bradford or BCA assay is a routine example of a standard series.

Mixing between steps and changing tips

Each calculation assumes the transferred aliquot has the average concentration of its well. The American Society for Microbiology method names accurate pipetting and adequate agitation of dilution tubes as the determinants of accuracy [1], and statistical models of dilution series assume homogenization at each step [9]. The Hanson model explicitly left out incomplete mixing [5], so the bias figures above do not include it.

• Mix before the next transfer. Mix the well thoroughly by repeated aspiration or on a plate shaker, using the same method every time, and avoid foam.
• Change tips between steps where carryover matters. The same method advises using different pipettes between dilutions to avoid carrying cells across [1]. For analytes, liquid left on the outside or inside of a tip moves from a higher concentration into a lower one, and the relative effect grows as concentrations fall.
• Check fixed tips separately. Washed fixed tips can retain system liquid, the source of the bias in the Hanson model [5].
• Do not assume the tip is inert. Ekins and colleagues found tip-based serial dilution gave weaker potency values than acoustic dispensing for all 14 compounds in a small data set; they suggested tip leaching but did not measure it [10].

Peptide adsorption and what the stock concentration means

Loss to tube and plate walls

Kristensen and colleagues measured three cationic membrane-active peptides (mastoparan X, melittin and magainin 2) in glass and plastic. At 1 µM in 220 µL, only 10–20% of the peptide was recovered from borosilicate glass or standard polypropylene after one hour. Adsorption occurred within seconds, and recovery rose with concentration, consistent with the walls saturating. Most relevant here, passing a 5 µM solution through successive containers took recovery close to zero after four containers [11]. A real series also lowers concentration at each step, but the repeated exposure to fresh surfaces is similar.

Protein LoBind tubes lost less peptide in that study [11], but low-bind labware is not a universal answer. Goebel-Stengel and colleagues tested radiolabelled endocrine peptides including ghrelin, GLP-1, insulin and leptin in a range of glass and plastic tubes. Recovery depended on the container, siliconization reduced it, and adding bovine serum albumin improved it [12]. Weikart and colleagues tested dye-labelled peptides at 1.5 nM: standard polypropylene recovered about 20% of epidermal growth factor while low-binding plates recovered nearly all, yet glucagon on one commercial low-bind plate did no better than on standard plastic. Those authors work for the maker of another plate in the comparison [13].

The practical message is to run a recovery check for your own peptide, labware and assay buffer before trusting the low end of a curve. Any additive must also be compatible with the assay, and pre-saturating a container can backfire because adsorbed peptide can later come off [11].

Why the nominal stock concentration can mislead

Every well in a series is a multiple of the stock value, so an error in the stock scales the whole row. Peptide powders commonly hold water and counter-ions as well as peptide, which is why net peptide content and TFA or acetate salts change what a nominal concentration means. A series built on gross weight keeps correct ratios between wells but wrong absolute values.

Chemical change matters too; our guide to how peptides degrade covers the routes. A working rule, not a cited finding, is to prepare the series close to the assay.

Choosing twofold, half-log or tenfold spacing

A large factor covers more range with fewer wells, and a small factor gives finer resolution.

Twofold steps are the familiar pattern in broth microdilution, where they run through a 96-well plate [14]. Half-log spacing is easy to plan: 100 µL into about 216 µL of assay buffer. Log-spaced points suit sigmoidal curves, and the range has to reach both plateaus. Our guide on how to read a peptide study covers how to weigh reported values.

Serial or independent dilutions: which to use

In a parallel or independent design, each concentration is prepared directly from the stock or from an intermediate, rather than from the previous well. The two designs fail differently.

In the Hanson model, direct acoustic dispensing avoided accumulated bias but showed potency variation nearly twice that of the tip-based method [5]. A middle route is to prepare two or three independent intermediates and run short series from each, which limits how far a single error can travel. Replicate series from separate starting points also expose drift that replicate wells cannot.

What’s established and what isn’t

Sourcing peptide reagents for concentration work: notes for Canadian labs

When comparing sources of research peptides in Canada, ask for documentation that speaks to the issues above.

• Identity and purity. Our guides on reading a peptide analysis report and HPLC testing explain those results.
• Net peptide content. HPLC purity does not state how much of the powder’s weight is peptide, so ask whether that figure is available if absolute concentration matters.
• Traceability. Record the lot behind each series; see lot number traceability.
• Storage. Our peptide storage guide covers lyophilized material.
• Supplier comparison. See how to compare Canadian peptide suppliers and our overview of research peptides in Canada.

These catalogue peptides illustrate the properties discussed above. None of the adsorption studies cited here tested them, so recovery would need its own check.

LL-37A human cathelicidin from the cationic host-defence peptide class [15].View product GHRP-6A six-residue ghrelin-receptor agonist; reference [12] tested native ghrelin, not this peptide.View product SNAP-8An eight-residue peptide containing one methionine, a residue open to oxidation.View product

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