# Decomposing information into copying versus transformation

## TL;DR

This paper introduces a novel information-theoretic decomposition distinguishing between copying and transformation in information transfer, with implications for understanding biological replication and other systems.

## Contribution

It derives a formal decomposition of mutual information into copying and transformation components, generalizes it to various channels, and links copy information to physical work in copying processes.

## Key findings

- Decomposition applies to channels with identical source and destination messages.
- Copy information correlates with minimal work in physical copying processes.
- Model analysis of amino acid substitution rates demonstrates practical relevance.

## Abstract

In many real-world systems, information can be transmitted in two qualitatively different ways: by copying or by transformation. Copying occurs when messages are transmitted without modification, e.g., when an offspring receives an unaltered copy of a gene from its parent. Transformation occurs when messages are modified systematically during transmission, e.g., when mutational biases occur during genetic replication. Standard information-theoretic measures do not distinguish these two modes of information transfer, although they may reflect different mechanisms and have different functional consequences. Starting from a few simple axioms, we derive a decomposition of mutual information into the information transmitted by copying versus the information transmitted by transformation. We begin with a decomposition that applies when the source and destination of the channel have the same set of messages and a notion of message identity exists. We then generalize our decomposition to other kinds of channels, which can involve different source and destination sets and broader notions of similarity. In addition, we show that copy information can be interpreted as the minimal work needed by a physical copying process, which is relevant for understanding the physics of replication. We use the proposed decomposition to explore a model of amino acid substitution rates. Our results apply to any system in which the fidelity of copying, rather than simple predictability, is of critical relevance.