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Equation 7 Terms - How to Obtain Each One (Integration Method)

Paper's Equation 7

[M]₁ = ([M]ref × [TSP]₁ × scale(M)) / ([TSP]ref × scale(TSP))

Term-by-Term Breakdown

1. [M]ref - Reference Concentration

Aspect Value How Obtained
What Known concentration of Alanine in reference sample Sample preparation records
Value 40.00 mM Excel file: "Reference Library Data Analysis"
Source File 10.dx Highest concentration reference

In our data:

  • Row 34 in Excel: "40mM Ala 01/03/2021"
  • Concentration: 40.633 mM (prepared)

2. [TSP]₁ - TSP Concentration in Sample

Aspect Value How Obtained
What TSP concentration in test sample Sample preparation records
Value 1.0 mM (assumed constant) Same TSP added to all samples

Key Point:

  • TSP is added as internal standard at same concentration to all samples
  • [TSP]₁ = [TSP]ref (they cancel out in the equation)

3. [TSP]ref - TSP Concentration in Reference

Aspect Value How Obtained
What TSP concentration in reference sample Sample preparation records
Value 1.0 mM (same as sample) Same TSP concentration

Result: [TSP]₁/[TSP]ref = 1.0 (cancels out)

4. scale(M) - Metabolite Scaling Factor

Aspect Calculation How Obtained
What Ratio of metabolite integrals Peak integration
Formula scale(M) = ∫(Ala in sample) / ∫(Ala in reference) Numerical integration

Step-by-step:

# Reference (40 mM)
ref_ala_integral = integrate(ref_ppm, ref_data, region=(1.4, 1.55))
# Result: 3.0131e+06

# Sample (e.g., 20 mM)
sample_ala_integral = integrate(sample_ppm, sample_data, region=(1.4, 1.55))
# Result: 2.9092e+06

# scale(M)
scale_M = sample_ala_integral / ref_ala_integral
# Result: 0.9655

Physical meaning:

  • scale(M) ≈ 0.96 means sample has ~96% of the alanine signal compared to reference
  • Expected: 20/40 = 0.5 (but we get 0.96 due to varying TSP areas)

5. scale(TSP) - TSP Scaling Factor

Aspect Calculation How Obtained
What Ratio of TSP integrals Peak integration
Formula scale(TSP) = ∫(TSP in sample) / ∫(TSP in reference) Numerical integration

Step-by-step:

# Reference (40 mM)
ref_tsp_integral = integrate(ref_ppm, ref_data, region=(-0.2, 0.2))
# Result: 2.3327e+05

# Sample (e.g., 20 mM)
sample_tsp_integral = integrate(sample_ppm, sample_data, region=(-0.2, 0.2))
# Result: 3.6650e+05

# scale(TSP)
scale_TSP = sample_tsp_integral / ref_tsp_integral
# Result: 1.5711

Physical meaning:

  • scale(TSP) = 1.57 means sample has 57% MORE TSP signal than reference
  • This should be ~1.0 (same TSP concentration)
  • The variation indicates instrument instability (what we're correcting for!)

Simplified Equation (When [TSP] is constant)

Since [TSP]₁ = [TSP]ref:

[M]₁ = [M]ref × scale(M) / scale(TSP)

Or equivalently:

[M]₁ = [M]ref × (Ala/TSP)_sample / (Ala/TSP)_reference

Complete Example Calculation (File 20)

Term Value Source
[M]ref 40.00 mM Known
scale(M) 0.9655 ∫Ala_sample / ∫Ala_ref = 2.91e6 / 3.01e6
scale(TSP) 1.5711 ∫TSP_sample / ∫TSP_ref = 3.67e5 / 2.33e5

Calculation:

[M]₁ = 40.00 × 0.9655 / 1.5711
     = 40.00 × 0.6144
     = 24.58 mM

Comparison:

  • Calculated: 24.58 mM
  • Known: 20.32 mM
  • Recovery: 121%

Visual Summary

┌─────────────────────────────────────────────────────────────────┐
│                    EQUATION 7 FLOW                               │
├─────────────────────────────────────────────────────────────────┤
│                                                                  │
│  REFERENCE (40 mM)              SAMPLE (20 mM)                  │
│  ┌──────────────┐              ┌──────────────┐                 │
│  │ ∫Ala = 3.01e6 │              │ ∫Ala = 2.91e6 │                 │
│  │ ∫TSP = 2.33e5 │              │ ∫TSP = 3.67e5 │                 │
│  └──────────────┘              └──────────────┘                 │
│         │                              │                        │
│         ▼                              ▼                        │
│  scale(M)_ref = 1.0           scale(M) = 0.9655                 │
│  scale(TSP)_ref = 1.0         scale(TSP) = 1.5711               │
│                                                                  │
│  ┌──────────────────────────────────────────────────────────┐   │
│  │  [M]₁ = 40.00 × (0.9655 / 1.5711) = 24.58 mM            │   │
│  └──────────────────────────────────────────────────────────┘   │
│                                                                  │
└─────────────────────────────────────────────────────────────────┘

Key Takeaway

Using integration method (Figure S9):

  1. [M]ref → From sample preparation (Excel file)
  2. [TSP]₁, [TSP]ref → Same value, cancels out
  3. scale(M) → Peak area of metabolite (trapezoidal integration)
  4. scale(TSP) → Peak area of TSP (trapezoidal integration)

All terms obtained from simple numerical integration of peak areas!