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Conversation has been a primary means for the exchange of information since ancient times. Understanding patterns of information flow in conversations is a critical step in assessing and improving communication quality. In this paper, we describe COnversational DYnamics Model (CODYM) analysis, a novel approach for studying patterns of information flow in conversations. CODYMs are Markov Models that capture sequential dependencies in the lengths of speaker turns. The proposed method is automated and scalable, and preserves the privacy of the conversational participants. The primary function of CODYM analysis is to quantify and visualize patterns of information flow, concisely summarized over sequential turns from one or more conversations. Our approach is general and complements existing methods, providing a new tool for use in the analysis of any type of conversation. As an important first application, we demonstrate the model on transcribed conversations between palliative care clinicians and seriously ill patients. These conversations are dynamic and complex, taking place amidst heavy emotions, and include difficult topics such as end-of-life preferences and patient values. We perform a versatile set of CODYM analyses that (a) establish the validity of the model by confirming known patterns of conversational turn-taking and word usage, (b) identify normative patterns of information flow in serious illness conversations, and (c) show how these patterns vary across narrative time and differ under expressions of anger, fear and sadness. Potential applications of CODYMs range from assessment and training of effective healthcare communication to comparing conversational dynamics across language and culture, with the prospect of identifying universal similarities and unique "fingerprints" of information flow.