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 Articles in journal or book chapters
1. S. J. Rahi, J. Larsch, K. Pecani, N. Mansouri, A. Y. Katsov, K. Tsaneva-Atanasova, E. D. Sontag, and F. R. Cross. Oscillatory stimuli differentiate adapting circuit topologies. Nature Methods, 14:1010-1016, 2017. [PDF] Keyword(s): biochemical networks, periodic behaviors, monotone systems, entrainment, oscillations.
Abstract:
 Elucidating the structure of biological intracellular networks from experimental data remains a major challenge. This paper studies two types of response signatures'' to identify specific circuit motifs, from the observed response to periodic inputs. In particular, the objective is to distinguish negative feedback loops (NFLs) from incoherent feedforward loops (IFFLs), which are two types of circuits capable of producing exact adaptation. The theory of monotone systems with inputs is used to show that period skipping'' (non-harmonic responses) is ruled out in IFFL's, and a notion called refractory period stabilization'' is also analyzed. The approach is then applied to identify a circuit dominating cell cycle timing in yeast, and to uncover a calcium-mediated NFL circuit in \emph{C.elegans} olfactory sensory neurons.

2. E.D. Sontag. A dynamical model of immune responses to antigen presentation predicts different regions of tumor or pathogen elimination. Cell Systems, 4:231-241, 2017. [PDF] Keyword(s): scale invariance, fold change detection, T cells, incoherent feedforward loops, immunology, cancer.
Abstract:
 Since the early 1990s, many authors have independently suggested that self/nonself recognition by the immune system might be modulated by the rates of change of antigen challenges. This paper introduces an extremely simple and purely conceptual mathematical model that allows dynamic discrimination of immune challenges. The main component of the model is a motif which is ubiquitous in systems biology, the incoherent feedforward loop, which endows the system with the capability to estimate exponential growth exponents, a prediction which is consistent with experimental work showing that exponentially increasing antigen stimulation is a determinant of immune reactivity. Combined with a bistable system and a simple feedback repression mechanism, an interesting phenomenon emerges as a tumor growth rate increases: elimination, tolerance (tumor growth), again elimination, and finally a second zone of tolerance (tumor escape). This prediction from our model is analogous to the two-zone tumor tolerance'' phenomenon experimentally validated since the mid 1970s. Moreover, we provide a plausible biological instantiation of our circuit using combinations of regulatory and effector T cells.

3. L. Bleris, Z. Xie, D. Glass, A. Adadey, E.D. Sontag, and Y. Benenson. Synthetic incoherent feed-forward circuits show adaptation to the amount of their genetic template. Molecular Systems Biology, 7:519-, 2011. [PDF] Keyword(s): adaptation, feedforward loops, systems biology, synthetic biology.
Abstract:
 Natural and synthetic biological networks must function reliably in the face of fluctuating stoichiometry of their molecular components. These fluctuations are caused in part by changes in relative expression efficiency and the DNA template amount of the network-coding genes. Gene product levels could potentially be decoupled from these changes via built-in adaptation mechanisms, thereby boosting network reliability. Here we show that a mechanism based on an incoherent feed-forward motif enables adaptive gene expression in mammalian cells. We modeled, synthesized, and tested transcriptional and post-transcriptional incoherent loops and found that in all cases the gene product adapts to changes in DNA template abundance. We also observed that the post-transcriptional form results in superior adaptation behavior, higher absolute expression levels, and lower intrinsic fluctuations. Our results support a previously-hypothesized endogenous role in gene dosage compensation for such motifs and suggest that their incorporation in synthetic networks will improve their robustness and reliability.

4. O. Shoval, U. Alon, and E.D. Sontag. Symmetry invariance for adapting biological systems. SIAM Journal on Applied Dynamical Systems, 10:857-886, 2011. Note: See here for a small typo: http://www.math.rutgers.edu/(tilde)sontag/FTPDIR/shoval.alon.sontag.erratum.pdf.[PDF] Keyword(s): adaptation, feedforward loops, integral feedback, scale invariance, systems biology, transient behavior, symmetries, fcd, fold-change detection.
Abstract:
 Often, the ultimate goal of regulation is to maintain a narrow range of concentration levels of vital quantities (homeostasis, adaptation) while at the same time appropriately reacting to changes in the environment (signal detection or sensitivity). Much theoretical, modeling, and analysis effort has been devoted to the understanding of these questions, traditionally in the context of steady-state responses to constant or step-changing stimuli. In this paper, we present a new theorem that provides a necessary and sufficient characterization of invariance of transient responses to symmetries in inputs. A particular example of this property, scale invariance (a.k.a. "fold change detection"), appears to be exhibited by biological sensory systems ranging from bacterial chemotaxis pathways to signal transduction mechanisms in eukaryotes. The new characterization amounts to the solvability of an associated partial differential equation. It is framed in terms of a notion which considerably extends equivariant actions of compact Lie groups. For several simple system motifs that are recurrent in biology, the solvability criterion may be checked explicitly.

5. O. Shoval, L. Goentoro, Y. Hart, A. Mayo, E.D. Sontag, and U. Alon. Fold change detection and scalar symmetry of sensory input fields. Proc Natl Acad Sci USA, 107:15995-16000, 2010. [PDF] Keyword(s): adaptation, feedforward loops, integral feedback, scale invariance, systems biology, transient behavior, symmetries, fcd, fold-change detection.
Abstract:
 Certain cellular sensory systems display fold-change detection (FCD): a response whose entire shape, including amplitude and duration, depends only on fold-changes in input, and not on absolute changes. Thus, a step change in input from, say, level 1 to 2, gives precisely the same dynamical output as a step from level 2 to 4, since the steps have the same fold-change. We ask what is the benefit of FCD, and show that FCD is necessary and sufficient for sensory search to be independent of multiplying the input-field by a scalar. Thus the FCD search pattern depends only on the spatial profile of the input, and not on its amplitude. Such scalar symmetry occurs in a wide range of sensory inputs, such as source strength multiplying diffusing/convecting chemical fields sensed in chemotaxis, ambient light multiplying the contrast field in vision, and protein concentrations multiplying the output in cellular signaling-systems.Furthermore, we demonstrate that FCD entails two features found across sensory systems, exact adaptation and Weber's law, but that these two features are not sufficient for FCD. Finally, we present a wide class of mechanisms that have FCD, including certain non-linear feedback and feedforward loops.. We find that bacterial chemotaxis displays feedback within the present class, and hence is expected to show FCD. This can explain experiments in which chemotaxis searches are insensitive to attractant source levels. This study thus suggests a connection between properties of biological sensory systems and scalar symmetry stemming from physical properties of their input-fields.

6. E.D. Sontag. Remarks on Feedforward Circuits, Adaptation, and Pulse Memory. IET Systems Biology, 4:39-51, 2010. [PDF] Keyword(s): adaptation, feedforward loops, integral feedback, systems biology, transient behavior.
Abstract:
 This note studies feedforward circuits as models for perfect adaptation to step signals in biological systems. A global convergence theorem is proved in a general framework, which includes examples from the literature as particular cases. A notable aspect of these circuits is that they do not adapt to pulse signals, because they display a memory phenomenon. Estimates are given of the magnitude of this effect.

 Conference articles
1. E.D. Sontag. Some remarks on immune control of infections and tumors. In Proc. IEEE Conf. Decision and Control, Dec. 2016, pages 2476-2480, 2016. Keyword(s): scale invariance, fold change detection, T cells, incoherent feedforward loops, immunology, cancer.

2. O. Shoval, U. Alon, and E.D. Sontag. Input symmetry invariance, and applications to biological systems. In Proc. IEEE Conf. Decision and Control, Orlando, Dec. 2011, pages TuA02.5, 2011. Keyword(s): adaptation, feedforward loops, integral feedback, scale invariance, systems biology, transient behavior, symmetries, fcd, fold-change detection, jump Markov processes.
Abstract:
 This paper studies invariance with respect to symmetries in sensory fields, a particular case of which, scale invariance, has recently been found in certain eukaryotic as well as bacterial cell signaling systems. We describe a necessary and sufficient characterization of symmetry invariance in terms of equivariant transformations, show how this characterization helps find all possible symmetries in standard models of biological adaptation, and discuss symmetry-invariant searches.

 Internal reports
1. E.D. Sontag. A remark on incoherent feedforward circuits as change detectors and feedback controllers. Technical report, arXiv:1602.00162, 2016. [PDF] Keyword(s): scale invariance, fold change detection, T cells, incoherent feedforward loops, immunology.
Abstract:
 This note analyzes incoherent feedforward loops in signal processing and control. It studies the response properties of IFFL's to exponentially growing inputs, both for a standard version of the IFFL and for a variation in which the output variable has a positive self-feedback term. It also considers a negative feedback configuration, using such a device as a controller. It uncovers a somewhat surprising phenomenon in which stabilization is only possible in disconnected regions of parameter space, as the controlled system's growth rate is varied.

2. E.D. Sontag. Two-zone tumor tolerance can arise from a simple immunological feedforward motif that estimates tumor growth rates. Technical report, bioRxiv https://doi.org/10.1101/095455, 2016. [PDF] Keyword(s): scale invariance, fold change detection, T cells, incoherent feedforward loops, immunology, cancer.
Abstract:
 Preprint version of "A dynamical model of immune responses to antigen presentation predicts different regions of tumor or pathogen elimination", to appear in Cell Systems 2017. However, the journal version does not include Section 9 on degradation-based IFFL's from this preprint.

3. E.D. Sontag. Incoherent feedforward motifs as immune change detectors. Technical report, bioRxiv http://dx.doi.org/10.1101/035600, December 2015. [PDF] Keyword(s): scale invariance, fcd, fold change detection, T cells, incoherent feedforward loops, immunology.
Abstract:
 We speculate that incoherent feedforward loops may be phenomenologically involved in self/nonself discrimination in immune-infection and immune-tumor interactions, acting as "change detectors". In turn, this may result in logarithmic sensing (Weber phenomenon) and even scale invariance (fold-change detection).

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