The Analysis of Human Serum Albumin Proteoforms Using Compositional Framework

Abstract

Mass spectrometric immuno assays (MSIA) can now measure multiple modified forms of a protein in large cohorts of patients. These measurements consist of the relative abundances of proteoforms, and are well-suited for the compositional data analysis statistical framework. In this article, we describe an approach to the analysis of relative abundance of proteoforms from MSIA data using the compositional framework. We demonstrate the application of these concepts by exploring the association of human serum albumin’s posttranslational modifications and kidney function in patients with Type 2 diabetes mellitus. Finally, we discuss the pitfalls of ignoring the compositional nature of such data, and highlight emerging applications demonstrating the generality of the framework.

Appendix

1.1 A Synopsis on Compositional Framework

Compositional data describe the proportion that each of D components contributes to the whole. Coherence of inference between subset of components and the whole composition, also called subcompositional coherence, is an important feature of the analysis. Scale invariance is essential for such analysis.

A D part composition is defined as an element of the d-dimensional positive simplex

$$\displaystyle{ \mathcal{S}^{\mathrm{d}} = \left \{\mathbf{x} = (x_{ 1},x_{2},\ldots,x_{D}): x_{i}> 0(i = 1,2,\ldots,D),\sum \limits _{i=1}^{D}x_{ i} = 1\right \} }$$

(5)

where d = D − 1. We will use the convention d = D − 1 in the rest of Appendix. Thus the sample space is a subset of the space $\mathcal{S}^{\mathrm{d}}$.

Let C denote the closure operator on $\mathbb{R}^{\mathrm{D}}$ which normalizes the vector to a unit sum. That is, for $z \in \mathbb{R}^{\mathrm{D}}$,

$$\displaystyle{ C(z) = \left ( \frac{z_{1}} {\sum _{i=1}^{D}z_{i}}, \frac{z_{2}} {\sum _{i=1}^{D}z_{i}},\ldots, \frac{z_{D}} {\sum _{i=1}^{D}z_{i}}\right ) \in \mathcal{S}^{\mathrm{d}} }$$

(6)

Also, we define two additional operations. For any two elements x = (x ₁, x ₂, …, x _D) and $y = (y_{1},y_{2},\ldots,y_{\mathrm{D}}) \in \mathcal{S}^{\mathrm{d}}$ and for $\alpha \in \mathbb{R}$, we define

$$\displaystyle{ x \oplus y = C((x_{1} \cdot y_{1},x_{2} \cdot y_{2},\ldots,x_{D} \cdot y_{D})) }$$

(7)

$$\displaystyle{ \alpha \odot x = C((x_{1}^{\alpha },x_{ 2}^{\alpha },\ldots,x_{ D}^{\alpha })). }$$

(8)

The operations in Eqs. (7) and (8) are called the perturbation and power operators, respectively. With perturbation operator as addition and power operator as the scalar multiplication, $\mathcal{S}^{\mathrm{d}}$ acquires the structure of a d-dimensional Hilbert space [6, 21] with a metric given by (2).

The additive log ratio transform defined as:

$$\displaystyle{ \begin{array}{rl} \phi: \mathcal{S}^{\mathrm{d}} & \rightarrow \mathbb{R}^{\mathrm{d}} \\ x&\longmapsto \left (\log \left ( \dfrac{x_{1}} {x_{D}}\right ),\log \left ( \dfrac{x_{2}} {x_{D}}\right ),\ldots,\log \left ( \dfrac{x_{d}} {x_{D}}\right )\right ) \end{array} }$$

(9)

and the centered log ratio (3) are alternative co-ordinate systems on this space.

The dependence structure induced by the unit sum (compositional) constraint is often addressed by using the class of logistic normal distributions as appropriate models of the data. For illustration, consider an element $x = (x_{1},x_{2},\ldots,x_{\mathrm{D}}) \in \mathcal{S}^{\mathrm{d}}$. Following Aitchison [1] the pullback of the multivariate normal distribution using ϕ from Eq. (9) is the logistic normal density function given by:

$$\displaystyle{ f(x\vert \mu,\varSigma ) = (2\pi )^{d/2}\vert \varSigma \vert ^{-1/2}\left ( \dfrac{1} {\prod _{i=1}^{k}x_{i}}\right )\mathrm{exp}{\biggl [ -\dfrac{1} {2}(\phi (x)-\mu )'\varSigma ^{-1}(\phi (x)-\mu )\biggr ]} }$$

(10)

where μ is the location parameter in $\mathbb{R}^{\mathrm{d}}$ and Σ is the d × d variance–covariance matrix, (∏ _i = 1 ^D x _i )⁻¹ is the Jacobian of the transformation. In the following, we will denote this d-dimensional logistic normal distribution by $\mathcal{L}\mathcal{N}_{\mathrm{d}}$ and A′ will denote the transpose if A is a matrix.

The part S composition is orthogonal projection of the full composition, with respect to the inner product on $\mathcal{S}^{\mathrm{d}}$ [12]. The class preserving property of logistic normal distributions, i.e., if $x \in \mathcal{L}\mathcal{N}_{D}(\mu,\varSigma )$ and A is an n × D matrix, then $Ax \in \mathcal{L}\mathcal{N}_{n}(A\mu,A\varSigma A')$ [5], ensures that the orthogonal projections satisfy the property of subcompositional coherence.

A comprehensive review of analytical techniques and applications of compositional framework is available in the book Pawlowsky-Glahn et al. [20].