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M.S. Bauer
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1
Unifying theories in high-dimensional biophysics
Approaches, challenges and opportunities
Journal article
(2026)
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Marianne Bauer, Akshit Goyal, Sidhartha Goyal, Gautam Reddy, Shaon Chakrabarti, Michael M. Desai, William Gilpin, Jacopo Grilli, Sanjay Jain, More Authors
Across biological subdisciplines, the last decade has seen an explosion of high-dimensional datasets. At the ICTS workshop ‘Unifying Theories in High-Dimensional Biophysics’, we discussed whether this high dimensionality poses a challenge or an opportunity for theoretically describing, understanding and predicting biological systems. We discussed methods, models and frameworks that can be used for this purpose. This Comment summarizes our discussions from the perspectives of individual participants.
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Across biological subdisciplines, the last decade has seen an explosion of high-dimensional datasets. At the ICTS workshop ‘Unifying Theories in High-Dimensional Biophysics’, we discussed whether this high dimensionality poses a challenge or an opportunity for theoretically describing, understanding and predicting biological systems. We discussed methods, models and frameworks that can be used for this purpose. This Comment summarizes our discussions from the perspectives of individual participants.
Journal article
(2026)
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Tamara Mijatović, Aimée R. Kok, Merlijn Brüggen, Jos W. Zwanikken, Marianne Bauer
Transcription factor concentrations provide signals to cells that allow them to regulate gene expression to make correct cell fate decisions. Calculations for noise bounds in gene regulation suggest that clustering or cooperative binding of transcription factors decreases signal-to-noise ratios at binding sites. However, clustering of transcription factor molecules around binding sites is frequently observed. We develop two complementary models for clustering transcription factors at binding site sensors that allow us to study information transfer from a signal, the morphogen Bicoid, to a variable relevant to development, namely, future cell fates. We find that weak cooperativity or clustering can allow for maximal information transfer, especially about the relevant variable. The timescale of measurement is crucial for predicting the optimal clustering strength: for short measurements, finite clustering is optimal because it allows for the implementation of a switch, while for long measurements, a range of weak clustering strengths allow binding site sensors to access near-maximal developmental information. Weak transcription factor clustering also helps binding site sensors achieve optimality consistent with the information bottleneck bound, which encodes an optimal trade-off between conveying relevant information and making costly measurements: changes in clustering in conjunction with changes in the binding energy can shift the binding site sensor along the optimal bound, and towards an optimal trade-off between obtaining information about the signal and obtaining relevant information.
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Transcription factor concentrations provide signals to cells that allow them to regulate gene expression to make correct cell fate decisions. Calculations for noise bounds in gene regulation suggest that clustering or cooperative binding of transcription factors decreases signal-to-noise ratios at binding sites. However, clustering of transcription factor molecules around binding sites is frequently observed. We develop two complementary models for clustering transcription factors at binding site sensors that allow us to study information transfer from a signal, the morphogen Bicoid, to a variable relevant to development, namely, future cell fates. We find that weak cooperativity or clustering can allow for maximal information transfer, especially about the relevant variable. The timescale of measurement is crucial for predicting the optimal clustering strength: for short measurements, finite clustering is optimal because it allows for the implementation of a switch, while for long measurements, a range of weak clustering strengths allow binding site sensors to access near-maximal developmental information. Weak transcription factor clustering also helps binding site sensors achieve optimality consistent with the information bottleneck bound, which encodes an optimal trade-off between conveying relevant information and making costly measurements: changes in clustering in conjunction with changes in the binding energy can shift the binding site sensor along the optimal bound, and towards an optimal trade-off between obtaining information about the signal and obtaining relevant information.
Journal article
(2026)
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O.Z.J. Witteveen, Samuel J. Rosen, Ryan S. Lach, Maxwell Z. Wilson, M.S. Bauer
Populations of cells regulate gene expression in response to external signals, but their ability to make reliable collective decisions is limited by both intrinsic noise in molecular signaling and variability between individual cells. In this work, we use optogenetic control of the canonical Wnt pathway as an example to study how reliably information about an external signal is transmitted to a population of cells, and determine an optimal encoding strategy to maximize information transmission from Wnt signals to gene expression. We find that it is possible to reach an information capacity beyond 1 bit only through an appropriate, discrete encoding of signals: using no Wnt, a short Wnt pulse, or a sustained Wnt signal. By averaging over an increasing number of outputs, we systematically vary the effective noise in the pathway. As the effective noise decreases, the optimal encoding comprises more discrete input signals. These signals do not need to be fine-tuned to achieve near-optimal information transmission. The optimal code transitions into a continuous code in the small-noise limit, which can be shown to be consistent with the Jeffreys prior. We visualize the performance of different signal encodings using decoding maps. Our results suggest that optogenetic Wnt signaling allows for regulatory control beyond a simple binary switch and provide a framework to apply ideas from information processing to single-cell in vitro experiments.
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Populations of cells regulate gene expression in response to external signals, but their ability to make reliable collective decisions is limited by both intrinsic noise in molecular signaling and variability between individual cells. In this work, we use optogenetic control of the canonical Wnt pathway as an example to study how reliably information about an external signal is transmitted to a population of cells, and determine an optimal encoding strategy to maximize information transmission from Wnt signals to gene expression. We find that it is possible to reach an information capacity beyond 1 bit only through an appropriate, discrete encoding of signals: using no Wnt, a short Wnt pulse, or a sustained Wnt signal. By averaging over an increasing number of outputs, we systematically vary the effective noise in the pathway. As the effective noise decreases, the optimal encoding comprises more discrete input signals. These signals do not need to be fine-tuned to achieve near-optimal information transmission. The optimal code transitions into a continuous code in the small-noise limit, which can be shown to be consistent with the Jeffreys prior. We visualize the performance of different signal encodings using decoding maps. Our results suggest that optogenetic Wnt signaling allows for regulatory control beyond a simple binary switch and provide a framework to apply ideas from information processing to single-cell in vitro experiments.
Journal article
(2026)
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Samuel J. Rosen, O.Z.J. Witteveen, Naomi Baxter, Ryan S. Lach, Erik Hopkins, M.S. Bauer, Maxwell Z. Wilson
Cells process dynamic signaling inputs to regulate fate decisions during development. While oscillations or waves in key developmental pathways, such as Wnt, have been widely observed, the principles governing how cells decode these signals remain unclear. By leveraging optogenetic control of the Wnt signaling pathway in both HEK293T cells and H9 human embryonic stem cells, we systematically map the relationship between signal frequency and downstream pathway activation. We find that cells exhibit a minimal response to Wnt at certain frequencies, a behavior we term anti-resonance. We developed both detailed biochemical and simplified hidden variable models that explain how anti-resonance emerges from the interplay between fast and slow pathway dynamics. Remarkably, we find that frequency directly influences cell fate decisions involved in human gastrulation; signals delivered at anti-resonant frequencies result in dramatically reduced mesoderm differentiation. Our work reveals a previously unknown mechanism of how cells decode dynamic signals and how anti-resonance may filter against spurious activation. These findings establish new insights into how cells decode dynamic signals with implications for tissue engineering, regenerative medicine, and cancer biology.
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Cells process dynamic signaling inputs to regulate fate decisions during development. While oscillations or waves in key developmental pathways, such as Wnt, have been widely observed, the principles governing how cells decode these signals remain unclear. By leveraging optogenetic control of the Wnt signaling pathway in both HEK293T cells and H9 human embryonic stem cells, we systematically map the relationship between signal frequency and downstream pathway activation. We find that cells exhibit a minimal response to Wnt at certain frequencies, a behavior we term anti-resonance. We developed both detailed biochemical and simplified hidden variable models that explain how anti-resonance emerges from the interplay between fast and slow pathway dynamics. Remarkably, we find that frequency directly influences cell fate decisions involved in human gastrulation; signals delivered at anti-resonant frequencies result in dramatically reduced mesoderm differentiation. Our work reveals a previously unknown mechanism of how cells decode dynamic signals and how anti-resonance may filter against spurious activation. These findings establish new insights into how cells decode dynamic signals with implications for tissue engineering, regenerative medicine, and cancer biology.
Journal article
(2025)
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Maximilian F. Madern, Sora Yang, Olivier Witteveen, Hendrika A. Segeren, Marianne Bauer, Marvin E. Tanenbaum
The genetic information stored in mRNAs is decoded by ribosomes during mRNA translation. mRNAs are typically translated by multiple ribosomes simultaneously, but it is unclear whether and how the activity of different ribosomes on an mRNA is coordinated. Here, we develop an imaging approach based on stopless-ORF circular RNAs (socRNAs) to monitor translation of individual ribosomes in either monosomes or polysomes with very high resolution. Using experiments and simulations, we find that translating ribosomes frequently undergo transient collisions. However, unlike persistent collisions, such transient collisions escape detection by cellular quality control pathways. Rather, transient ribosome collisions promote productive translation by reducing ribosome pausing on problematic sequences, a process we term ribosome cooperativity. Ribosome cooperativity also reduces recycling of ribosomes by quality control pathways, thus enhancing processive translation. Together, our single-ribosome imaging approach reveals that ribosomes cooperate during translation to ensure fast and efficient translation.
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The genetic information stored in mRNAs is decoded by ribosomes during mRNA translation. mRNAs are typically translated by multiple ribosomes simultaneously, but it is unclear whether and how the activity of different ribosomes on an mRNA is coordinated. Here, we develop an imaging approach based on stopless-ORF circular RNAs (socRNAs) to monitor translation of individual ribosomes in either monosomes or polysomes with very high resolution. Using experiments and simulations, we find that translating ribosomes frequently undergo transient collisions. However, unlike persistent collisions, such transient collisions escape detection by cellular quality control pathways. Rather, transient ribosome collisions promote productive translation by reducing ribosome pausing on problematic sequences, a process we term ribosome cooperativity. Ribosome cooperativity also reduces recycling of ribosomes by quality control pathways, thus enhancing processive translation. Together, our single-ribosome imaging approach reveals that ribosomes cooperate during translation to ensure fast and efficient translation.
The SARS-CoV-2 nucleocapsid protein, or N-protein, is a structural protein that plays an important role in the SARS-CoV-2 life cycle. The N-protein takes part in the regulation of viral RNA replication and drives highly specific packaging of full-length genomic RNA prior to virion formation. One regulatory mechanism that is proposed to drive the switch between these two operating modes is the phosphorylation state of the N-protein. Here, we assess the dynamic behavior of non-phosphorylated and phosphorylated versions of the N-protein homodimer through atomistic molecular dynamics simulations. We show that the introduction of phosphorylation yields a more dynamic protein structure and decreases the binding affinity between the N-protein and RNA. Furthermore, we find that secondary structure is essential for the preferential binding of particular RNA elements from the 5′ UTR of the viral genome to the N-terminal domain of the N-protein. Altogether, we provide detailed molecular insights into N-protein dynamics, N-protein:RNA interactions, and phosphorylation. Our results corroborate the hypothesis that phosphorylation of the N-protein serves as a regulatory mechanism that determines N-protein function.
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The SARS-CoV-2 nucleocapsid protein, or N-protein, is a structural protein that plays an important role in the SARS-CoV-2 life cycle. The N-protein takes part in the regulation of viral RNA replication and drives highly specific packaging of full-length genomic RNA prior to virion formation. One regulatory mechanism that is proposed to drive the switch between these two operating modes is the phosphorylation state of the N-protein. Here, we assess the dynamic behavior of non-phosphorylated and phosphorylated versions of the N-protein homodimer through atomistic molecular dynamics simulations. We show that the introduction of phosphorylation yields a more dynamic protein structure and decreases the binding affinity between the N-protein and RNA. Furthermore, we find that secondary structure is essential for the preferential binding of particular RNA elements from the 5′ UTR of the viral genome to the N-terminal domain of the N-protein. Altogether, we provide detailed molecular insights into N-protein dynamics, N-protein:RNA interactions, and phosphorylation. Our results corroborate the hypothesis that phosphorylation of the N-protein serves as a regulatory mechanism that determines N-protein function.
Journal article
(2024)
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Yasmine El Azhar, Pascal Schulthess, Marek J. van Oostrom, Sonja D.C. Weterings, Wilke H.M. Meijer, Nobuko Tsuchida-Straeten, Wouter M. Thomas, Marianne Bauer, Katharina F. Sonnen
The intricate dynamics of Hes expression across diverse cell types in the developing vertebrate embryonic tail have remained elusive. To address this, we have developed an endogenously tagged Hes1-Achilles mouse line, enabling precise quantification of dynamics at the single-cell resolution across various tissues. Our findings reveal striking disparities in Hes1 dynamics between presomitic mesoderm (PSM) and preneural tube (pre-NT) cells. While pre-NT cells display variable, low-amplitude oscillations, PSM cells exhibit synchronized, high-amplitude oscillations. Upon the induction of differentiation, the oscillation amplitude increases in pre-NT cells. Additionally, our study of Notch inhibition on Hes1 oscillations unveils distinct responses in PSM and pre-NT cells, corresponding to differential Notch ligand expression dynamics. These findings suggest the involvement of separate mechanisms driving Hes1 oscillations. Thus, Hes1 demonstrates dynamic behaviour across adjacent tissues of the embryonic tail, yet the varying oscillation parameters imply differences in the information that can be transmitted by these dynamics.
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The intricate dynamics of Hes expression across diverse cell types in the developing vertebrate embryonic tail have remained elusive. To address this, we have developed an endogenously tagged Hes1-Achilles mouse line, enabling precise quantification of dynamics at the single-cell resolution across various tissues. Our findings reveal striking disparities in Hes1 dynamics between presomitic mesoderm (PSM) and preneural tube (pre-NT) cells. While pre-NT cells display variable, low-amplitude oscillations, PSM cells exhibit synchronized, high-amplitude oscillations. Upon the induction of differentiation, the oscillation amplitude increases in pre-NT cells. Additionally, our study of Notch inhibition on Hes1 oscillations unveils distinct responses in PSM and pre-NT cells, corresponding to differential Notch ligand expression dynamics. These findings suggest the involvement of separate mechanisms driving Hes1 oscillations. Thus, Hes1 demonstrates dynamic behaviour across adjacent tissues of the embryonic tail, yet the varying oscillation parameters imply differences in the information that can be transmitted by these dynamics.
How does an organism regulate its genes? The involved regulation typically occurs in terms of a signal processing chain: an externally applied stimulus or a maternally supplied transcription factor leads to the expression of some downstream genes, which, in turn, are transcription factors for further genes. Especially during development, these transcription factors are frequently expressed in amounts where noise is still important; yet, the signals that they provide must not be lost in the noise. Thus, the organism needs to extract exactly relevant information in the signal. New experimental approaches involving single-molecule measurements at high temporal precision as well as increased precision in manipulations directly on the genome are allowing us to tackle this question anew. These new experimental advances mean that also from the theoretical side, theoretical advances should be possible. In this review, I will describe, specifically on the example of fly embryo gene regulation, how theoretical approaches, especially from inference and information theory, can help in understanding gene regulation. To do so, I will first review some more traditional theoretical models for gene regulation, followed by a brief discussion of information-theoretical approaches and when they can be applied. I will then introduce early fly development as an exemplary system where such information-theoretical approaches have traditionally been applied and can be applied; I will specifically focus on how one such method, namely the information bottleneck approach, has recently been used to infer structural features of enhancer architecture.
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How does an organism regulate its genes? The involved regulation typically occurs in terms of a signal processing chain: an externally applied stimulus or a maternally supplied transcription factor leads to the expression of some downstream genes, which, in turn, are transcription factors for further genes. Especially during development, these transcription factors are frequently expressed in amounts where noise is still important; yet, the signals that they provide must not be lost in the noise. Thus, the organism needs to extract exactly relevant information in the signal. New experimental approaches involving single-molecule measurements at high temporal precision as well as increased precision in manipulations directly on the genome are allowing us to tackle this question anew. These new experimental advances mean that also from the theoretical side, theoretical advances should be possible. In this review, I will describe, specifically on the example of fly embryo gene regulation, how theoretical approaches, especially from inference and information theory, can help in understanding gene regulation. To do so, I will first review some more traditional theoretical models for gene regulation, followed by a brief discussion of information-theoretical approaches and when they can be applied. I will then introduce early fly development as an exemplary system where such information-theoretical approaches have traditionally been applied and can be applied; I will specifically focus on how one such method, namely the information bottleneck approach, has recently been used to infer structural features of enhancer architecture.
Journal article
(2021)
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Marianne Bauer, Mariela D. Petkova, Thomas Gregor, Eric F. Wieschaus, William Bialek
In the regulation of gene expression, information of relevance to the organism is represented by the concentrations of transcription factor molecules. To extract this information the cell must effectively “measure” these concentrations, but there are physical limits to the precision of these measurements. We use the gap gene network in the early fly embryo as an example of the tradeoff between the precision of concentration measurements and the transmission of relevant information. For thresholded measurements we find that lower thresholds are more important, and fine tuning is not required for near-optimal information transmission. We then consider general sensors, constrained only by a limit on their information capacity, and find that thresholded sensors can approach true information theoretic optima. The information theoretic approach allows us to identify the optimal sensor for the entire gap gene network and to argue that the physical limitations of sensing necessitate the observed multiplicity of enhancer elements, with sensitivities to combinations rather than single transcription factors.
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In the regulation of gene expression, information of relevance to the organism is represented by the concentrations of transcription factor molecules. To extract this information the cell must effectively “measure” these concentrations, but there are physical limits to the precision of these measurements. We use the gap gene network in the early fly embryo as an example of the tradeoff between the precision of concentration measurements and the transmission of relevant information. For thresholded measurements we find that lower thresholds are more important, and fine tuning is not required for near-optimal information transmission. We then consider general sensors, constrained only by a limit on their information capacity, and find that thresholded sensors can approach true information theoretic optima. The information theoretic approach allows us to identify the optimal sensor for the entire gap gene network and to argue that the physical limitations of sensing necessitate the observed multiplicity of enhancer elements, with sensitivities to combinations rather than single transcription factors.
Journal article
(2018)
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Marianne Bauer, Erwin Frey
Multiple scales in metapopulations can give rise to paradoxical behavior: in a conceptual model for a public goods game, the species associated with a fitness cost due to the public good production can be stabilized in the well-mixed limit due to the mere existence of these scales. The scales in this model involve a length scale corresponding to separate patches, coupled by mobility, and separate time scales for reproduction and interaction with a local environment. Contrary to the well-mixed high mobility limit, we find that for low mobilities, the interaction rate progressively stabilizes this species due to stochastic effects, and that the formation of spatial patterns is not crucial for this stabilization.
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Multiple scales in metapopulations can give rise to paradoxical behavior: in a conceptual model for a public goods game, the species associated with a fitness cost due to the public good production can be stabilized in the well-mixed limit due to the mere existence of these scales. The scales in this model involve a length scale corresponding to separate patches, coupled by mobility, and separate time scales for reproduction and interaction with a local environment. Contrary to the well-mixed high mobility limit, we find that for low mobilities, the interaction rate progressively stabilizes this species due to stochastic effects, and that the formation of spatial patterns is not crucial for this stabilization.
Journal article
(2018)
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Marianne Bauer, Erwin Frey
Organisms that exploit different environments may experience a stochastic delay in adjusting their fitness when they switch habitats. We study two such organisms whose fitness is determined by the species composition of the local environment, as they interact through a public good. We show that a delay in the fitness adjustment can lead to the coexistence of the two species in a metapopulation, although the faster-growing species always wins in well-mixed competition experiments. Coexistence is favored over wide parameter ranges and is independent of spatial clustering. It arises when species are heterogeneous in their fitness and can keep each other balanced.
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Organisms that exploit different environments may experience a stochastic delay in adjusting their fitness when they switch habitats. We study two such organisms whose fitness is determined by the species composition of the local environment, as they interact through a public good. We show that a delay in the fitness adjustment can lead to the coexistence of the two species in a metapopulation, although the faster-growing species always wins in well-mixed competition experiments. Coexistence is favored over wide parameter ranges and is independent of spatial clustering. It arises when species are heterogeneous in their fitness and can keep each other balanced.
Journal article
(2018)
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Marianne Bauer, Erwin Frey
How does delayed fitnesses adaptation after local habitat changes affect survival of species metapopulation? We study this question in a two-species model system, where the species composition of a local patch determines the reference fitness of all its individuals. When individuals move, this local species composition changes. As the local environment on the patch might adapt slowly to this change, we assume that individuals in turn adapt their fitness with a stochastic delay. We show that the combination of delay and spatial substructure can yield significantly different phase diagrams for the survival of these species with respect to models with immediate response. We investigate this exemplarily for the case where the two species interact via an exoproduct: thus, our population consists of a slow-growing producer species and a fast-growing dominant species. We provide a conceptual understanding of the role of delay by presenting analytic results in the well-mixed and low-mobility limit. By studying intermediate mobilities numerically, we ensure that our results are robust, and may be relevant to different ecological situations as well as microbial metapopulation experiments.
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How does delayed fitnesses adaptation after local habitat changes affect survival of species metapopulation? We study this question in a two-species model system, where the species composition of a local patch determines the reference fitness of all its individuals. When individuals move, this local species composition changes. As the local environment on the patch might adapt slowly to this change, we assume that individuals in turn adapt their fitness with a stochastic delay. We show that the combination of delay and spatial substructure can yield significantly different phase diagrams for the survival of these species with respect to models with immediate response. We investigate this exemplarily for the case where the two species interact via an exoproduct: thus, our population consists of a slow-growing producer species and a fast-growing dominant species. We provide a conceptual understanding of the role of delay by presenting analytic results in the well-mixed and low-mobility limit. By studying intermediate mobilities numerically, we ensure that our results are robust, and may be relevant to different ecological situations as well as microbial metapopulation experiments.
Journal article
(2017)
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Marianne Bauer, Isabella R. Graf, Vudtiwat Ngampruetikorn, Greg J. Stephens, Erwin Frey
A deterministic population dynamics model involving birth and death for a two-species system, comprising a wild-type and more resistant species competing via logistic growth, is subjected to two distinct stress environments designed to mimic those that would typically be induced by temporal variation in the concentration of a drug (antibiotic or chemotherapeutic) as it permeates through the population and is progressively degraded. Different treatment regimes, involving single or periodical doses, are evaluated in terms of the minimal population size (a measure of the extinction probability), and the population composition (a measure of the selection pressure for resistance or tolerance during the treatment). We show that there exist timescales over which the low-stress regime is as effective as the high-stress regime, due to the competition between the two species. For multiple periodic treatments, competition can ensure that the minimal population size is attained during the first pulse when the high-stress regime is short, which implies that a single short pulse can be more effective than a more protracted regime. Our results suggest that when the duration of the high-stress environment is restricted, a treatment with one or multiple shorter pulses can produce better outcomes than a single long treatment. If ecological competition is to be exploited for treatments, it is crucial to determine these timescales, and estimate for the minimal population threshold that suffices for extinction. These parameters can be quantified by experiment.
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A deterministic population dynamics model involving birth and death for a two-species system, comprising a wild-type and more resistant species competing via logistic growth, is subjected to two distinct stress environments designed to mimic those that would typically be induced by temporal variation in the concentration of a drug (antibiotic or chemotherapeutic) as it permeates through the population and is progressively degraded. Different treatment regimes, involving single or periodical doses, are evaluated in terms of the minimal population size (a measure of the extinction probability), and the population composition (a measure of the selection pressure for resistance or tolerance during the treatment). We show that there exist timescales over which the low-stress regime is as effective as the high-stress regime, due to the competition between the two species. For multiple periodic treatments, competition can ensure that the minimal population size is attained during the first pulse when the high-stress regime is short, which implies that a single short pulse can be more effective than a more protracted regime. Our results suggest that when the duration of the high-stress environment is restricted, a treatment with one or multiple shorter pulses can produce better outcomes than a single long treatment. If ecological competition is to be exploited for treatments, it is crucial to determine these timescales, and estimate for the minimal population threshold that suffices for extinction. These parameters can be quantified by experiment.
Journal article
(2014)
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Marianne Bauer, Meera M. Parish, Tilman Enss
We determine the thermodynamic properties and the spectral function for a homogeneous two-dimensional Fermi gas in the normal state using the Luttinger-Ward, or self-consistent T-matrix, approach. The density equation of state deviates strongly from that of the ideal Fermi gas even for moderate interactions, and our calculations suggest that temperature has a pronounced effect on the pressure in the crossover from weak to strong coupling, consistent with recent experiments. We also compute the superfluid transition temperature for a finite system in the crossover region. There is a pronounced pseudogap regime above the transition temperature: the spectral function shows a Bogoliubov-like dispersion with backbending, and the density of states is significantly suppressed near the chemical potential. The contact density at low temperatures increases with interaction and compares well with both experiment and zero-temperature Monte Carlo results.
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We determine the thermodynamic properties and the spectral function for a homogeneous two-dimensional Fermi gas in the normal state using the Luttinger-Ward, or self-consistent T-matrix, approach. The density equation of state deviates strongly from that of the ideal Fermi gas even for moderate interactions, and our calculations suggest that temperature has a pronounced effect on the pressure in the crossover from weak to strong coupling, consistent with recent experiments. We also compute the superfluid transition temperature for a finite system in the crossover region. There is a pronounced pseudogap regime above the transition temperature: the spectral function shows a Bogoliubov-like dispersion with backbending, and the density of states is significantly suppressed near the chemical potential. The contact density at low temperatures increases with interaction and compares well with both experiment and zero-temperature Monte Carlo results.