Visualizing infection spread: dual-color fluorescent reporting of virus-host interactions

Oscillations in a Spatial Oncolytic Virus Model

Virotherapy treatment is a new and promising target therapy that selectively attacks cancer cells without harming normal cells. Mathematical models of oncolytic viruses have shown predator-prey like oscillatory patterns as result of an underlying Hopf bifurcation. In a spatial context, these oscillations can lead to different spatio-temporal phenomena such as hollow-ring patterns, target patterns, and dispersed patterns. In this paper we continue the systematic analysis of these spatial oscillations and discuss their relevance in the clinical context. We consider a bifurcation analysis of a spatially explicit reaction-diffusion model to find the above mentioned spatio-temporal virus infection patterns. The desired pattern for tumor eradication is the hollow ring pattern and we find exact conditions for its occurrence. Moreover, we derive the minimal speed of travelling invasion waves for the cancer and for the oncolytic virus. Our numerical simulations in 2-D reveal complex spatial interactions of the virus infection and a new phenomenon of a periodic peak splitting. An effect that we cannot explain with our current methods.

Incorporation of 5 methylcytidine alleviates innate immune response to self-amplifying RNA vaccine

In order to improve vaccine effectiveness and safety profile of existing synthetic RNA-based vaccines, we have developed a self-amplifying RNA (saRNA)-based vaccine expressing membrane-anchored receptor binding domain (RBD) of SARS-CoV-2 S protein (S-RBD) and have demonstrated that a minimal dose of this saRNA vaccine elicits robust immune responses. Results from a recent clinical trial with 5-methylcytidine (5mC) incorporating saRNA vaccine demonstrated reduced vaccine-induced adverse effects while maintaining robust humoral responses. In this study, we investigate the mechanisms accounting for induction of efficient innate and adaptive immune responses and attenuated adverse effects induced by the 5mC-incorporated saRNA. We show that the 5mC-incorporating saRNA platform leads to prolonged and robust expression of antigen, while induction of type-I interferon (IFN-I), a key driver of reactogenicity, is attenuated in peripheral blood mononuclear cells (PBMCs), but not in macrophages and dendritic cells. Interestingly, we find that the major cellular source of IFN-I production in PBMCs is plasmacytoid dendritic cells (pDCs), which is attenuated upon 5mC incorporation in saRNA. In addition, we demonstrate that monocytes also play an important role in amplifying proinflammatory responses. Furthermore, we show that the detection of saRNA is mediated by a host cytosolic RNA sensor, RIG-I. Importantly, 5mC-incorporating saRNA vaccine candidate produced robust IgG responses against S-RBD upon injection in mice, thus providing strong support for the potential clinical use of 5mC-incorporating saRNA vaccines.

Monitoring Virus-Induced Stress Granule Dynamics Using Long-Term Live-Cell Imaging

The integrated stress response is a highly regulated signaling cascade that allows cells to react to a variety of external and internal stimuli. Activation of different stress-responsive kinases leads to the phosphorylation of their common downstream target, the eukaryotic translation initiation factor 2 alpha (eIF2α), which is a critical component of functional translation preinitiation complexes. As a consequence, stalled ribonucleoprotein complexes accumulate in the cytoplasm and condense into microscopically visible cytoplasmic stress granules (SGs). Over the past years, numerous microscopy approaches have been developed to study the spatiotemporal control of SG formation in response to a variety of stressors. Here, we apply long-term live-cell microscopy to monitor the dynamic cellular stress response triggered by infection with chronic hepatitis C virus (HCV) at single-cell level and study the behavior of infected cells that repeatedly switch between a stressed and unstressed state. We describe in detail the engineering of fluorescent SG-reporter cells expressing enhanced yellow fluorescent protein (YFP)-tagged T cell internal antigen 1 (TIA-1) using lentiviral delivery, as well as the production of mCherry-tagged HCV trans-complemented particles, which allow live tracking of SG assembly and disassembly, SG number and size in single infected cells over time.

Cell culture-based production and in vivo characterization of purely clonal defective interfering influenza virus particles

BACKGROUND

Infections with influenza A virus (IAV) cause high morbidity and mortality in humans. Additional to vaccination, antiviral drugs are a treatment option. Besides FDA-approved drugs such as oseltamivir or zanamivir, virus-derived defective interfering (DI) particles (DIPs) are considered promising new agents. IAV DIPs typically contain a large internal deletion in one of their eight genomic viral RNA (vRNA) segments. Consequently, DIPs miss the genetic information necessary for replication and can usually only propagate by co-infection with infectious standard virus (STV), compensating for their defect. In such a co-infection scenario, DIPs interfere with and suppress STV replication, which constitutes their antiviral potential.

RESULTS

In the present study, we generated a genetically engineered MDCK suspension cell line for production of a purely clonal DIP preparation that has a large deletion in its segment 1 (DI244) and is not contaminated with infectious STV as egg-derived material. First, the impact of the multiplicity of DIP (MODIP) per cell on DI244 yield was investigated in batch cultivations in shake flasks. Here, the highest interfering efficacy was observed for material produced at a MODIP of 1E-2 using an in vitro interference assay. Results of RT-PCR suggested that DI244 material produced was hardly contaminated with other defective particles. Next, the process was successfully transferred to a stirred tank bioreactor (500 mL working volume) with a yield of 6.0E+8 PFU/mL determined in genetically modified adherent MDCK cells. The produced material was purified and concentrated about 40-fold by membrane-based steric exclusion chromatography (SXC). The DI244 yield was 92.3% with a host cell DNA clearance of 97.1% (99.95% with nuclease digestion prior to SXC) and a total protein reduction of 97.2%. Finally, the DIP material was tested in animal experiments in D2(B6).A2G-Mx1^r/r mice. Mice infected with a lethal dose of IAV and treated with DIP material showed a reduced body weight loss and all animals survived.

CONCLUSION

In summary, experiments not only demonstrated that purely clonal influenza virus DIP preparations can be obtained with high titers from animal cell cultures but confirmed the potential of cell culture-derived DIPs as an antiviral agent.

OP7, a novel influenza A virus defective interfering particle: production, purification, and animal experiments demonstrating antiviral potential

Abstract

The novel influenza A virus (IAV) defective interfering particle “OP7” inhibits IAV replication in a co-infection and was previously suggested as a promising antiviral agent. Here, we report a batch-mode cell culture-based production process for OP7. In the present study, a seed virus containing standard virus (STV) and OP7 was used. The yield of OP7 strongly depended on the production multiplicity of infection. To inactivate infectious STV in the OP7 material, which may cause harm in a potential application, UV irradiation was used. The efficacy of OP7 in this material was preserved, as shown by an in vitro interference assay. Next, steric exclusion chromatography was used to purify and to concentrate (~ 13-fold) the UV-treated material. Finally, administration of produced OP7 material in mice did not show any toxic effects. Furthermore, all mice infected with a lethal dose of IAV survived the infection upon OP7 co-treatment. Thus, the feasibility of a production workflow for OP7 and its potential for antiviral treatment was demonstrated.

Key points

• OP7 efficacy strongly depended on the multiplicity of infection used for production

• Purification by steric exclusion chromatography increased OP7 efficacy

• OP7-treated mice were protected against a lethal infection with IAV

Supplementary Information

The online version contains supplementary material available at 10.1007/s00253-020-11029-5.

Microfluidics in Single-Cell Virology: Technologies and Applications

Microfluidics has proven to be a powerful tool for probing biology at the single-cell level. However, it is only in the past 5 years that single-cell microfluidics has been used in the field of virology. An array of strategies based on microwells, microvalves, and droplets is now available for tracking viral infection dynamics, identifying cell subpopulations with particular phenotypes, as well as high-throughput screening. The insights into the virus-host interactions gained at the single-cell level are unprecedented and usually inaccessible by population-based experiments. Therefore, single-cell microfluidics, which opens new avenues for mechanism elucidation and development of antiviral therapeutics, would be a valuable tool for the study of viral pathogenesis.

Production of Defective Interfering Particles of Influenza A Virus in Parallel Continuous Cultures at Two Residence Times-Insights From qPCR Measurements and Viral Dynamics Modeling

Defective interfering particles (DIPs) are a natural byproduct of influenza A virus (IAV) replication. DIPs interfere with the propagation and spread of infectious standard virus (STV), reduce virus yields by competing for viral and cellular resources, and induce antiviral responses. These properties open exciting possibilities for the development of DIP-based antivirals. Exploring options for cell culture-based DIP production, we have established a fully continuous cultivation process, where one bioreactor is used to grow cells that are fed to two bioreactors operated in parallel for virus production. This system allows head-to-head comparisons of STV and DIP replication dynamics over extended time periods. Cultivations were performed at two residence times (RT, 22 and 36 h) using MDCK suspension cells grown in a fully defined medium. For infection, we used a virus seed generated by reverse genetics containing STVs and a known DIP carrying a deletion in segment 1 (delS1(1)). Four days post infection, DIPs achieved maximum concentrations of 7.0·10⁹ virions/mL and 8.4·10⁹ virions/mL for RTs of 22 and 36 h, respectively. Furthermore, oscillations in virus titers with two to three maxima were found for DIP accumulation at 36 and 22 h RT, respectively. To complement the study, a basic mathematical model using simple kinetics and a reasonable number of parameters to describe DIP-propagation in continuous cultures was established. Upon fitting the model individually to each of the two data sets, oscillations in the viral dynamics and the cell population dynamics were described well. Modeling suggests that both STV inactivation and virus degradation have to be taken into account to achieve good agreement of simulations and experimental data for longer RTs. Together, the high DIP titers obtained, and the successful simulation of the experimental data showed that the combination of continuous bioreactors and mathematical models can enable studies regarding DIP dynamics over extended time periods and allow large scale manufacturing of DIP-based antivirals.

Investigating Functional Roles for Positive Feedback and Cellular Heterogeneity in the Type I Interferon Response to Viral Infection

Secretion of type I interferons (IFN) by infected cells mediates protection against many viruses, but prolonged or excessive type I IFN secretion can lead to immune pathology. A proper type I IFN response must therefore maintain a balance between protection and excessive IFN secretion. It has been widely noted that the type I IFN response is driven by positive feedback and is heterogeneous, with only a fraction of infected cells upregulating IFN expression even in clonal cell lines, but the functional roles of feedback and heterogeneity in balancing protection and excessive IFN secretion are not clear. To investigate the functional roles for feedback and heterogeneity, we constructed a mathematical model coupling IFN and viral dynamics that extends existing mathematical models by accounting for feedback and heterogeneity. We fit our model to five existing datasets, reflecting different experimental systems. Fitting across datasets allowed us to compare the IFN response across the systems and suggested different signatures of feedback and heterogeneity in the different systems. Further, through numerical experiments, we generated hypotheses of functional roles for IFN feedback and heterogeneity consistent with our mathematical model. We hypothesize an inherent tradeoff in the IFN response: a positive feedback loop prevents excessive IFN secretion, but also makes the IFN response vulnerable to viral antagonism. We hypothesize that cellular heterogeneity of the IFN response functions to protect the feedback loop from viral antagonism. Verification of our hypotheses will require further experimental studies. Our work provides a basis for analyzing the type I IFN response across systems.

Quantitative profiling of innate immune activation by viral infection in single cells

Cells infected by viruses can exhibit diverse patterns of viral and cellular gene expression. The patterns arise in part from the stochastic or noisy reaction kinetics associated with the small number of genomes, enzymes, and other molecules that typically initiate virus replication and activate cellular anti-viral defenses. It is not known what features, if any, of the early viral or cellular gene expression correlate with later processes of viral replication or cell survival. Here we used two fluorescent reporters to visualize innate immune activation of human prostate cancer (PC3) cells against infection by vesicular stomatitis virus. The cells were engineered to express green-fluorescent protein under control of the promoter for IFIT2, an interferon-sensitive component of the anti-viral response, while red-fluorescent protein was expressed as a byproduct of virus infection. To isolate and quantitatively analyze single-cells, we used a unique microwell array device and open-source image processing software. Kinetic analysis of viral and cellular reporter profiles from hundreds of cells revealed novel relationships between gene expression and the outcome of infection. Specifically, the relative timing rather than the magnitude of the viral gene expression and innate immune activation correlated with the infection outcome. Earlier viral or anti-viral gene expression favored or hindered virus growth, respectively. Further, analysis of kinetic parameters estimated from these data suggests a trade-off between robust antiviral signaling and cell death, as indicated by a higher rate of detectable cell lysis in infected cells with a detectable immune response. In short, cells that activate an immune response lyse at a higher rate. More broadly, we demonstrate how the intrinsic heterogeneity of individual cell behaviors can be exploited to discover features of viral and host gene expression that correlate with single-cell outcomes, which will ultimately impact whether or not infections spread.

Inhibition of infection spread by co-transmitted defective interfering particles

Although virus release from host cells and tissues propels the spread of many infectious diseases, most virus particles are not infectious; many are defective, lacking essential genetic information needed for replication. When defective and viable particles enter the same cell, the defective particles can multiply while interfering with viable particle production. Defective interfering particles (DIPs) occur in nature, but their role in disease pathogenesis and spread is not known. Here, we engineered an RNA virus and its DIPs to express different fluorescent reporters, and we observed how DIPs impact viral gene expression and infection spread. Across thousands of host cells, co-infected with infectious virus and DIPs, gene expression was highly variable, but average levels of viral reporter expression fell at higher DIP doses. In cell populations spatial patterns of infection spread provided the first direct evidence for the co-transmission of DIPs with infectious virus. Patterns of spread were highly sensitive to the behavior of initial or early co-infected cells, with slower overall spread stemming from higher early DIP doses. Under such conditions striking patterns of patchy gene expression reflected localized regions of DIP or virus enrichment. From a broader perspective, these results suggest DIPs contribute to the ecological and evolutionary persistence of viruses in nature.

Single particle labeling of RNA virus in live cells

Real-time and visual tracking of viral infection is crucial for elucidating the infectious and pathogenesis mechanisms. To track the virus successfully, an efficient labeling method is necessary. In this review, we first discuss the practical labeling techniques for virus tracking in live cells. We then describe the current knowledge of interactions between RNA viruses (especially influenza viruses, immunodeficiency viruses, and Flaviviruses) and host cellular structures, obtained using single particle labeling techniques combined with real-time fluorescence microscopy. Single particle labeling provides an easy system for understanding the RNA virus life cycle.

Modeling Viral Spread

The way in which a viral infection spreads within a host is a complex process that is not well understood. Different viruses, such as human immunodeficiency virus type 1 and hepatitis C virus, have evolved different strategies, including direct cell-to-cell transmission and cell-free transmission, to spread within a host. To what extent these two modes of transmission are exploited in vivo is still unknown. Mathematical modeling has been an essential tool to get a better systematic and quantitative understanding of viral processes that are difficult to discern through strictly experimental approaches. In this review, we discuss recent attempts that combine experimental data and mathematical modeling in order to determine and quantify viral transmission modes. We also discuss the current challenges for a systems-level understanding of viral spread, and we highlight the promises and challenges that novel experimental techniques and data will bring to the field.

Spatial-Temporal Patterns of Viral Amplification and Interference Initiated by a Single Infected Cell

When viruses infect their host cells, they can make defective virus-like particles along with intact virus. Cells coinfected with virus and defective particles often exhibit interference with virus growth caused by the competition for resources by defective genomes. Recent reports of the coexistence and cotransmission of such defective interfering particles (DIPs) in vivo, across epidemiological length and time scales, suggest a role in viral pathogenesis, but it is not known how DIPs impact infection spread, even under controlled culture conditions. Using fluorescence microscopy, we quantified coinfections of vesicular stomatitis virus (VSV) expressing a fluorescent reporter protein and its DIPs on BHK-21 host cell monolayers. We found that viral gene expression was more delayed, infections spread more slowly, and patterns of spread became more “patchy” with higher DIP inputs to the initial cell. To examine how infection spread might depend on the behavior of the initial coinfected cell, we built a computational model, adapting a cellular automaton (CA) approach to incorporate kinetic data on virus growth for the first time. Specifically, changes in observed patterns of infection spread could be directly linked to previous high-throughput single-cell measures of virus-DIP coinfection. The CA model also provided testable hypotheses on the spatial-temporal distribution of the DIPs, which remain governed by their predator-prey interaction. More generally, this work offers a data-driven computational modeling approach for better understanding of how single infected cells impact the multiround spread of virus infections across cell populations.

IMPORTANCE Defective interfering particles (DIPs) compete with intact virus, depleting host cell resources that are essential for virus growth and infection spread. However, it is not known how such competition, strong or weak, ultimately affects the way in which infections spread and cause disease. In this study, we address this unmet need by developing an integrated experimental-computational approach, which sheds new light on how infections spread. We anticipate that our approach will also be useful in the development of DIPs as therapeutic agents to manage the spread of viral infections.

Rapid induction and persistence of paracrine-induced cellular antiviral states arrest viral infection spread in A549 cells

The virus/host interaction is a complex interplay between pro- and anti-viral factors that ultimately determines the spread or halt of virus infections in tissues. This interplay develops over multiple rounds of infection. The purpose of this study was to determine how cellular-level processes combine to impact the spatial spread of infection. We measured the kinetics of virus replication (VSV), antiviral paracrine signal upregulation and secretion, spatial spread of virus and paracrine antiviral signaling, and inhibition of virus production in antiviral-exposed A549 human lung epithelial cells. We found that initially infected cells released antiviral signals 4-to-7h following production of virus. However, the subsequent rapid dissemination of signal and fast induction of a robust and persistent antiviral state ultimately led to a suppression of infection spread. This work shows how cellular responses to infection and activation of antiviral responses can integrate to ultimately control infection spread across host cell populations.

Neucode Labels for Multiplexed, Absolute Protein Quantification

We describe a new method to accomplish multiplexed, absolute protein quantification in a targeted fashion. The approach draws upon the recently developed neutron encoding (NeuCode) metabolic labeling strategy and parallel reaction monitoring (PRM). Since PRM scanning relies upon high-resolution tandem mass spectra for targeted protein quantification, incorporation of multiple NeuCode labeled peptides permits high levels of multiplexing that can be accessed from high-resolution tandem mass spectra. Here we demonstrate this approach in cultured cells by monitoring a viral infection and the corresponding viral protein production over many infection time points in a single experiment. In this context the NeuCode PRM combination affords up to 30 channels of quantitative information in a single MS experiment.

High-Throughput Single-Cell Kinetics of Virus Infections in the Presence of Defective Interfering Particles

Defective interfering particles (DIPs) are virus mutants that lack essential genes for growth. In coinfections with helper virus, the diversion of viral proteins to the replication and packaging of DIP genomes can interfere with virus production. Mounting cases of DIPs and DIP-like genomes in clinical and natural isolates, as well as growing interest in DIP-based therapies, underscore a need to better elucidate how DIPs work. DIP activity is primarily measured by its inhibition of virus infection yield, an endpoint that masks the dynamic and potentially diverse individual cell behaviors. Using vesicular stomatitis virus (VSV) as a model, we coinfected BHK cells with VSV DIPs and recombinant helper virus carrying a gene encoding a red fluorescent protein (RFP) whose expression correlates with the timing and level of virus release. For single cells within a monolayer, 10 DIPs per cell suppressed the reporter expression in only 1.2% of the cells. In most cells, it slowed and reduced viral gene expression, manifested as a shift in mean latent time from 4 to 6 h and reduced virus yields by 10-fold. For single cells isolated in microwells, DIP effects were more pronounced, reducing virus yields by 100-fold and extending latent times to 12 h, including individual instances above 20 h. Together, these results suggest that direct or indirect cell-cell interactions prevent most coinfected cells from being completely suppressed by DIPs. Finally, a gamma distribution model captures well how the infection kinetics quantitatively depends on the DIP dose. Such models will be useful for advancing a predictive biology of DIP-associated virus growth and infection spread.

IMPORTANCE

During the last century, basic studies in virology have focused on developing a molecular mechanistic understanding of how infectious viruses reproduce in their living host cells. However, over the last 10 years, the advent of deep sequencing and other powerful technologies has revealed in natural and patient infections that viruses do not act alone. Instead, viruses are often accompanied by defective virus-like particles that carry large deletions in their genomes and fail to replicate on their own. Coinfections of viable and defective viruses behave in unpredictable ways, but they often interfere with normal virus growth, potentially enabling infections to evade host immune surveillance. In the current study, controlled levels of defective viruses are coinfected with viable viruses that have been engineered to express a fluorescent reporter protein during infection. Unique profiles of reporter expression acquired from thousands of coinfected cells reveal how interference acts at multiple stages of infection.

Tools for Single-Cell Kinetic Analysis of Virus-Host Interactions

Measures of cellular gene expression or behavior, when performed on individual cells, inevitably reveal a diversity of behaviors and outcomes that can correlate with normal or diseased states. For virus infections, the potential diversity of outcomes are pushed to an extreme, where measures of infection reflect features of the specific infecting virus particle, the individual host cell, as well as interactions between viral and cellular components. Single-cell measures, while revealing, still often rely on specialized fluid handling capabilities, employ end-point measures, and remain labor-intensive to perform. To address these limitations, we consider a new microwell-based device that uses simple pipette-based fluid handling to isolate individual cells. Our design allows different experimental conditions to be implemented in a single device, permitting easier and more standardized protocols. Further, we utilize a recently reported dual-color fluorescent reporter system that provides dynamic readouts of viral and cellular gene expression during single-cell infections by vesicular stomatitis virus. In addition, we develop and show how free, open-source software can enable streamlined data management and batch image analysis. Here we validate the integration of the device and software using the reporter system to demonstrate unique single-cell dynamic measures of cellular responses to viral infection.

Sterol Carrier Protein 2, a Critical Host Factor for Dengue Virus Infection, Alters the Cholesterol Distribution in Mosquito Aag2 Cells

Host factors that enable dengue virus (DENV) to propagate in the mosquito host cells are unclear. It is known that cellular cholesterol plays an important role in the life cycle of DENV in human host cells but unknown if the lipid requirements differ for mosquito versus mammalian. In mosquito Aedes aegypti, sterol carrier protein 2 (SCP-2) is critical for cellular cholesterol homeostasis. In this study, we identified SCP-2 as a critical host factor for DENV production in mosquito Aag2 cells. Treatment with a small molecule commonly referred to as SCPI-1, (N-(4-{[4-(3,4-dichlorophenyl)-1,3-thiazol-2-yl]amino}phenyl)acetamide hydrobromide, a known inhibitor of SCP-2, or knockdown of SCP-2 dramatically repressed the virus production in mosquito but not mammalian cells. We showed that the intracellular cholesterol distribution in mosquito cells was altered by SCP-2 inhibitor treatment, suggesting that SCP-2-mediated cholesterol trafficking pathway is important for DENV viral production. A comparison of the effect of SCP-2 on mosquito and human cells suggests that SCPI-1 treatment decreases cholesterol in both cell lines, but this decrease in cholesterol only leads to a decline in viral titer in mosquito host cells, perhaps, owing to a more drastic effect on perinuclear cholesterol storages in mosquito cells that was absent in human cells. SCP-2 had no inhibitory effect on another enveloped RNA virus grown in mosquito cells, suggesting that SCP-2 does not have a generalized anti-cellular or antiviral effect. Our cell culture results imply that SCP-2 may play a limiting role in mosquito-dengue vector competence.

Genome rearrangement affects RNA virus adaptability on prostate cancer cells

Gene order is often highly conserved within taxonomic groups, such that organisms with rearranged genomes tend to be less fit than wild type gene orders, and suggesting natural selection favors genome architectures that maximize fitness. But it is unclear whether rearranged genomes hinder adaptability: capacity to evolutionarily improve in a new environment. Negative-sense non-segmented RNA viruses (order Mononegavirales) have specific genome architecture: 3' UTR - core protein genes - envelope protein genes - RNA-dependent RNA-polymerase gene - 5' UTR. To test how genome architecture affects RNA virus evolution, we examined vesicular stomatitis virus (VSV) variants with the nucleocapsid (N) gene moved sequentially downstream in the genome. Because RNA polymerase stuttering in VSV replication causes greater mRNA production in upstream genes, N gene translocation toward the 5' end leads to stepwise decreases in N transcription, viral replication and progeny production, and also impacts the activation of type 1 interferon mediated antiviral responses. We evolved VSV gene-order variants in two prostate cancer cell lines: LNCap cells deficient in innate immune response to viral infection, and PC-3 cells that mount an IFN stimulated anti-viral response to infection. We observed that gene order affects phenotypic adaptability (reproductive growth; viral suppression of immune function), especially on PC-3 cells that strongly select against virus infection. Overall, populations derived from the least-fit ancestor (most-altered N position architecture) adapted fastest, consistent with theory predicting populations with low initial fitness should improve faster in evolutionary time. Also, we observed correlated responses to selection, where viruses improved across both hosts, rather than suffer fitness trade-offs on unselected hosts. Whole genomics revealed multiple mutations in evolved variants, some of which were conserved across selective environments for a given gene order.