PCR Fold Amplification Calculator
PCR Fold Amplification Calculator
Theoretical yield as (1 + E) raised to the number of cycles — and what the exponent does to a small efficiency deficit, which is why qPCR efficiency is measured rather than assumed.
PCR Fold Amplification
Efficiency, cycles → fold30 cycles at 100% amplification efficiency
Formula
E = efficiency as a fraction, so 1.0 is 100%
at E = 1, fold = 2^n
- E
- amplification efficiency as a fraction. E = 1 means every template molecule present at the start of a cycle is copied during it, so the quantity doubles; E = 0.8 means only 80% are copied and the quantity multiplies by 1.8. The page takes it as a percentage and divides by 100
- n
- the number of cycles. Because it is the exponent, it multiplies the effect of any efficiency difference — which is the point of the page
- (1 + E)
- the per-cycle multiplier. The whole behaviour of the formula is in the fact that this number is raised to a power: a 10% shortfall in the multiplier becomes a 20-fold shortfall in yield over thirty cycles, and a 40-fold shortfall over thirty-five
- why efficiency is measured
- because it cannot be inferred from a successful reaction. An assay at 80% efficiency still produces abundant product and a clean curve; it simply produces 24 times less of it than a 100% assay over 30 cycles, and quantification that assumes doubling gets the wrong answer
- what the formula ignores
- plateau. Exponential growth is a good description of the early and middle cycles only. Once reagents deplete and product re-anneals in competition with the primers, amplification slows and stops — which is why quantitative PCR measures the cycle at which signal crosses a threshold rather than the amount of product at the end
Worked example
30 cycles at 100% amplification efficiency
Per-cycle multiplier = 1 + 1.00 = 2, so fold = 2³⁰ = 1,073,741,824, about 1.07 × 10⁹
Now drop the efficiency to 80% and change nothing else. The multiplier is 1.8, and 1.8³⁰ = 45,517,160, about 4.55 × 10⁷
The ratio is 1.07 × 10⁹ ÷ 4.55 × 10⁷ = 23.6. A 20 percentage point efficiency deficit costs a factor of nearly 24 in yield over the same 30 cycles
Extend both to 35 cycles and the gap widens further: 2³⁵ is 34,359,738,368 against 1.8³⁵ of 860,077,682, a ratio of just under 40. The same deficit, more cycles, a bigger penalty — that is what an exponent does
The practically important range is subtler. At 95% efficiency, 30 cycles gives 502,386,940 — less than half the yield at 100%, from a 5 percentage point difference that would be easy to miss on a standard curve
This is the assumption the ΔΔCt method makes when it uses a base of 2: that both the target and the reference assay amplify at exactly 100%. Where they do not, the base is wrong for both and the error compounds with the size of the ΔΔCt
Yield over 30 cycles at different efficiencies
| Efficiency | Per-cycle multiplier | Fold amplification | Relative to 100% |
|---|---|---|---|
| 100% | 2.00 | 1,073,741,824 | 1 |
| 95% | 1.95 | 502,386,940 | about 1/2 |
| 90% | 1.90 | 230,466,618 | about 1/5 |
| 85% | 1.85 | 103,550,417 | about 1/10 |
| 80% | 1.80 | 45,517,160 | about 1/24 |
| 70% | 1.70 | 8,193,466 | about 1/131 |
Yield at 100% efficiency by cycle number
| Cycles | Fold amplification |
|---|---|
| 10 | 1,024 |
| 20 | 1,048,576 |
| 25 | 33,554,432 |
| 30 | 1,073,741,824 |
| 35 | 34,359,738,368 |
| 40 | 1,099,511,627,776 |
What an exponent does to a small deficit
The polymerase chain reaction is described by one of the simplest growth models there is. If a fraction E of the template molecules present at the start of a cycle are successfully copied during it, then the quantity is multiplied by (1 + E) each cycle, and after n cycles it has been multiplied by (1 + E) raised to the power n. At perfect efficiency, E = 1, the multiplier is 2 and thirty cycles give 2³⁰ — a little over a thousand million fold. That is the number that makes the technique work: a handful of starting molecules becomes a detectable quantity.
The interesting behaviour is in what happens when E is not 1. At 80% efficiency the multiplier is 1.8 rather than 2, which sounds like a modest shortfall. Over thirty cycles it is not: 1.8³⁰ is about 4.6 × 10⁷ against 1.07 × 10⁹, a factor of nearly 24. Extend both to thirty-five cycles and the same deficit costs a factor of 40. A per-cycle difference of ten per cent becomes an order-of-magnitude difference in yield, because the exponent applies it again and again — and the penalty grows with the number of cycles, not just with the size of the deficit.
The range that matters in practice is narrower and less obvious. At 95% efficiency, thirty cycles gives a little under half the yield of a 100% assay. Five percentage points, and half the product. An assay at 95% or even 90% produces plenty of material and a perfectly convincing amplification curve, so nothing about its appearance announces the deficit. That is exactly why efficiency has to be measured from a standard curve rather than inferred from a reaction that worked, and why the MIQE guidelines ask for the efficiency figure to be published: without it, a quantitative result cannot be evaluated by anyone else.
The quantitative consequence is specific. The ΔΔCt shortcut computes relative expression as 2 raised to the negative ΔΔCt, and the 2 in that expression is precisely the assumption that E = 1 exactly, for the target assay and the reference assay alike. Where an assay runs at 90%, its true base is 1.9, and because the term is exponentiated the error compounds with the size of the ΔΔCt rather than staying a fixed percentage. This page quantifies that assumption; the efficiency calculator measures the quantity it needs; and where the measured efficiencies are not close to 100% and to each other, the Pfaffl method takes them as explicit inputs instead. One caveat applies to everything above: exponential growth describes the early and middle cycles only. Reagents deplete, product re-anneals in competition with the primers, polymerase activity falls, and a real reaction plateaus — which is why quantification reads the cycle at which signal crosses a threshold during the exponential phase and never the amount of product at the end.
Frequently asked questions
How do you calculate fold amplification in PCR?
Fold amplification = (1 + E)^n, where E is the efficiency as a fraction and n the number of cycles. At 100% efficiency (E = 1) over 30 cycles that is 2³⁰ = 1,073,741,824. At 80% efficiency the multiplier is 1.8 and the same 30 cycles give about 4.55 × 10⁷.
How much does low PCR efficiency reduce yield?
Far more than the efficiency figure suggests, because the deficit compounds. Over 30 cycles, 95% efficiency gives about half the yield of 100%, 90% about a fifth, and 80% about a twenty-fourth. Over 35 cycles the 80% penalty widens to about a fortieth.
Why does qPCR efficiency have to be measured rather than assumed?
Because an inefficient assay still looks like a working one. An assay at 90% efficiency produces abundant product and a clean curve, but yields a fifth of what a 100% assay would over 30 cycles — and any quantification assuming doubling gets the wrong answer. Efficiency comes from a standard curve.
What does the 2 in 2^-ΔΔCt assume?
That one cycle doubles the product, for both the target and the reference assay — that is, that both run at exactly 100% efficiency. This page quantifies what that assumption costs when it is wrong: at 90% the true base is 1.9, and the error compounds with the size of the ΔΔCt.
Can amplification efficiency exceed 100%?
Not really. An apparent efficiency above 100% usually means primer-dimer or non-specific product contributing signal, an inhibitor being diluted out along the standard curve and lifting its low points, or a pipetting error in the dilution series. Investigate it rather than reporting it.
Related calculators
References
- Bustin SA, Benes V, Garson JA, et al. The MIQE guidelines: minimum information for publication of quantitative real-time PCR experiments. Clin Chem. 2009;55(4):611–622.
- Livak KJ, Schmittgen TD. Analysis of relative gene expression data using real-time quantitative PCR and the 2(−ΔΔC(T)) method. Methods. 2001;25(4):402–408.
- Pfaffl MW. A new mathematical model for relative quantification in real-time RT-PCR. Nucleic Acids Res. 2001;29(9):e45.
- Svec D, Tichopad A, Novosadova V, Pfaffl MW, Kubista M. How good is a PCR efficiency estimate: recommendations for precise and robust qPCR efficiency assessments. Biomol Detect Quantif. 2015;3:9–16.
Medical Disclaimer: The tools and content provided here are for educational and reference purposes only. They are not intended to substitute for professional medical advice, diagnosis, or treatment. Clinical decisions should always be based on the comprehensive assessment of a qualified healthcare professional.