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Equation 7 Terms - Lineshape Fitting Method (Figure S8)

Paper's Equation 7

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

How to Obtain Each Term with Lineshape Fitting

1. [M]ref - Reference Concentration

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

2. [TSP]₁ - TSP Concentration in Sample

Aspect Value Source
What TSP concentration in test sample Sample preparation records
Value 0.95 mM Excel: "Sample Preparation" sheet
Details Stock solution: 0.9508 mM TSP + 6 mM NaN₃ Same in all samples

3. [TSP]ref - TSP Concentration in Reference

Aspect Value Source
What TSP concentration in reference sample Sample preparation records
Value 0.95 mM Same as sample (constant)

Note: Since [TSP]₁ = [TSP]ref, these cancel out in Equation 7!

4. scale(M) - Metabolite Scaling Factor (LINESHAPE FITTING)

Method: Fit reference spectrum to sample spectrum

Procedure:

# 1. Extract region of interest
sample_region = sample_spectrum[ppm_region]
reference_region = reference_spectrum[ppm_region]

# 2. Find scale factor that minimizes residuals
#    (optimize scale to minimize Σ|sample - scale×reference|²)
scale(M) = argmin(||sample - scale×reference||²)

What it represents:

  • The scaling factor that makes the reference spectrum match the sample spectrum
  • Accounts for the full lineshape (including artifacts)
  • Physically: scale(M) ∝ [M]₁/[M]ref × (spectrometer stability factors)

Difference from Integration:

  • Integration: scale(M) = ∫(Ala_sample) / ∫(Ala_ref)
  • Lineshape fitting: scale(M) = optimal fit parameter from full spectrum matching

5. scale(TSP) - TSP Scaling Factor

Method: Peak integration (same as Figure S9)

Calculation:

# Integrate TSP in sample and reference
∫TSP_sample = integrate(sample, region=(-0.2, 0.2))
∫TSP_ref = integrate(reference, region=(-0.2, 0.2))

scale(TSP) = ∫TSP_sample / ∫TSP_ref

Purpose:

  • Corrects for spectrometer stability variations (λSS factor in paper)
  • Accounts for different TSP peak intensities due to instrument drift

Two Approaches for Lineshape Fitting

APPROACH 1: TSP-Normalize First, Then Fit ⭐ Recommended

Steps:

  1. TSP-normalize both spectra: spectrum / ∫TSP
  2. Fit normalized reference to normalized sample
  3. The fitted scale = scale(M)/scale(TSP) combined

Equation:

[M]₁ = [M]ref × fitted_scale

Advantages:

  • Single fitting step
  • TSP normalization handles instrument stability
  • Direct concentration calculation

APPROACH 2: Fit Raw, Then TSP-Correct

Steps:

  1. Fit raw reference to raw sample → get scale(M)
  2. Calculate scale(TSP) from TSP integration
  3. Apply full Equation 7

Equation:

[M]₁ = [M]ref × scale(M) / scale(TSP)

Advantages:

  • Separates lineshape fitting from TSP correction
  • Matches paper's equation structure exactly

Complete Example (Approach 1)

Term Value How Obtained
[M]ref 40.00 mM Excel preparation data
[TSP]₁ 0.95 mM Sample prep (cancels out)
[TSP]ref 0.95 mM Sample prep (cancels out)
scale(M)/scale(TSP) 0.508 From TSP-normalized lineshape fitting
[M]₁ 20.32 mM 40.00 × 0.508

Physical Meaning:

  • Expected: 20 mM / 40 mM = 0.5
  • Fitted: 0.508
  • Recovery: 101.6% ✓

Visual Comparison

Integration Method (S9):

┌─────────────────────────────────────┐
│  scale(M) = Area(sample) / Area(ref) │
│  scale(TSP) = Area(TSPs) / Area(TSPr)│
└─────────────────────────────────────┘

Lineshape Fitting Method (S8):

┌────────────────────────────────────────────┐
│  Fit: sample ≈ scale × reference          │
│                                            │
│  ┌──────────┐      ┌──────────┐           │
│  │ REFERENCE│ ───→ │  sample  │  (match)  │
│  │  (40mM)  │scale │  (20mM)  │           │
│  └──────────┘      └──────────┘           │
│                                            │
│  scale(M) = fitted_scale                   │
│  scale(TSP) = TSP area ratio (integration) │
└────────────────────────────────────────────┘

Key Differences

Aspect Integration (S9) Lineshape Fitting (S8)
scale(M) Area ratio Fit parameter
Data used Peak area only Full spectrum
Artifacts Ignored Accounted for
Overlaps Difficult Better
Speed Fast Slower (optimization)
Accuracy Good for pure peaks Better for complex peaks

Summary Table

Equation 7 Term Integration (S9) Lineshape Fitting (S8)
[M]ref 40.00 mM (Excel) 40.00 mM (Excel)
[TSP]₁ 0.95 mM (Sample Prep) 0.95 mM (Sample Prep)
[TSP]ref 0.95 mM (Sample Prep) 0.95 mM (Sample Prep)
scale(M) ∫Ala_s / ∫Ala_r Fit parameter
scale(TSP) ∫TSP_s / ∫TSP_r ∫TSP_s / ∫TSP_r

Bottom Line: The main difference is how scale(M) is obtained:

  • Integration: Simple area ratio
  • Lineshape fitting: Full spectrum optimization